Thermodynamics and Statistical Mechanics, Exercises of Quantum Mechanics

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Thermodynamics and Statistical Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 7 1.1 Intended Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Major Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Why Study Thermodynamics? . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Atomic Theory of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 What is Thermodynamics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Need for Statistical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Microscopic and Macroscopic Systems . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Classical and Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . 10 1.9 Classical and Quantum Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Probability Theory 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 What is Probability? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Combining Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Two-State System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Combinatorial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Binomial Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Mean, Variance, and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . 18 2.8 Application to Binomial Probability Distribution . . . . . . . . . . . . . . . . . . 19 2.9 Gaussian Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.10 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Statistical Mechanics 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Specification of State of Many-Particle System . . . . . . . . . . . . . . . . . . . 29 2 CONTENTS 3.3 Principle of Equal A Priori Probabilities . . . . . . . . . . . . . . . . . . . . . . 31 3.4 H-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Reversibility and Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Probability Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 Behavior of Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Heat and Work 43 4.1 Brief History of Heat and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Macrostates and Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Microscopic Interpretation of Heat and Work . . . . . . . . . . . . . . . . . . . . 45 4.4 Quasi-Static Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.5 Exact and Inexact Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 Statistical Thermodynamics 53 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Thermal Interaction Between Macrosystems . . . . . . . . . . . . . . . . . . . . 53 5.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Mechanical Interaction Between Macrosystems . . . . . . . . . . . . . . . . . . 59 5.5 General Interaction Between Macrosystems . . . . . . . . . . . . . . . . . . . . 60 5.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.7 Properties of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.8 Uses of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.9 Entropy and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.10 Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Classical Thermodynamics 77 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Ideal Gas Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.3 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.4 Calculation of Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.5 Isothermal and Adiabatic Expansion . . . . . . . . . . . . . . . . . . . . . . . . 84 6.6 Hydrostatic Equilibrium of Atmosphere . . . . . . . . . . . . . . . . . . . . . . 85 6.7 Isothermal Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.8 Adiabatic Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.9 Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.10 Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.11 Helmholtz Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.12 Gibbs Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.13 General Relation Between Specific Heats . . . . . . . . . . . . . . . . . . . . . . 95 CONTENTS 5 C.10 Three-Dimensional Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 278 C.11 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 C.12 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6 CONTENTS Introduction 7 1 Introduction 1.1 Intended Audience These lecture notes outline a single-semester course intended for upper-division physics majors. 1.2 Major Sources The textbooks that I have consulted most frequently while developing course material are: Fundamentals of Statistical and Thermal Physics: F. Reif (McGraw-Hill, New York NY, 1965). Introduction to Quantum Theory: D. Park, 3rd Edition (McGraw-Hill, New York NY, 1992). Classical and Statistical Thermodynamics: A.S. Carter (Prentice-Hall, Upper Saddle River NJ, 2001). 1.3 Why Study Thermodynamics? In a nutshell, thermodynamics is the study of the internal motions of many-body systems. Virtu- ally all physical entities that we encounter in everyday life are many-body systems of some type or other (e.g., solids, liquids, gases, and even electromagnetic radiation). Not surprisingly, therefore, thermodynamics is a discipline with an exceptionally wide range of applicability. Indeed, ther- modynamics is one of the most extensively used subfields of physics outside physics departments. Prospective engineers, chemists, and material scientists generally do not study relativity or particle physics, but thermodynamics is an integral, and very important, part of their degree courses. Many people are drawn to physics because they wish to understand why the world around us is like it is. For instance, why the sky is blue, why raindrops are spherical, why we do not fall through the floor, et cetera. It turns out that thermodynamics is a very powerful tool for accounting for the observed features of the physical world. For example, in this course, we shall explain why heat spontaneously flows from hot to cold bodies; why it is impossible to measure a temperature below -273◦ centigrade; why there is a maximum theoretical efficiency of a power generation unit that can never be exceeded, no matter what the design; why the Earth’s atmosphere becomes thinner and colder at higher altitudes; why the Sun appears yellow, whereas colder stars appear red, and hotter stars appear bluish-white; and why high mass stars must ultimately collapse to form black-holes. 1.4 Atomic Theory of Matter According to the well-known atomic theory of matter, the familiar objects that make up the world around us, such as tables and chairs, are themselves made up of a great many microscopic particles. 10 1.7. Microscopic and Macroscopic Systems 1.7 Microscopic and Macroscopic Systems It is useful, at this stage, to make a distinction between the different sizes of the systems that we are going to examine. We shall call a system microscopic if it is roughly of atomic dimensions, or smaller. On the other hand, we shall call a system macroscopic if it is large enough to be visible in the ordinary sense. This is a rather inexact definition. The exact definition depends on the number of particles in the system, which we shall call N. A system is macroscopic if 1/N 1/2  1, which means that statistical arguments can be applied to reasonable accuracy. For instance, if we wish to keep the relative statistical error below one percent then a macroscopic system would have to contain more than about ten thousand particles. Any system containing less than this number of particles would be regarded as essentially microscopic, and, hence, statistical arguments could not be applied to such a system without unacceptable error. 1.8 Classical and Statistical Thermodynamics In this course, we are going to develop some machinery for interrelating the statistical properties of a system containing a very large number of particles, via a statistical treatment of the laws of atomic or molecular motion. It turns out that, once we have developed this machinery, we can obtain some very general results that do not depend on the exact details of the statistical treatment. These results can be described without reference to the underlying statistical nature of the system, but their validity depends ultimately on statistical arguments. They take the form of general state- ments regarding heat and work, and are usually referred to as classical thermodynamics, or just thermodynamics, for short. Historically, classical thermodynamics was the first type of thermody- namics to be discovered. In fact, for many years, the laws of classical thermodynamics seemed rather mysterious, because their statistical justification had yet to be discovered. The strength of classical thermodynamics is its great generality, which comes about because it does not depend on any detailed assumptions about the statistical properties of the system under investigation. This generality is also the principle weakness of classical thermodynamics. Only a relatively few state- ments can be made on such general grounds, so many interesting properties of the system remain outside the scope of classical thermodynamics. If we go beyond classical thermodynamics, and start to investigate the statistical machinery that underpins it, then we get all of the results of classical thermodynamics, plus a large number of other results that enable the macroscopic parameters of the system to be calculated from a knowledge of its microscopic constituents. This approach is known as statistical thermodynamics, and is extremely powerful. The only drawback is that the further we delve inside the statistical machinery of thermodynamics, the harder it becomes to perform the necessary calculations. Note that both classical and statistical thermodynamics are only valid for systems in equilib- rium. If the system is not in equilibrium then the problem becomes considerably more difficult. In fact, the thermodynamics of non-equilibrium systems, which is generally called irreversible thermodynamics, is a graduate-level subject. Introduction 11 1.9 Classical and Quantum Approaches We mentioned earlier that the motions (by which we really meant the translational motions) of atoms and molecules are described exactly by quantum mechanics, and only approximately by classical mechanics. It turns out that the non-translational motions of molecules, such as their rotation and vibration, are very poorly described by classical mechanics. So, why bother using classical mechanics at all? Unfortunately, quantum mechanics deals with the translational mo- tions of atoms and molecules (via wave mechanics) in a rather awkward manner. The classical approach is far more straightforward, and, under most circumstances, yields the same statistical results. Hence, throughout the first part of this course, we shall use classical mechanics, as much as possible, to describe the translational motion of atoms and molecules, and will reserve quantum mechanics for dealing with non-translational motions. However, towards the end of this course, in Chapter 8, we shall switch to a purely quantum-mechanical approach. 12 1.9. Classical and Quantum Approaches Probability Theory 15 2.4 Two-State System The simplest non-trivial system that we can investigate using probability theory is one for which there are only two possible outcomes. (There would obviously be little point in investigating a one-outcome system.) Let us suppose that there are two possible outcomes to an observation made on some system, S . Let us denote these outcomes 1 and 2, and let their probabilities of occurrence be P(1) = p, (2.9) P(2) = q. (2.10) It follows immediately from the normalization condition, Equation (2.5), that p + q = 1, (2.11) so q = 1 − p. The best known example of a two-state system is a tossed coin. The two outcomes are “heads” and “tails,” each with equal probabilities 1/2. So, p = q = 1/2 for this system. Suppose that we make N statistically independent observations of S . Let us determine the probability of n1 occurrences of the outcome 1, and N − n1 occurrences of the outcome 2, with no regard as to the order of these occurrences. Denote this probability PN(n1). This type of calculation crops up very often in probability theory. For instance, we might want to know the probability of getting nine “heads” and only one “tails” in an experiment where a coin is tossed ten times, or where ten coins are tossed simultaneously. Consider a simple case in which there are only three observations. Let us try to evaluate the probability of two occurrences of the outcome 1, and one occurrence of the outcome 2. There are three different ways of getting this result. We could get the outcome 1 on the first two observations, and the outcome 2 on the third. Or, we could get the outcome 2 on the first observation, and the outcome 1 on the latter two observations. Or, we could get the outcome 1 on the first and last observations, and the outcome 2 on the middle observation. Writing this symbolically, we have P3(2) = P(1 ⊗ 1 ⊗ 2 |2 ⊗ 1 ⊗ 1 |1 ⊗ 2 ⊗ 1). (2.12) Here, the symbolic operator ⊗ stands for “and,” whereas the symbolic operator | stands for “or.” This symbolic representation is helpful because of the two basic rules for combining probabilities that we derived earlier in Equations (2.4) and (2.8): P(X |Y) = P(X) + P(Y), (2.13) P(X ⊗ Y) = P(X) P(Y). (2.14) The straightforward application of these rules gives P3(2) = p p q + q p p + p q p = 3 p 2 q (2.15) for the case under consideration. 16 2.5. Combinatorial Analysis The probability of obtaining n1 occurrences of the outcome 1 in N observations is given by PN(n1) = C N n1, N−n1 p n1 q N−n1 , (2.16) where C N n1, N−n1 is the number of ways of arranging two distinct sets of n1 and N − n1 indistin- guishable objects. Hopefully, this is, at least, plausible from the previous example. There, the probability of getting two occurrences of the outcome 1, and one occurrence of the outcome 2, was obtained by writing out all of the possible arrangements of two p s (the probability of outcome 1) and one q (the probability of outcome 2), and then adding them all together. 2.5 Combinatorial Analysis The branch of mathematics that studies the number of different ways of arranging things is called combinatorial analysis. We need to know how many different ways there are of arranging N objects that are made up of two groups of n1 and N − n1 indistinguishable objects. This is a rather difficult problem. Let us start off by tackling a slightly easier problem. How many ways are there of arranging N distinguishable objects? For instance, suppose that we have six pool balls, numbered one through six, and we pot one each into every one of the six pockets of a pool table (that is, top-left, top-right, middle-left, middle-right, bottom-left, and bottom-right). How many different ways are there of doing this? Let us start with the top-left pocket. We could pot any one of the six balls into this pocket, so there are 6 possibilities. For the top-right pocket we only have 5 possibilities, because we have already potted a ball into the top-left pocket, and it cannot be in two pockets simultaneously. So, our 6 original possibilities combined with these 5 new possibilities gives 6 × 5 ways of potting two balls into the top two pockets. For the middle-left pocket we have 4 possibilities, because we have already potted two balls. These possibilities combined with our 6 × 5 possibilities gives 6 × 5 × 4 ways of potting three balls into three pockets. At this stage, it should be clear that the final answer is going to be 6 × 5 × 4 × 3 × 2 × 1. The factorial of a general positive integer, n, is defined n! = n (n − 1) (n − 2) · · · 3 · 2 · 1. (2.17) So, 1! = 1, and 2! = 2 × 1 = 2, and 3! = 3 × 2 × 1 = 6, and so on. Clearly, the number of ways of potting six distinguishable pool balls into six pockets is 6! (which incidentally equals 720). Because there is nothing special about pool balls, or the number six, we can safely infer that the number of different ways of arranging N distinguishable objects, denoted C N , is given by C N = N! . (2.18) Suppose that we take the number four ball off the pool table, and replace it by a second number five ball. How many different ways are there of potting the balls now? Consider a previous arrange- ment in which the number five ball was potted into the top-left pocket, and the number four ball was potted into the top-right pocket, and then consider a second arrangement that only differs from the first because the number four and five balls have been swapped around. These arrangements are now indistinguishable, and are therefore counted as a single arrangement, whereas previously they Probability Theory 17 were counted as two separate arrangements. Clearly, the previous arrangements can be divided into two groups, containing equal numbers of arrangements, that differ only by the permutation of the number four and five balls. Because these balls are now indistinguishable, we conclude that there are only half as many different arrangements as there were before. If we take the number three ball off the table, and replace it by a third number five ball, then we can split the original arrangements into six equal groups of arrangements that differ only by the permutation of the number three, four, and five balls. There are six groups because there are 3! = 6 separate permutations of these three balls. Because the number three, four, and five balls are now indistinguishable, we conclude that there are only 1/6 the number of original arrangements. Generalizing this result, we conclude that the number of arrangements of n1 indistinguishable and N − n1 distinguishable objects is C N n1 = N! n1! . (2.19) We can see that if all the balls on the table are replaced by number five balls then there is only N!/N! = 1 possible arrangement. This corresponds, of course, to a number five ball in each pocket. A further straightforward generalization tells us that the number of arrangements of two groups of n1 and N − n1 indistinguishable objects is C N n1, N−n1 = N! n1! (N − n1)! . (2.20) 2.6 Binomial Probability Distribution It follows from Equations (2.16) and (2.20) that the probability of obtaining n1 occurrences of the outcome 1 in N statistically independent observations of a two-state system is PN(n1) = N! n1! (N − n1)! p n1 q N−n1 . (2.21) This probability function is called the binomial probability distribution. The reason for this name becomes obvious if we tabulate the probabilities for the first few possible values of N, as is done in Table 2.1. Of course, we immediately recognize the expressions appearing in the first four rows of this table: they appear in the standard algebraic expansions of (p + q), (p + q) 2, (p + q) 3, and (p+q) 4, respectively. In algebra, the expansion of (p+q) N is called the binomial expansion (hence, the name given to the probability distribution function), and is written (p + q) N ≡ ∑ n=0,N N! n! (N − n)! p n q N−n. (2.22) Equations (2.21) and (2.22) can be used to establish the normalization condition for the binomial distribution function:∑ n1=0,N PN(n1) = ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1 ≡ (p + q) N = 1, (2.23) because p + q = 1. [See Equation (2.11).] 20 2.8. Application to Binomial Probability Distribution and 2, with respective probabilities p and q = 1−p, then the probability of obtaining n1 occurrences of outcome 1 in N observations is PN(n1) = N! n1! (N − n1)! p n1 q N−n1 . (2.38) Thus, making use of Equation (2.27), the mean number of occurrences of outcome 1 in N obser- vations is given by n1 = ∑ n1=0,N PN(n1) n1 = ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1 n1. (2.39) We can see that if the final factor n1 were absent on the right-hand side of the previous expression then it would just reduce to the binomial expansion, which we know how to sum. [See Equa- tion (2.23).] We can take advantage of this fact using a rather elegant mathematical sleight of hand. Observe that because n1 p n1 ≡ p ∂ ∂p p n1 , (2.40) the previous summation can be rewritten as ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1 n1 ≡ p ∂ ∂p  ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1  . (2.41) The term in square brackets is now the familiar binomial expansion, and can be written more succinctly as (p + q) N . Thus, ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1 n1 = p ∂ ∂p (p + q) N = p N (p + q) N−1. (2.42) However, p + q = 1 for the case in hand [see Equation (2.11)], so n1 = N p. (2.43) In fact, we could have guessed the previous result. By definition, the probability, p, is the number of occurrences of the outcome 1 divided by the number of trials, in the limit as the number of trials goes to infinity: p = lt N→∞ n1 N . (2.44) If we think carefully, however, we can see that taking the limit as the number of trials goes to infinity is equivalent to taking the mean value, so that p = (n1 N ) = n1 N . (2.45) But, this is just a simple rearrangement of Equation (2.43). Probability Theory 21 Let us now calculate the variance of n1. Recall, from Equation (2.36), that (∆n1) 2 = (n1) 2 − (n1) 2. (2.46) We already know n1, so we just need to calculate (n1) 2. This average is written (n1) 2 = ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1 (n1) 2. (2.47) The sum can be evaluated using a simple extension of the mathematical trick that we used previ- ously to evaluate n1. Because (n1) 2 p n1 ≡ ( p ∂ ∂p ) 2 p n1 , (2.48) then ∑ n1=0,N N! n1! (N − n1)! p n1 q N−n1 (n1) 2 ≡ ( p ∂ ∂p )2 ∑ n1=0,N N! n1! (N − n1)! p n1q N−n1 = ( p ∂ ∂p )2 (p + q) N = ( p ∂ ∂p ) [ p N (p + q) N−1 ] = p [ N (p + q) N−1 + p N (N − 1) (p + q) N−2 ] . (2.49) Using p + q = 1, we obtain (n1) 2 = p [ N + p N (N − 1) ] = N p [ 1 + p N − p ] = (N p) 2 + N p q = (n1) 2 + N p q, (2.50) because n1 = N p. [See Equation (2.43).] It follows that the variance of n1 is given by (∆n1) 2 = (n1) 2 − (n1) 2 = N p q. (2.51) The standard deviation of n1 is the square root of the variance [see Equation (2.37)], so that ∆∗n1 = √ N p q. (2.52) Recall that this quantity is essentially the width of the range over which n1 is distributed around its mean value. The relative width of the distribution is characterized by ∆∗n1 n1 = √ N p q N p = √ q p 1√ N . (2.53) It is clear, from this formula, that the relative width decreases with increasing N like N −1/2. So, the greater the number of trials, the more likely it is that an observation of n1 will yield a result that is relatively close to the mean value, n1. 22 2.9. Gaussian Probability Distribution 2.9 Gaussian Probability Distribution Consider a very large number of observations, N 1, made on a system with two possible out- comes. Suppose that the probability of outcome 1 is sufficiently large that the average number of occurrences after N observations is much greater than unity: that is, n1 = N p 1. (2.54) In this limit, the standard deviation of n1 is also much greater than unity, ∆∗n1 = √ N p q 1, (2.55) implying that there are very many probable values of n1 scattered about the mean value, n1. This suggests that the probability of obtaining n1 occurrences of outcome 1 does not change significantly in going from one possible value of n1 to an adjacent value. In other words, |PN(n1 + 1) − PN(n1)| PN(n1)  1. (2.56) In this situation, it is useful to regard the probability as a smooth function of n1. Let n be a contin- uous variable that is interpreted as the number of occurrences of outcome 1 (after N observations) whenever it takes on a positive integer value. The probability that n lies between n and n + dn is defined P(n, n + dn) = P(n) dn, (2.57) where P(n) is called the probability density, and is independent of dn. The probability can be written in this form because P(n, n+dn) can always be expanded as a Taylor series in dn, and must go to zero as dn→ 0. We can write ∫ n1+1/2 n1−1/2 P(n) dn = PN(n1), (2.58) which is equivalent to smearing out the discrete probability PN(n1) over the range n1 ± 1/2. Given Equations (2.38) and (2.56), the previous relation can be approximated as P(n)  PN(n) = N! n! (N − n)! p n q N−n. (2.59) For large N, the relative width of the probability distribution function is small: that is, ∆∗n1 n1 = √ q p 1√ N  1. (2.60) This suggests that P(n) is strongly peaked around the mean value, n = n1. Suppose that lnP(n) attains its maximum value at n = ñ (where we expect ñ ∼ n1). Let us Taylor expand lnP(n) around Probability Theory 25 This is the famous Gaussian probability distribution, named after the German mathematician Carl Friedrich Gauss, who discovered it while investigating the distribution of errors in measurements. The Gaussian distribution is only valid in the limits N 1 and n1 1. Suppose we were to plot the probability PN(n1) against the integer variable n1, and then fit a continuous curve through the discrete points thus obtained. This curve would be equivalent to the continuous probability density curve P(n), where n is the continuous version of n1. According to Equation (2.81), the probability density attains its maximum value when n equals the mean of n1, and is also symmetric about this point. In fact, when plotted with the appropriate ratio of vertical to horizontal scalings, the Gaussian probability density curve looks rather like the outline of a bell centered on n = n1. Hence, this curve is sometimes called a bell curve. At one standard deviation away from the mean value—that is n = n1 ± ∆∗n1—the probability density is about 61% of its peak value. At two standard deviations away from the mean value, the probability density is about 13.5% of its peak value. Finally, at three standard deviations away from the mean value, the probability density is only about 1% of its peak value. We conclude that there is very little chance that n1 lies more than about three standard deviations away from its mean value. In other words, n1 is almost certain to lie in the relatively narrow range n1 ± 3∆∗n1. In the previous analysis, we went from a discrete probability function, PN(n1), to a continuous probability density, P(n). The normalization condition becomes 1 = ∑ n1=0,N PN(n1)  ∫ ∞ −∞ P(n) dn (2.82) under this transformation. Likewise, the evaluations of the mean and variance of the distribution are written n1 = ∑ n1=0,N PN(n1) n1  ∫ ∞ −∞ P(n) n dn, (2.83) and (∆n1) 2 ≡ (∆∗n1) 2 = ∑ n1=0,N PN(n1) (n1 − n1) 2  ∫ ∞ −∞ P(n) (n − n1) 2 dn, (2.84) respectively. These results follow as simple generalizations of previously established results for the discrete function PN(n1). The limits of integration in the previous expressions can be approx- imated as ±∞ because P(n) is only non-negligible in a relatively narrow range of n. Finally, it is easily demonstrated that Equations (2.82)–(2.84) are indeed true by substituting in the Gaus- sian probability density, Equation (2.81), and then performing a few elementary integrals. (See Exercise 2.3.) 2.10 Central Limit Theorem It may appear, at first sight, that the Gaussian distribution function is only relevant to two-state systems. In fact, as we shall see, the Gaussian probability distribution is of crucial importance to statistical physics because, under certain circumstances, it applies to all types of system. 26 2.10. Central Limit Theorem Let us briefly review how we obtained the Gaussian distribution function in the first place. We started from a very simple system with only two possible outcomes. Of course, the probability distribution function (for n1) for this system did not look anything like a Gaussian. However, when we combined very many of these simple systems together, to produce a complicated system with a great number of possible outcomes, we found that the resultant probability distribution function (for n1) reduced to a Gaussian in the limit that the number of simple systems tended to infinity. We started from a two outcome system because it was easy to calculate the final probability distribution function when a finite number of such systems were combined together. Clearly, if we had started from a more complicated system then this calculation would have been far more difficult. Suppose that we start from a general system, with a general probability distribution function (for some measurable quantity x). It turns out that if we combine a sufficiently large number of such systems together then the resultant distribution function (for x) is always Gaussian. This as- tonishing result is known as the central limit theorem. Unfortunately, the central limit theorem is notoriously difficult to prove. A somewhat restricted proof is presented in Sections 1.10 and 1.11 of Reif. The central limit theorem guarantees that the probability distribution of any measurable quantity is Gaussian, provided that a sufficiently large number of statistically independent obser- vations are made. We can, therefore, confidently predict that Gaussian probability distributions are going to crop up very frequently in statistical thermodynamics. Exercises 2.1 Let Ix = ∫ ∞ ∞ e−x 2 dx and Iy = ∫ ∞ ∞ e−y 2 dy. Show that Ix Iy = ∫ ∞ 0 2π r e−r 2 dr, where r 2 = x 2 + y 2. Hence, deduce that Ix = Iy = π 1/2. 2.2 Show that ∫ ∞ −∞ e−β x 2 dx = √ π β . Hence, deduce that ∫ ∞ −∞ x 2 e−β x 2 dx = π 1/2 2 β 3/2 . 2.3 Confirm that ∫ ∞ −∞ P(n) dn = 1,∫ ∞ −∞ P(n) n dn = n̄,∫ ∞ −∞ P(n) (n − n̄) 2 dn = ∆∗n, Probability Theory 27 where P(n) = 1√ 2π ∆∗n exp [ − (n − n̄) 2 2 (∆∗n) 2 ] . 2.4 Show that the probability of throwing 6 points or less with three (six-sided) dice is 5/54. 2.5 Consider a game in which six (six-sided) dice are rolled. Find the probability of obtaining: (a) exactly one ace. (b) at least one ace. (c) exactly two aces. 2.6 In the game of Russian roulette, one inserts a single cartridge into the drum of a revolver, leaving the other five chambers of the drum empty. One then spins the drum, aims at one’s head, and pulls the trigger. (a) Show that the probability of still being alive after playing the game N times is (5/6) N . (b) Show that the probability of surviving (N − 1) turns in this game, and then being shot the Nth times one pulls the trigger, is (5/6) N (1/6). (c) Show that the mean number of times a player gets to pull the trigger is 6. 2.7 A battery of total emf V is connected to a resistor R. As a result, an amount of power P = V 2/R is dissipated in the resistor. The battery itself consists of N individual cells connected in series, so that V is equal to the sum of the emf’s of all these cells. The battery is old, however, so that not all cells are in perfect condition. Thus, there is a probability p that the emf of any individual cell has its normal value v; and a probability 1 − p that the emf of any individual cell is zero because the cell has become internally shorted. The individual cells are statistically independent of each other. Under these conditions, show that the mean power, P, dissipated in the resistor, is P = p 2 V 2 R [ 1 − (1 − p) N p ] . 2.8 A drunk starts out from a lamppost in the middle of a street, taking steps of uniform length l to the right or to the left with equal probability. (a) Show that the average distance from the lamppost after N steps is zero. (b) Show that the root-mean-square distance (i.e. the square-root of the mean of the dis- tance squared) from the lamppost after N steps is √ N l. (c) Show that the probability that the drunk will return to the lamppost after N steps is zero if N is odd, and PN = N! (N/2)! (N/2)! ( 1 2 )N if N is even. 30 3.2. Specification of State of Many-Particle System automatically ensures that we do not attempt to specify q and p to an accuracy greater than our experimental error. Finally, let us consider a system consisting of N spinless particles moving classically in three dimensions. In order to specify the state of the system, we need to specify a large number of q-p pairs. The requisite number is simply the number of degrees of freedom, f . For the present case, f = 3N, because each particle needs three q-p pairs. Thus, phase-space (i.e., the space of all the q-p pairs) now possesses 2 f = 6N dimensions. Consider a particular pair of conjugate (see Section B.4) phase-space coordinates, qi and pi. As before, we divide the qi-pi plane into rectangular cells of uniform dimensions δq and δp. This is equivalent to dividing phase-space into regular 2 f dimensional cells of volume h f 0 . The state of the system is specified by indicating which cell it occupies in phase-space at any given time. In principle, we can specify the state of the system to arbitrary accuracy, by taking the limit h0 → 0. In reality, we know from Heisenberg’s uncertainty principle (see Section C.8) that it is impossible to simultaneously measure a coordinate, qi, and its conjugate momentum, pi, to greater accuracy than δqi δpi =
/2. Here,
is Planck’s constant divided by 2π. This implies that h0 ≥
/2. (3.2) In other words, the uncertainty principle sets a lower limit on how finely we can chop up classical phase-space. In quantum mechanics, we can specify the state of the system by giving its wavefunction at time t, ψ(q1, · · · , qf , s1, · · · , sg, t), (3.3) where f is the number of translational degrees of freedom, and g the number of internal (e.g., spin) degrees of freedom. (See Appendix C.) For instance, if the system consists of N spin-one-half particles then there will be 3N translational degrees of freedom, and N spin degrees of freedom (because the spin of each particle can either be directed up or down along the z-axis). Alternatively, if the system is in a stationary state (i.e., an eigenstate of the Hamiltonian) then we can just specify f + g quantum numbers. (See Sections C.9 and C.10.) Either way, the future time evolution of the wavefunction is fully determined by Schrödinger’s equation. In reality, this approach is not practical because the Hamiltonian of the system is only known approximately. Typically, we are dealing with a system consisting of many weakly-interacting particles. We usually know the Hamiltonian for completely non-interacting particles, but the component of the Hamiltonian associated with particle interactions is either impossibly complicated, or not very well known. We can define approximate stationary eigenstates using the Hamiltonian for non-interacting particles. The state of the system is then specified by the quantum numbers identifying these eigenstates. In the absence of particle interactions, if the system starts off in a stationary state then it stays in that state for ever, so its quantum numbers never change. The interactions allow the system to make transitions between different “stationary” states, causing its quantum numbers to change in time. Statistical Mechanics 31 3.3 Principle of Equal A Priori Probabilities We now know how to specify the instantaneous state of a many-particle system. In principle, such a system is completely deterministic. Once we know the initial state, and the equations of motion (or the Hamiltonian), we can evolve the system forward in time, and, thereby, determine all future states. In reality, it is quite impossible to specify the initial state, or the equations of motion, to sufficient accuracy for this method to have any chance of working. Furthermore, even if it were possible, it would still not be a practical proposition to evolve the equations of motion. Remember that we are typically dealing with systems containing Avogadro’s number of particles: that is, about 10 24 particles. We cannot evolve 10 24 simultaneous differential equations. Even if we could, we would not want to. After all, we are not particularly interested in the motions of individual particles. What we really want is statistical information regarding the motions of all particles in the system. Clearly, what is required here is a statistical treatment of the problem. Instead of focusing on a single system, let us proceed, in the usual manner, and consider a statistical ensemble consisting of a large number of identical systems. (See Section 2.2.) In general, these systems are distributed over many different states at any given time. In order to evaluate the probability that the system possesses a particular property, we merely need to find the number of systems in the ensemble that exhibit this property, and then divide by the total number of systems, in the limit as the latter number tends to infinity. We can usually place some general constraints on the system. Typically, we know the total energy, E, the total volume, V , and the total number of particles, N. To be more exact, we can only really say that the total energy lies between E and E + δE, et cetera, where δE is an experimental error. Thus, we need only concern ourselves with those systems in the ensemble exhibiting states that are consistent with the known constraints. We shall call these the states accessible to the system. In general, there are a great many such states. We now need to calculate the probability of the system being found in each of its accessible states. In fact, the only way that we could “calculate” these probabilities would be to evolve all of the systems in the ensemble in time, and observe how long, on average, they spend in each accessible state. But, as we have already discussed, such a calculation is completely out of the question. Instead, we shall effectively guess the probabilities. Let us consider an isolated system in equilibrium. In this situation, we would expect the prob- ability of the system being found in one of its accessible states to be independent of time. This implies that the statistical ensemble does not evolve with time. Individual systems in the ensemble will constantly change state, but the average number of systems in any given state should remain constant. Thus, all macroscopic parameters describing the system, such as the energy and the vol- ume, should also remain constant. There is nothing in the laws of mechanics that would lead us to suppose that the system will be found more often in one of its accessible states than in another. We assume, therefore, that the system is equally likely to be found in any of its accessible states. This is called the assumption of equal a priori probabilities, and lies at the heart of statistical mechanics. In fact, we use assumptions like this all of the time without really thinking about them. Suppose that we were asked to pick a card at random from a well-shuffled pack of ordinary playing cards. Most people would accept that we have an equal probability of picking any card in the pack. There 32 3.4. H-Theorem is nothing that would favor one particular card over all of the others. Hence, because there are fifty-two cards in a normal pack, we would expect the probability of picking the ace of spades, say, to be 1/52. We could now place some constraints on the system. For instance, we could only count red cards, in which case the probability of picking the ace of hearts, say, would be 1/26, by the same reasoning. In both cases, we have used the principle of equal a priori probabilities. People really believe that this principle applies to games of chance, such as cards, dice, and roulette. In fact, if the principle were found not to apply to a particular game then most people would conclude that the game was crooked. But, imagine trying to prove that the principle actually does apply to a game of cards. This would be a very difficult task. We would have to show that the way most people shuffle cards is effective at randomizing their order. A convincing study would have to be part mathematics and part psychology. In statistical mechanics, we treat a many-particle system a little like an extremely large game of cards. Each accessible state corresponds to one of the cards in the pack. The interactions between particles cause the system to continually change state. This is equivalent to constantly shuffling the pack. Finally, an observation of the state of the system is like picking a card at random from the pack. The principle of equal a priori probabilities then boils down to saying that we have an equal chance of choosing any particular card. It is, unfortunately, impossible to prove with mathematical rigor that the principle of equal a priori probabilities applies to many-particle systems. Over the years, many people have attempted this proof, and all have failed. Not surprisingly, therefore, statistical mechanics was greeted with a great deal of scepticism when it was first proposed in the late 1800’s. One of the its main proponents, Ludvig Boltzmann, became so discouraged by all of the criticism that he eventually committed suicide. Nowadays, statistical mechanics is completely accepted into the cannon of physics—quite simply because it works. It is actually possible to formulate a reasonably convincing scientific case for the principle of equal a priori probabilities. To achieve this we have to make use of the so-called H-theorem. 3.4 H-Theorem Consider a system of weakly-interacting particles. In quantum mechanics, we can write the Hamil- tonian for such a system as H = H0 + H1, (3.4) where H0 is the Hamiltonian for completely non-interacting particles, and H1 a small correction due to the particle interactions. We can define approximate stationary eigenstates of the system using H0. Thus, H0 Ψr = Er Ψr, (3.5) where the index r labels a state of energy Er and eigenstate Ψr. (See Section C.9.) In general, there are many different eigenstates with the same energy—these are called degenerate states. (See Section C.10.) For example, consider N non-interacting spinless particles of mass m confined in a cubic box of dimension L. According to standard wave-mechanics, the energy levels of the ith particle are Statistical Mechanics 35 This quantity changes as the individual probabilities Pr vary in time. Straightforward differentia- tion of the previous equation yields dH dt = ∑ r ( dPr dt ln Pr + dPr dt ) = ∑ r dPr dt (ln Pr + 1). (3.13) According to Equation (3.11), this can be written dH dt = ∑ r ∑ s Wrs (Ps − Pr) (ln Pr + 1). (3.14) We can now interchange the dummy summations indices r and s to give dH dt = ∑ r ∑ s Wsr (Pr − Ps) (ln Ps + 1). (3.15) Finally, we can write dH/dt in a more symmetric form by adding the previous two equations, and then making use of Equation (3.9): dH dt = −1 2 ∑ r ∑ s Wrs (Pr − Ps) (ln Pr − ln Ps). (3.16) Note, however, that ln Pr is a monotonically increasing function of Pr. It follows that ln Pr > ln Ps whenever Pr > Ps, and vice versa. Thus, in general, the right-hand side of the previous equation is the sum of many negative contributions. Hence, we conclude that dH dt ≤ 0. (3.17) The equality sign only holds in the special case where all accessible states are equally probable, so that Pr = Ps for all r and s. This result is called the H-theorem, and was first proved by the unfortunate Professor Boltzmann. The H-theorem tells us that if an isolated system is initially not in equilibrium then it will evolve in time, under the influence of particle interactions, in such a manner that the quantity H always decreases. This process will continue until H reaches its minimum possible value, at which point dH/dt = 0, and there is no further evolution of the system. According to Equation (3.16), in this final equilibrium state, the system is equally likely to be found in any one of its accessible states. This is, of course, the situation predicted by the principle of equal a priori probabilities. The previous argument does not constitute a mathematically rigorous proof that the principle of equal a priori probabilities applies to many-particle systems, because we tacitly made an un- warranted assumption. That is, we assumed that the probability of the system making a transition from some state r to another state s is independent of the past history of the system. In general, this is not the case in physical systems, although there are many situations in which it is a fairly good approximation. Thus, the epistemological status of the principle of equal a priori probabilities is that it is plausible, but remains unproven. As we have already mentioned, the ultimate justifica- tion for this principle is empirical—it leads to theoretical predictions that are in accordance with experimental observations. 36 3.5. Relaxation Time 3.5 Relaxation Time The H-theorem guarantees that an isolated many-particle system will eventually reach an equilib- rium state, irrespective of its initial state. The typical time required for this process to take place is called the relaxation time, and depends, in detail, on the nature of the inter-particle interactions. The principle of equal a priori probabilities is only valid for equilibrium states. It follows that we can only safely apply this principle to systems that have remained undisturbed for many relaxation times since they were setup, or last interacted with the outside world. The relaxation time for the air in a typical classroom is very much less than one second. This suggests that such air is probably in equilibrium most of the time, and should, therefore, be gov- erned by the principle of equal a priori probabilities. In fact, this is known to be the case. Consider another example. Our galaxy, the “Milky Way,” is an isolated dynamical system made up of about 10 11 stars. In fact, it can be thought of as a self-gravitating “gas” of stars. At first sight, the Milky Way would seem to be an ideal system on which to test out the ideas of statistical mechanics. Stars in the Milky Way interact via occasional near-miss events in which they exchange energy and momentum. Actual collisions are very rare indeed. Unfortunately, such interactions take place very infrequently, because there is a lot of empty space between the stars. The best estimate for the relaxation time of the Milky Way is about 10 13 years. This should be compared with the estimated age of the Milky Way, which is only about 10 10 years. It is clear that, despite its great age, the Milky Way has not been around long enough to reach an equilibrium state. This suggests that the principle of equal a priori probabilities cannot be used to describe stellar dynamics. Not surprisingly, the observed velocity distribution of the stars in the vicinity of the Sun is not governed by this principle. 3.6 Reversibility and Irreversibility Previously, we mentioned that, on a microscopic level, the laws of physics are invariant under time- reversal. In other words, microscopic phenomena look physically plausible when run in reverse. We usually say that these phenomena are reversible. Does this imply that macroscopic phenomena are also reversible? Consider an isolated many-particle system that starts off far from equilibrium. According to the H-theorem, it will evolve towards equilibrium and, as it does so, the macroscopic quantity H will decrease. But, if we run this process backwards in time then the system will appear to evolve away from equilibrium, and the quantity H will increase. This type of behavior is not physical because it violates the H-theorem. In other words, if we saw a film of a macroscopic process then we could very easily tell if it was being run backwards. For instance, suppose that, by some miracle, we were able to move all of the oxygen molecules in the air in some classroom to one side of the room, and all of the nitrogen molecules to the opposite side. We would not expect this state to persist for very long. Pretty soon the oxygen and nitrogen molecules would start to intermingle, and this process would continue until they were thoroughly mixed together throughout the room. This, of course, is the equilibrium state for air. In reverse, this process appears completely unphysical. We would start off from perfectly normal air, and suddenly, for no good reason, the air’s constituent oxygen and nitrogen molecules would Statistical Mechanics 37 appear to separate, and move to opposite sides of the room. This scenario is not impossible, but, from everything we know about the world around us, it is spectacularly unlikely. We conclude, therefore, that macroscopic phenomena are generally irreversible, because they appear unphysical when run in reverse. How does the irreversibility of macroscopic phenomena arise? It certainly does not come from the fundamental laws of physics, because these laws are all reversible. In the previous example, the oxygen and nitrogen molecules intermingled by continually scattering off one another. Each individual scattering event would look perfectly reasonable viewed in reverse. However, the net result of these scattering event appears unphysical when run backwards. How can we obtain an irreversible process from the combined effects of very many reversible processes? This is a vitally important question. Unfortunately, we are not quite at the stage where we can formulate a con- vincing answer. (We shall answer the question in Section 5.6.) Note, however, that the essential irreversibility of macroscopic phenomena is one of the key results of statistical thermodynamics. 3.7 Probability Calculations The principle of equal a priori probabilities is fundamental to all of statistical mechanics, and al- lows a complete description of the properties of macroscopic systems in equilibrium. In principle, statistical mechanics calculations are very simple. Consider a system in equilibrium that is iso- lated, so that its total energy is known to have a constant value lying somewhere in the range E to E + δE. In order to make statistical predictions, we focus attention on an ensemble of such sys- tems, all of which have their energy in this range. Let Ω(E) be the total number of different states in the ensemble with energies in the specified range. Suppose that, among these states, there are a number Ω(E; yk) for which some parameter, y, of the system assumes the discrete value yk. (This discussion can easily be generalized to deal with a parameter that can assume a continuous range of values.) The principle of equal a priori probabilities tells us that all of the Ω(E) accessible states of the system are equally likely to occur in the ensemble. It follows that the probability, P(yk), that the parameter y of the system assumes the value yk is simply P(yk) = Ω(E; yk) Ω(E) . (3.18) Clearly, the mean value of y for the system is given by ȳ = ∑ k Ω(E; yk) yk Ω(E) , (3.19) where the sum is over all possible values that y can assume. In the previous formula, it is tacitly assumed that Ω(E) →∞, which is generally the case in thermodynamic systems. It can be seen that, using the principle of equal a priori probabilities, all calculations in statisti- cal mechanics reduce to counting states, subject to various constraints. In principle, this is a fairly straightforward task. In practice, problems arise if the constraints become too complicated. These problems can usually be overcome with a little mathematical ingenuity. Nevertheless, there is no doubt that this type of calculation is far easier than trying to solve the classical equations of motion (or Schrödinger’s equation) directly for a many-particle system. 40 3.8. Behavior of Density of States (b) Explicitly calculate the force per unit area (or pressure) acting on this wall. By aver- aging over all possible states, find an expression for the mean pressure on this wall. (Hint: exploit the fact that n 2 x = n 2 y = n 2 z must all be equal, by symmetry.) Show that this mean pressure can be written p = 2 3 E V , where E is the mean energy of the particle, and V = Lx Ly Lz the volume of the box. 3.2 The state of a system with f degrees of freedom at time t is specified by its generalized coordinates, q1, · · · , qf , and conjugate momenta, p1, · · · , pf . These evolve according to Hamilton’s equations (see Section B.9): q̇i = ∂H ∂pi , ṗi = −∂H ∂qi . Here, H(q1, · · · , qf , p1, · · · , pf , t) is the Hamiltonian of the system. Consider a statistical ensemble of such systems. Let ρ(q1, · · · , qf , p1, · · · , pf , t) be the number density of systems in phase-space. In other words, let ρ(q1, · · · , qf , p1, · · · , pf , t) dq1 dq2 · · · dqf dp1 dp2 · · · dpf be the number of states with q1 lying between q1 and q1 + dq1, p1 lying between p1 and p1 + dp1, et cetera, at time t. (a) Show that ρ evolves in time according to Liouville’s theorem: ∂ρ ∂t + ∑ i=1, f ( q̇i ∂ρ ∂qi + ṗi ∂ρ ∂pi ) = 0. [Hint: Consider how the the flux of systems into a small volume of phase-space causes the number of systems in the volume to change in time.] (b) By definition, N = ∫ ∞ −∞ ρ dq1 · · · dqf dp1 · · · dpf is the total number of systems in the ensemble. The integral is over all of phase- space. Show that Liouville’s theorem conserves the total number of systems (i.e., dN/dt = 0). You may assume that ρ becomes negligibly small if any of its arguments (i.e., q1, · · · , qf and p1, · · · , pf ) becomes very large. This is equivalent to assuming that all of the systems are localized to some region of phase-space. (c) Suppose that H has no explicit time dependence (i.e., ∂H/∂t = 0). Show that the ensemble-averaged energy, H = ∫ ∞ −∞ H ρ dq1 · · · dqf dp1 · · · dpf , is a constant of the motion. Statistical Mechanics 41 (d) Show that if H is also not an explicit function of the coordinate qj then the ensemble average of the conjugate momentum, pj = ∫ ∞ −∞ pj ρ dq1 · · · dqf dp1 · · · dpf , is a constant of the motion. 3.3 Consider a system consisting of very many particles. Suppose that an observation of a macroscopic variable, x, can result in any one of a great many closely-spaced values, xr. Let the (approximately constant) spacing between adjacent values be δx. The probability of occurrence of the value xr is denoted Pr. The probabilities are assumed to be properly normalized, so that ∑ r Pr  ∫ ∞ −∞ Pr(xr) δx dxr = 1, where the summation is over all possible values. Suppose that we know the mean and the variance of x, so that x = ∑ r xr Pr and (∆x) 2 = ∑ r (xr − x) 2 Pr are both fixed. According to the H-theorem, the system will naturally evolve towards a final equilibrium state in which the quantity H = ∑ r Pr ln Pr is minimized. Used the method of Lagrange multipliers to minimixe H with respect to the Pr, subject to the constraints that the probabilities remain properly normalized, and that the mean and variance of x remain constant. (See Section B.6.) Show that the most general form for the Pr which can achieve this goal is Pr(xr)  δx√ 2π (∆x) 2 −(xr − x) 2 2 (∆x) 2  . This result demonstrates that the system will naturally evolve towards a final equilibrium state in which all of its macroscopic variables have Gaussian probability distributions, which is in accordance with the central limit theorem. (See Section 2.10.) 42 3.8. Behavior of Density of States Heat and Work 45 4.2 Macrostates and Microstates In describing a system made up of a great many particles, it is usually possible to specify some macroscopically measurable independent parameters, x1, x2, · · · , xn, that affect the particles’ equa- tions of motion. These parameters are termed the external parameters of the system. Examples of such parameters are the volume (this gets into the equations of motion because the potential energy becomes infinite when a particle strays outside the available volume), and any applied elec- tric and magnetic fields. A microstate of the system is defined as a state for which the motions of the individual particles are completely specified (subject, of course, to the unavoidable limitations imposed by the uncertainty principle of quantum mechanics). In general, the overall energy of a given microstate, r, is a function of the external parameters: Er ≡ Er(x1, x2, · · · , xn). (4.6) A macrostate of the system is defined by specifying the external parameters, and any other con- straints to which the system is subject. For example, if we are dealing with an isolated system (i.e., one that can neither exchange heat with, nor do work on, its surroundings) then the macrostate might be specified by giving the values of the volume and the constant total energy. For a many- particle system, there are generally a very great number of microstates that are consistent with a given macrostate. 4.3 Microscopic Interpretation of Heat and Work Consider a macroscopic system, A, that is known to be in a given macrostate. To be more exact, consider an ensemble of similar macroscopic systems, A, where each system in the ensemble is in one of the many microstates consistent with the given macrostate. There are two fundamentally different ways in which the average energy of A can change due to interaction with its surroundings. If the external parameters of the system remain constant then the interaction is termed a purely thermal interaction. Any change in the average energy of the system is attributed to an exchange of heat with its environment. Thus, ∆E = Q, (4.7) where Q is the heat absorbed by the system. On a microscopic level, the energies of the individual microstates are unaffected by the absorption of heat. In fact, it is the distribution of the systems in the ensemble over the various microstates that is modified. Suppose that the system A is thermally insulated from its environment. This can be achieved by surrounding it by an adiabatic envelope (i.e., an envelope fabricated out of a material that is a poor conductor of heat, such a fiber glass). Incidentally, the term adiabatic is derived from the Greek adiabatos, which means “impassable.” In scientific terminology, an adiabatic process is one in which there is no exchange of heat. The system A is still capable of interacting with its environment via its external parameters. This type of interaction is termed mechanical interaction, and any change in the average energy of the system is attributed to work done on it by its surroundings. Thus, ∆E = −W, (4.8) 46 4.4. Quasi-Static Processes where W is the work done by the system on its environment. On a microscopic level, the energy of the system changes because the energies of the individual microstates are functions of the external parameters. [See Equation (4.6).] Thus, if the external parameters are changed then, in general, the energies of all of the systems in the ensemble are modified (because each is in a specific mi- crostate). Such a modification usually gives rise to a redistribution of the systems in the ensemble over the accessible microstates (without any heat exchange with the environment). Clearly, from a microscopic viewpoint, performing work on a macroscopic system is quite a complicated process. Nevertheless, macroscopic work is a quantity that is easy to measure experimentally. For instance, if the system A exerts a force F on its immediate surroundings, and the change in external parame- ters corresponds to a displacement x of the center of mass of the system, then the work done by A on its surroundings is simply W = F·x : (4.9) that is, the product of the force and the displacement along the line of action of the force. In a general interaction of the system A with its environment there is both heat exchange and work performed. We can write Q ≡ ∆E +W, (4.10) which serves as the general definition of the absorbed heat Q. (Hence, the equivalence sign.) The quantity Q is simply the change in the mean energy of the system that is not due to the modification of the external parameters. Note that the notion of a quantity of heat has no independent meaning apart from Equation (4.10). The mean energy, E, and work performed, W, are both physical quantities that can be determined experimentally, whereas Q is merely a derived quantity. 4.4 Quasi-Static Processes Consider the special case of an interaction of the system A with its surroundings that is carried out so slowly that A remains arbitrarily close to equilibrium at all times. Such a process is said to be quasi-static for the system A. In practice, a quasi-static process must be carried out on a timescale that is much longer than the relaxation time of the system. Recall that the relaxation time is the typical timescale for the system to return to equilibrium after being suddenly disturbed. (See Section 3.5.) A finite quasi-static change can be built up out of many infinitesimal changes. The infinitesimal heat, d̄Q, absorbed by the system when infinitesimal work, d̄W, is done on its environment, and its average energy changes by dE, is given by d̄Q ≡ dE + d̄W. (4.11) The special symbols d̄W and d̄Q are introduced to emphasize that the work done, and the heat absorbed, are infinitesimal quantities that do not correspond to the difference between two works or two heats. Instead, the work done, and the heat absorbed, depend on the interaction process itself. Thus, it makes no sense to talk about the work in the system before and after the process, or the difference between these. Heat and Work 47 If the external parameters of the system have the values x1, · · · , xn then the energy of the system in a definite microstate, r, can be written Er = Er(x1, · · · , xn). (4.12) Hence, if the external parameters are changed by infinitesimal amounts, so that xα → xα + dxα for α in the range 1 to n, then the corresponding change in the energy of the microstate is dEr = ∑ α=1,n ∂Er ∂xα dxα. (4.13) The work, d̄W, done by the system when it remains in this particular state r is [see Equation (4.8)] d̄W = ∑ α=1,n Xα r dxα, (4.14) where Xα r ≡ −∂Er ∂xα (4.15) is termed the generalized force (conjugate to the external parameter xα) in the state r. (See Sec- tion B.2.) Note that if xα is a displacement then Xα r is an ordinary force. Consider, now, an ensemble of systems. Provided that the external parameters of the system are changed quasi-statically, the generalized forces Xα r have well-defined mean values that are calculable from the distribution of systems in the ensemble characteristic of the instantaneous macrostate. The macroscopic work, d̄W, resulting from an infinitesimal quasi-static change of the external parameters is obtained by calculating the decrease in the mean energy resulting from the parameter change. Thus, d̄W = ∑ α=1,n Xα dxα, (4.16) where Xα ≡ −∂Er ∂xα (4.17) is the mean generalized force conjugate to xα. The mean value is calculated from the equilibrium distribution of systems in the ensemble corresponding to the external parameter values xα. The macroscopic work, W, resulting from a finite quasi-static change of external parameters can be obtained by integrating Equation (4.16). The most well-known example of quasi-static work in thermodynamics is that done by pressure when the volume changes. For simplicity, suppose that the volume V is the only external parameter of any consequence. The work done in changing the volume from V to V+dV is simply the product of the force and the displacement (along the line of action of the force). By definition, the mean equilibrium pressure, p̄, of a given macrostate is equal to the normal force per unit area acting on any surface element. Thus, the normal force acting on a surface element dSi is p̄ dSi. Suppose that the surface element is subject to a displacement dxi. The work done by the element is p̄ dSi ·dxi. The total work done by the system is obtained by summing over all of the surface elements. Thus, d̄W = p̄ dV, (4.18) 50 4.5. Exact and Inexact Differentials The elimination of dy/dx between Equations (4.32) and (4.33) yields Y ′ ∂σ ∂x = X′ ∂σ ∂y = X′ Y ′ τ , (4.34) where τ(x, y) is function of x and y. The previous equation could equally well be written X′ = τ ∂σ ∂x , Y ′ = τ ∂σ ∂y . (4.35) Inserting Equation (4.35) into Equation (4.26) gives d̄G = τ ( ∂σ ∂x dx + ∂σ ∂y dy ) = τ dσ, (4.36) or d̄G τ = dσ. (4.37) Thus, dividing the inexact differential d̄G by τ yields the exact differential dσ. A factor τ that possesses this property is termed an integrating factor. Because the previous analysis is quite general, it is clear that an inexact differential involving two independent variables always admits of an integrating factor. Note, however, this is not generally the case for inexact differentials involving more than two variables. After this mathematical excursion, let us return to a physical situation of interest. The macrostate of a macroscopic system can be specified by the values of the external parameters (e.g., the vol- ume), and the mean energy, E. This, in turn, fixes other parameters, such as the mean pressure, p̄. Alternatively, we can specify the external parameters and the mean pressure, which fixes the mean energy. Quantities such as dp̄ and dE are infinitesimal differences between well-defined quantities: that is, they are exact differentials. For example, dE = E f − Ei is just the difference between the mean energy of the system in the final macrostate f and the initial macrostate i, in the limit where these two states are nearly the same. It follows that if the system is taken from an initial macrostate i to any final macrostate f then the mean energy change is given by ∆E = E f − Ei = ∫ f i dE. (4.38) However, because the mean energy is just a function of the macrostate under consideration, E f and Ei depend only on the initial and final states, respectively. Thus, the integral ∫ dE depends only on the initial and final states, and not on the particular process used to get between them. Consider, now, the infinitesimal work done by the system in going from some initial macrostate i to some neighbouring final macrostate f . In general, d̄W = ∑ Xα dxα is not the difference between two numbers referring to the properties of two neighboring macrostates. Instead, it is merely an infinitesimal quantity characteristic of the process of going from state i to state f . In other words, the work d̄W is in general an inexact differential. The total work done by the system in going from any macrostate i to some other macrostate f can be written as Wi f = ∫ f i d̄W, (4.39) Heat and Work 51 where the integral represents the sum of the infinitesimal amounts of work d̄W performed at each stage of the process. In general, the value of the integral does depend on the particular process used in going from macrostate i to macrostate f . Recall that, in going from macrostate i to macrostate f , the change in energy, ∆E, does not depend on the process used, whereas the work done, W, in general, does. Thus, it follows from the first law of thermodynamics, Equation (4.10), that the heat absorbed, Q, in general, also depends on the process used. It follows that d̄Q ≡ dE + d̄W (4.40) is an inexact differential. However, by analogy with the mathematical example discussed previ- ously, there must exist some integrating factor, T (say), that converts the inexact differential d̄Q into an exact differential. So, d̄Q T ≡ dS . (4.41) It will be interesting to discover the physical quantities that correspond to the functions T and S . (See Section 5.5.) Suppose that the system is thermally insulated, so that d̄Q = 0. In this case, the first law of thermodynamics implies that Wi f = −∆E. (4.42) Thus, in this special case, the work done depends only on the energy difference between the initial and final states, and is independent of the process. In fact, when Clausius first formulated the first law of thermodynamics, in 1850, he wrote: If a thermally isolated system is brought from some initial to some final state then the work done by the system is independent of the process used. If the external parameters of the system are kept fixed, so that no work is done, then d̄W = 0, Equation (4.11) reduces to d̄Q = dE, (4.43) and d̄Q becomes an exact differential. The amount of heat, Q, absorbed in going from one macrostate to another depends only on the mean energy difference between them, and is inde- pendent of the process used to effect the change. In this situation, heat is a conserved quantity, and acts very much like the invisible indestructible fluid of Lavoisier’s calorific theory. Exercises 4.1 The mean pressure, p̄, of a thermally insulated gas varies with volume according to the relation p̄ V γ = K, where γ > 1 and K are positive constants. Show that the work done by this gas in a quasi- static process in which the state of the gas evolves from an initial macrostate with pressure p̄i and volume Vi to a final macrostate with pressure p̄ f and volume Vf is Wi f = 1 γ − 1 ( p̄i Vi − p̄ f V f ) . 52 4.5. Exact and Inexact Differentials 4.2 Consider the infinitesimal quantity d̄F ≡ (x 2 − y) dx + x dy. Is this an exact differential? If not, find the integrating factor that converts it into an exact differential. 4.3 A system undergoes a quasi-static process that appears as a closed curve in a diagram of mean pressure, p̄, versus volume, V . Such a process is termed cyclic, because the system ends up in a final macrostate that is identical to its initial macrostate. Show that the work done by the system is given by the area contained within the closed curve in the p̄-V plane. Statistical Thermodynamics 55 where β′ and λ′ are defined in an analogous manner to the parameters β and λ. Equations (5.5) and (5.10) can be combined to give ln [Ω(E)Ω′(E′)] = ln [Ω(Ẽ)Ω′(Ẽ′)] + [β(Ẽ) − β′(Ẽ′)] η − 1 2 [λ(Ẽ) + λ′(Ẽ′)] η 2 + · · · . (5.11) At the maximum of ln [Ω(E)Ω′(E′)], the linear term in the Taylor expansion must vanish, so β(Ẽ) = β′(Ẽ′), (5.12) which enables us to determine Ẽ. It follows that ln P(E) = ln P(Ẽ) − 1 2 λ0 η 2, (5.13) or P(E) = P(Ẽ) exp [ −1 2 λ0 (E − Ẽ) 2 ] , (5.14) where λ0 = λ(Ẽ) + λ′(Ẽ′). (5.15) Now, the parameter λ0 must be positive, otherwise the probability P(E) does not exhibit a pro- nounced maximum value. That is, the combined system, A(0), does not possess a well-defined equilibrium state as, physically, we know it must. It is clear that λ(Ẽ) must also be positive, be- cause we could always choose for A′ a system with a negligible contribution to λ0, in which case the constraint λ0 > 0 would effectively correspond to λ(Ẽ) > 0. [A similar argument can be used to show that λ′(Ẽ′) must be positive.] The same conclusion also follows from the estimate Ω ∝ E f , which implies that λ(Ẽ) ∼ f Ẽ 2 > 0. (5.16) According to Equation (5.14), the probability distribution function, P(E), is Gaussian. (See Section 2.9.) This is hardly surprising, because the central limit theorem ensures that the probabil- ity distribution for any macroscopic variable, such as E, is Gaussian in nature. (See Section 2.10.) It follows that the mean value of E corresponds to the situation of maximum probability (i.e., the peak of the Gaussian curve), so that E = Ẽ. (5.17) The standard deviation of the distribution is ∆∗E = λ−1/2 0 ∼ E√ f , (5.18) where use has been made of Equation (5.16) (assuming that system A makes the dominant contri- bution to λ0). It follows that the fractional width of the probability distribution function is given by ∆∗E E ∼ 1√ f . (5.19) 56 5.3. Temperature Hence, if A contains 1 mole of particles then f ∼ NA  10 24, and ∆∗E/E ∼ 10−12. Clearly, the probability distribution for E has an exceedingly sharp maximum. Experimental measurements of this energy will almost always yield the mean value, and the underlying statistical nature of the distribution may not be apparent. 5.3 Temperature Suppose that the systems A and A′ are initially thermally isolated from one another, with respective energies Ei and E′i . (Because the energy of an isolated system cannot fluctuate, we do not have to bother with mean energies here.) If the two systems are subsequently placed in thermal contact, so that they are free to exchange heat energy, then, in general, the resulting state is an extremely im- probable one [i.e., P(Ei) is much less than the peak probability]. The configuration will, therefore, tend to evolve in time until the two systems attain final mean energies, E f and E ′ f , which are such that β f = β ′ f , (5.20) where β f ≡ β(E f ) and β′f ≡ β′(E ′ f ). This corresponds to the state of maximum probability. (See Section 5.2.) In the special case where the initial energies, Ei and E′i , lie very close to the final mean energies, E f and E ′ f , respectively, there is no change in the two systems when they are brought into thermal contact, because the initial state already corresponds to a state of maximum probability. It follows from energy conservation that E f + E ′ f = Ei + E′i . (5.21) The mean energy change in each system is simply the net heat absorbed, so that Q ≡ E f − Ei, (5.22) Q′ ≡ E ′ f − E′i . (5.23) The conservation of energy then reduces to Q + Q′ = 0. (5.24) In other words, the heat given off by one system is equal to the heat absorbed by the other. (In our notation, absorbed heat is positive, and emitted heat is negative.) It is clear that if the systems A and A′ are suddenly brought into thermal contact then they will only exchange heat, and evolve towards a new equilibrium state, if the final state is more probable than the initial one. In other words, the system will evolve if P(E f ) > P(Ei), (5.25) or ln P(E f ) > ln P(Ei), (5.26) Statistical Thermodynamics 57 because the logarithm is a monotonic function. The previous inequality can be written lnΩ(E f ) + lnΩ′(E ′ f ) > lnΩ(Ei) + lnΩ′(E′i ), (5.27) with the aid of Equation (5.3). Taylor expansion to first order yields ∂ lnΩ(Ei) ∂E (E f − Ei) + ∂ lnΩ′(E′i ) ∂E′ (E ′ f − E′i ) > 0, (5.28) which finally gives (βi − β′i) Q > 0, (5.29) where βi ≡ β(Ei), β′i ≡ β′(E′i ), and use has been made of Equations (5.22)–(5.24). It is clear, from the previous analysis, that the parameter β, defined β = ∂ lnΩ ∂E , (5.30) has the following properties: 1. If two systems separately in equilibrium have the same value of β then the systems will remain in equilibrium when brought into thermal contact with one another. 2. If two systems separately in equilibrium have different values of β then the systems will not remain in equilibrium when brought into thermal contact with one another. Instead, the system with the higher value of β will absorb heat from the other system until the two β values are the same. [See Equation (5.29).] Incidentally, a partial derivative is used in Equation (5.30) because, in a purely thermal interaction, the external parameters of the system are held constant as the energy changes. Let us define the dimensionless parameter T , such that 1 k T ≡ β = ∂ lnΩ ∂E , (5.31) where k is a positive constant having the dimensions of energy. The parameter T is termed the thermodynamic temperature, and controls heat flow in much the same manner as a conventional temperature. Thus, if two isolated systems in equilibrium possess the same thermodynamic tem- perature then they will remain in equilibrium when brought into thermal contact. However, if the two systems have different thermodynamic temperatures then heat will flow from the system with the higher temperature (i.e., the “hotter” system) to the system with the lower temperature, until the temperatures of the two systems are the same. In addition, suppose that we have three systems A, B, and C. We know that if A and B remain in equilibrium when brought into thermal contact then their temperatures are the same, so that TA = TB. Similarly, if B and C remain in equilibrium when brought into thermal contact, then TB = TC . But, we can then conclude that TA = TC , so systems A and C will also remain in equilibrium when brought into thermal contact. Thus, we arrive at the following statement, which is sometimes called the zeroth law of thermodynamics: 60 5.5. General Interaction Between Macrosystems where use has been made of Equation (5.35). Dividing both sides by Ω gives ∂ lnΩ ∂x = ∂ lnΩ ∂E X + ∂X ∂E . (5.39) However, according to the usual estimateΩ ∝ E f (see Section 3.8), the first term on the right-hand side is of order ( f /E) X, whereas the second term is only of order X/E. Clearly, for a macroscopic system with many degrees of freedom, the second term is utterly negligible, so we have ∂ lnΩ ∂x = ∂ lnΩ ∂E X = β X, (5.40) where use has been made of Equation (5.30). When there are several external parameters, x1, · · · , xn, so that Ω ≡ Ω(E, x1, · · · , xn), the pre- vious derivation is valid for each parameter taken in isolation. Thus, ∂ lnΩ ∂xα = β Xα, (5.41) where Xα is the mean generalized force conjugate to the parameter xα. (See Section B.2.) 5.5 General Interaction Between Macrosystems Consider two systems, A and A′, that can interact by exchanging heat energy and doing work on one another. Let the system A have energy E, and adjustable external parameters x1, · · · , xn. Likewise, let the system A′ have energy E′, and adjustable external parameters x′1, · · · , x′n. The combined system A(0) = A + A′ is assumed to be isolated. It follows from the first law of thermodynamics that E + E′ = E (0) = constant. (5.42) Thus, the energy E′ of system A′ is determined once the energy E of system A is given, and vice versa. In fact, E′ could be regarded as a function of E. Furthermore, if the two systems can interact mechanically then, in general, the parameters x′ are some function of the parameters x. As a simple example, if the two systems are separated by a movable partition, in an enclosure of fixed volume V (0), then V + V ′ = V (0) = constant, (5.43) where V and V ′ are the volumes of systems A and A′, respectively. The total number of microstates accessible to A(0) is clearly a function of E, and the parameters xα (where α runs from 1 to n), so Ω (0) ≡ Ω (0)(E, x1, · · · , xn). We have already demonstrated (in Section 5.2) that Ω (0) exhibits a very pronounced maximum at one particular value of the energy, E = Ẽ, when E is varied but the external parameters are held constant. This behavior comes about because of the very strong, Ω ∝ E f , (5.44) Statistical Thermodynamics 61 increase in the number of accessible microstates of A (or A′) with energy. However, according to Section 3.8, the number of accessible microstates exhibits a similar strong increase with the volume, which is a typical external parameter, so that Ω ∝ V f . (5.45) It follows that the variation of Ω (0) with a typical parameter, xα, when all the other parameters and the energy are held constant, also exhibits a very sharp maximum at some particular value, xα = x̃α. The equilibrium situation corresponds to the configuration of maximum probability, in which virtually all systems A(0) in the ensemble have values of E and xα very close to Ẽ and x̃α, respectively. The mean values of these quantities are thus given by E = Ẽ and x̄α = x̃α. Consider a quasi-static process in which the system A is brought from an equilibrium state described by E and x̄α to an infinitesimally different equilibrium state described by E + dE and x̄α + dx̄α. Let us calculate the resultant change in the number of microstates accessible to A. Because Ω ≡ Ω(E, x1, · · · , xn), the change in lnΩ follows from standard mathematics: d lnΩ = ∂ lnΩ ∂E dE + ∑ α=1,n ∂ lnΩ ∂xα dx̄α. (5.46) However, we have previously demonstrated that β = ∂ lnΩ ∂E , β Xα = ∂ lnΩ ∂xα (5.47) [in Equations (5.30) and (5.41), respectively], so Equation (5.46) can be written d lnΩ = β dE + ∑ α Xα dx̄α  . (5.48) Note that the temperature parameter, β, and the mean conjugate forces, Xα, are only well defined for equilibrium states. This is why we are only considering quasi-static changes in which the two systems are always arbitrarily close to equilibrium. Let us rewrite Equation (5.48) in terms of the thermodynamic temperature, T , using the relation β ≡ 1/k T . We obtain dS = dE + ∑ α Xα dx̄α  / T, (5.49) where S = k lnΩ. (5.50) Equation (5.49) is a differential relation that enables us to calculate the quantity S as a function of the mean energy, E, and the mean external parameters, x̄α, assuming that we can calculate the temperature, T , and mean conjugate forces, Xα, for each equilibrium state. The function S (E, x̄α) is termed the entropy of system A. The word entropy is derived from the Greek entropē, which means “a turning towards” or ”tendency.” The reason for this etymology will become apparent presently. 62 5.6. Entropy It can be seen from Equation (5.50) that the entropy is merely a parameterization of the number of accessible microstates. Hence, according to statistical mechanics, S (E, x̄α) is essentially a measure of the relative probability of a state characterized by values of the mean energy and mean external parameters E and x̄α, respectively. According to Equation (4.16), the net amount of work performed during a quasi-static change is given by d̄W = ∑ α Xα dx̄α. (5.51) It follows from Equation (5.49) that dS = dE + d̄W T = d̄Q T . (5.52) Thus, the thermodynamic temperature, T , is the integrating factor for the first law of thermody- namics, d̄Q = dE + d̄W, (5.53) which converts the inexact differential d̄Q into the exact differential dS . (See Section 4.5.) It follows that the entropy difference between any two macrostates i and f can be written S f − S i = ∫ f i dS = ∫ f i d̄Q T , (5.54) where the integral is evaluated for any process through which the system is brought quasi-statically via a sequence of near-equilibrium configurations from its initial to its final macrostate. The pro- cess has to be quasi-static because the temperature, T , which appears in the integrand, is only well defined for an equilibrium state. Because the left-hand side of the previous equation only depends on the initial and final states, it follows that the integral on the right-hand side is independent of the particular sequence of quasi-static changes used to get from i to f . Thus, ∫ f i d̄Q/T is independent of the process (provided that it is quasi-static). All of the concepts that we have encountered up to now in this course, such as temperature, heat, energy, volume, pressure, et cetera, have been fairly familiar to us from other branches of physics. However, entropy, which turns out to be of crucial importance in thermodynamics, is something quite new. Let us consider the following questions. What does the entropy of a thermodynamic system actually signify? What use is the concept of entropy? 5.6 Entropy Consider an isolated system whose energy is known to lie in a narrow range. Let Ω be the number of accessible microstates. According to the principle of equal a priori probabilities, the system is equally likely to be found in any one of these states when it is in thermal equilibrium. The ac- cessible states are just that set of microstates that are consistent with the macroscopic constraints imposed on the system. These constraints can usually be quantified by specifying the values of some parameters, y1, · · · , yn, that characterize the macrostate. Note that these parameters are not Statistical Thermodynamics 65 process. This process is irreversible on a microscopic level because the initial configuration cannot be recovered by simply replacing the partition. It is irreversible on a macroscopic level because it is obviously unphysical for the molecules of a gas to spontaneously distribute themselves in such a manner that they only occupy half of the available volume. It is actually possible to quantify irreversibility. In other words, in addition to stating that a given process is irreversible, we can also give some indication of how irreversible it is. The parameter that measures irreversibility is the number of accessible states, Ω. Thus, if Ω for an isolated system spontaneously increases then the process is irreversible, the degree of irreversibility being proportional to the amount of the increase. If Ω stays the same then the process is reversible. Of course, it is unphysical for Ω to ever spontaneously decrease. In symbols, we can write Ω f − Ωi ≡ ∆Ω ≥ 0, (5.61) for any physical process operating on an isolated system. In practice, Ω itself is a rather unwieldy parameter with which to measure irreversibility. For instance, in the previous example, where an ideal gas doubles in volume (at constant energy) due to the removal of a partition, the fractional increase in Ω is Ω f Ωi  10 2 ν×10 23 , (5.62) where ν is the number of moles. This is an extremely large number. It is far more convenient to measure irreversibility in terms of lnΩ. If Equation (5.61) is true then it is certainly also true that lnΩ f − lnΩi ≡ ∆ lnΩ ≥ 0 (5.63) for any physical process operating on an isolated system. The increase in lnΩ when an ideal gas doubles in volume (at constant energy) is lnΩ f − lnΩi = νNA ln 2, (5.64) where NA = 6 × 10 23. This is a far more manageable number. Because we usually deal with particles by the mole in laboratory physics, it makes sense to pre-multiply our measure of irre- versibility by a number of order 1/NA. For historical reasons, the number that is generally used for this purpose is the Boltzmann constant, k, which can be written k = R NA joules/kelvin, (5.65) where R = 8.3143 joules/kelvin/mole (5.66) is the ideal gas constant that appears in the well-known equation of state for an ideal gas, P V = νR T . Thus, the final form for our measure of irreversibility is S = k lnΩ. (5.67) 66 5.6. Entropy This quantity is termed “entropy”, and is measured in joules per degree kelvin. The increase in entropy when an ideal gas doubles in volume (at constant energy) is S f − S i = νR ln 2, (5.68) which is order unity for laboratory-scale systems (i.e., those containing about one mole of parti- cles). The essential irreversibility of macroscopic phenomena can be summed up as follows: S f − S i ≡ ∆S ≥ 0, (5.69) for a process acting on an isolated system. [This formula is equivalent to Equations (5.61) and (5.63).] Thus: The entropy of an isolated system tends to increase with time, and can never decrease. This proposition is known as the second law of thermodynamics. One way of thinking of the number of accessible states,Ω, is that it is a measure of the disorder associated with a macrostate. For a system exhibiting a high degree of order, we would expect a strong correlation between the motions of the individual particles. For instance, in a fluid there might be a strong tendency for the particles to move in one particular direction, giving rise to an ordered flow of the system in that direction. On the other hand, for a system exhibiting a low degree of order, we expect far less correlation between the motions of individual particles. It follows that, all other things being equal, an ordered system is more constrained than a disordered system, because the former is excluded from microstates in which there is not a strong correlation between individual particle motions, whereas the latter is not. Another way of saying this is that an ordered system has less accessible microstates than a corresponding disordered system. Thus, entropy is effectively a measure of the disorder in a system (the disorder increases with increasing S ). With this interpretation, the second law of thermodynamics reduces to the statement that isolated systems tend to become more disordered with time, and can never become more ordered. Note that the second law of thermodynamics only applies to isolated systems. The entropy of a non-isolated system can decrease. For instance, if a gas expands (at constant energy) to twice its initial volume after the removal of a partition, we can subsequently recompress the gas to its original volume. The energy of the gas will increase because of the work done on it during compression, but if we absorb some heat from the gas then we can restore it to its initial state. Clearly, in restoring the gas to its original state, we have restored its original entropy. This appears to violate the second law of thermodynamics, because the entropy should have increased in what is obviously an irreversible process. However, if we consider a new system consisting of the gas plus the compression and heat absorption machinery then it is still true that the entropy of this system (which is assumed to be isolated) must increase in time. Thus, the entropy of the gas is only kept the same at the expense of increasing the entropy of the rest of the system, and the total entropy is increased. If we consider the system of everything in the universe, which is certainly an isolated system because there is nothing outside it with which it could interact, then the second law of thermodynamics becomes: The disorder of the universe tends to increase with time, and can never decrease. Statistical Thermodynamics 67 5.7 Properties of Entropy Entropy, as we have defined it, has some dependence on the resolution, δE, to which the energy of macrostates is measured. Recall that Ω(E) is the number of accessible microstates with energy in the range E to E + δE. Suppose that we choose a new resolution δ∗E, and define a new density of states, Ω ∗(E), which is the number of states with energy in the range E to E + δ∗E. It can easily be seen that Ω ∗(E) = δ∗E δE Ω(E). (5.70) It follows that the new entropy, S ∗ = k lnΩ ∗, is related to the previous entropy S = k lnΩ via S ∗ = S + k ln ( δ∗E δE ) . (5.71) Now, our usual estimate that Ω ∼ E f (see Section 3.8) gives S ∼ k f , where f is the number of degrees of freedom. It follows that even if δ∗E were to differ from δE by of order f (i.e., twenty four orders of magnitude), which is virtually inconceivable, the second term on the right-hand side of the previous equation is still only of order k ln f , which is utterly negligible compared to k f . It follows that S ∗ = S (5.72) to an excellent approximation, so our definition of entropy is completely insensitive to the resolu- tion to which we measure energy (or any other macroscopic parameter). Note that, like the temperature, the entropy of a macrostate is only well defined if the macrostate is in equilibrium. The crucial point is that it only makes sense to talk about the number of accessible states if the systems in the ensemble are given sufficient time to thoroughly explore all of the possible microstates consistent with the known macroscopic constraints. In other words, we can only be sure that a given microstate is inaccessible when the systems in the ensemble have had ample opportunity to move into it, and yet have not done so. For an equilibrium state, the entropy is just as well defined as more familiar quantities such as the temperature and the mean pressure. Consider, again, two systems, A and A′, that are in thermal contact, but can do no work on one another. (See Section 5.2.) Let E and E′ be the energies of the two systems, and Ω(E) and Ω′(E′) the respective densities of states. Furthermore, let E (0) be the conserved energy of the system as a whole, and Ω (0) the corresponding density of states. We have from Equation (5.2) that Ω (0)(E) = Ω(E)Ω′(E′), (5.73) where E′ = E (0) − E. In other words, the number of states accessible to the whole system is the product of the numbers of states accessible to each subsystem, because every microstate of A can be combined with every microstate of A′ to form a distinct microstate of the whole system. We know, from Section 5.2, that in equilibrium the mean energy of A takes the value E = Ẽ for which Ω (0)(E) is maximum, and the temperatures of A and A′ are equal. The distribution of E around the mean value is of order ∆∗E = Ẽ/ √ f , where f is the number of degrees of freedom. It follows that 70 5.9. Entropy and Quantum Mechanics where the subscripts are to remind us what is held constant in the partial derivatives. We can write an analogous pair of equations for the system A′. The overall system is assumed to be isolated, so conservation of energy gives dE + dE′ = 0. Furthermore, Equation (5.81) implies that dV + dV ′ = 0. It follows that the total change in entropy is given by dS (0) = dS + dS ′ = ( 1 T − 1 T ′ ) dE + ( p̄ T − p̄′ T ′ ) dV. (5.87) The equilibrium state is the most probable state. (See Section 5.2.) According to statistical me- chanics, this is equivalent to the state with the largest number of accessible microstates. Finally, Equation (5.78) implies that this is the maximum entropy state. The system can never sponta- neously leave a maximum entropy state, because this would imply a spontaneous reduction in en- tropy, which is forbidden by the second law of thermodynamics. A maximum or minimum entropy state must satisfy dS (0) = 0 for arbitrary small variations of the energy and external parameters. It follows from Equation (5.87) that T = T ′, (5.88) p̄ = p̄′, (5.89) for such a state. This corresponds to a maximum entropy state (i.e., an equilibrium state) provided that ( ∂ 2S ∂E 2 ) V < 0, (5.90) ( ∂ 2S ∂V 2 ) E < 0, (5.91) with a similar pair of inequalities for system A′. The usual estimate Ω ∝ E f V f (see Section 3.8), giving S = k f ln E + k f ln V + · · · , ensures that the previous inequalities are satisfied in con- ventional macroscopic systems. In the maximum entropy state, the systems A and A′ have equal temperatures (i.e., they are in thermal equilibrium), and equal pressures (i.e., they are in mechani- cal equilibrium). The second law of thermodynamics implies that the two interacting systems will evolve towards this state, and will then remain in it indefinitely (if left undisturbed). 5.9 Entropy and Quantum Mechanics The entropy, S , of a system is defined in terms of the number, Ω, of accessible microstates consis- tent with an overall energy in the range E to E + δE via S = k lnΩ. (5.92) We have already demonstrated that this definition is utterly insensitive to the resolution, δE, to which the macroscopic energy is measured (See Section 5.7.) In classical mechanics, if a system Statistical Thermodynamics 71 possesses f degrees of freedom then phase-space is conventionally subdivided into cells of arbi- trarily chosen volume h f 0 . (See Section 3.2.) The number of accessible microstates is equivalent to the number of these cells in the volume of phase-space consistent with an overall energy of the system lying in the range E to E + δE. Thus, Ω = 1 h f 0 ∫ · · · ∫ dq1 · · · dqf dp1 · · · dpf , (5.93) giving S = k ln (∫ · · · ∫ dq1 · · · dqf dp1 · · · dpf ) − k f ln h0. (5.94) Thus, in classical mechanics, the entropy is undetermined to an arbitrary additive constant that depends on the size of the cells in phase-space. In fact, S increases as the cell size decreases. The second law of thermodynamics is only concerned with changes in entropy, and is, therefore, unaffected by an additive constant. Likewise, macroscopic thermodynamical quantities, such as the temperature and pressure, that can be expressed as partial derivatives of the entropy with re- spect to various macroscopic parameters [see Equations (5.85) and (5.86)] are unaffected by such a constant. So, in classical mechanics, the entropy is rather like a gravitational potential—it is undetermined to an additive constant, but this does not affect any physical laws. The non-unique value of the entropy comes about because there is no limit to the precision to which the state of a classical system can be specified. In other words, the cell size, h0, can be made arbitrarily small, which corresponds to specifying the particle coordinates and momenta to arbitrary accuracy. However, in quantum mechanics, the uncertainty principle sets a definite limit to how accurately the particle coordinates and momenta can be specified. (See Section C.8.) In general, δqi δpi ≥
/2, (5.95) where
is Planck’s constant over 2π, pi is the momentum conjugate to the generalized coordinate qi, and δqi, δpi are the uncertainties in these quantities, respectively. In fact, in quantum mechanics, the number of accessible quantum states with the overall energy in the range E to E + δE is completely determined. This implies that, in reality, the entropy of a system has a unique and unambiguous value. Quantum mechanics can often be “mocked up” in classical mechanics by setting the cell size in phase-space equal to Planck’s constant, so that h0 = h. This automatically enforces the most restrictive form of the uncertainty principle, δqi δpi 
/2. In many systems, the substitution h0 → h in Equation (5.94) gives the same, unique value for S as that obtained from a full quantum-mechanical calculation. (See Section 8.10.) Consider a simple quantum-mechanical system consisting of N non-interacting spinless parti- cles of mass m confined in a cubic box of dimension L. (See Section C.10.) The energy levels of the ith particle are given by ei =
2π 2 2 m L 2 ( n 2 1 i + n 2 2 i + n 2 3 i ) , (5.96) where n1 i, n2 i, and n3 i are three (positive) quantum numbers. The overall energy of the system is 72 5.9. Entropy and Quantum Mechanics the sum of the energies of the individual particles, so that for a general state r, Er = ∑ i=1,N ei. (5.97) The overall state of the system is completely specified by 3N quantum numbers, so the number of degrees of freedom is f = 3N. The classical limit corresponds to the situation where all of the quantum numbers are much greater than unity. In this limit, the number of accessible states varies with energy according to our usual estimateΩ ∝ E f . (See Section 3.8.) The lowest possible energy state of the system, the so-called ground-state, corresponds to the situation where all quantum numbers take their lowest possible value, unity. Thus, the ground-state energy, E0, is given by E0 = f
2 π 2 2 m L 2 . (5.98) There is only one accessible microstate at the ground-state energy (i.e., that where all quantum numbers are unity), so by our usual definition of entropy, S (E0) = k ln 1 = 0. (5.99) In other words, there is no disorder in the system when all the particles are in their ground-states. Clearly, as the energy approaches the ground-state energy, the number of accessible states becomes far less than the usual classical estimate E f . This is true for all quantum mechanical systems. In general, the number of microstates varies roughly like Ω(E) ∼ 1 +C (E − E0) f , (5.100) where C is a positive constant. According to Equation (5.31), the temperature varies approximately like T ∼ E − E0 k f , (5.101) provided Ω 1. Thus, as the absolute temperature of a system approaches zero, the internal energy approaches a limiting value E0 (the quantum-mechanical ground-state energy), and the entropy approaches the limiting value zero. This proposition is known as the third law of thermo- dynamics. At low temperatures, great care must be taken to ensure that equilibrium thermodynamical arguments are applicable, because the rate of attaining equilibrium may be very slow. Another difficulty arises when dealing with a system in which the atoms possess nuclear spins. Typically, when such a system is brought to a very low temperature, the entropy associated with the degrees of freedom not involving nuclear spins becomes negligible. Nevertheless, the number of microstates, Ωs, corresponding to the possible nuclear spin orientations may be very large. Indeed, it may be just as large as the number of states at room temperature. The reason for this is that nuclear magnetic moments are extremely small, and, therefore, have extremely weak mutual interactions. Thus, it only takes a tiny amount of heat energy in the system to completely randomize the spin orientations. Typically, a temperature as small as 10−3 degrees kelvin is sufficient to randomize the spins. Statistical Thermodynamics 75 5.4 The heat absorbed by a mole of ideal gas in a quasi-static process in which the temperature, T , changes by dT , and the volume, V , by dV , is given by d̄Q = c dT + p dV, where c is its constant molar specific heat at constant volume, and p = R T/V is its pressure. Find an expression for the change in entropy of the gas in a quasi-static process which takes it from the initial values of temperature and volume Ti and Vi, respectively, to the final values T f and Vf , respectively. Does the answer depend on the process involved in going from the initial to the final state? What is the relationship between the temperature and the volume in an adiabatic process (i.e. a quasi-static process in which no heat is absorbed)? What is the change in entropy in an adiabatic process in which the volume changes from an initial value Vi to a final value Vf ? 5.5 A solid contains N magnetic atoms having spin 1/2. At sufficiently high temperatures, each spin is completely randomly oriented. In other words, it is equally likely to be in either one of two possible states. But at sufficiently low temperature, the interactions be- tween the magnetic atoms causes them to exhibit ferromagnetism, with the result that their spins become oriented in the same direction. A very crude approximation suggests that the spin-dependent contribution, C(T ), to the heat capacity of the solid has an approximate temperature dependence given by C(T ) = C1 ( 2 T T1 − 1 ) for T1/2 < T < T1, and C(T ) = 0, otherwise. The abrupt increase in specific heat as T is reduced below T1 is due to the onset of ferromagnetic behavior. Find two expressions for the increase in entropy as the temperature of the system is raised from a value below T1/2 to one above T1. By equating these two expressions, show that C1 = 2.26 N k. 76 5.10. Laws of Thermodynamics Classical Thermodynamics 77 6 Classical Thermodynamics 6.1 Introduction We have already learned that macroscopic quantities, such as energy, temperature, and pressure, are, in fact, statistical in nature. In other words, in an equilibrium state, they exhibit random fluctuations about some mean value. If we were to plot the probability distribution for the energy (say) of a system in thermal equilibrium with its surroundings then we would obtain a Gaussian distribution with a very small fractional width. In fact, we expect ∆∗E E ∼ 1√ f , (6.1) where the number of degrees of freedom, f , is about 10 24 for laboratory-scale systems. This im- plies that the statistical fluctuations of macroscopic quantities about their mean values are typically only about 1 part in 10 12. Because the statistical fluctuations of equilibrium quantities are so small, it is an excellent approximation to neglect them altogether, and, thereby, to replace macroscopic quantities, such as energy, temperature, and pressure, by their mean values. In other words, E → E, T → T , p → p̄, et cetera. In the following discussion, we shall drop the overbars, so that E should be understood to represent the mean energy, E, et cetera. This prescription, which is the essence of classical thermodynamics, is equivalent to replacing all statistically-varying quantities by their most probable values. Although, formally, there are four laws of thermodynamics (i.e., the zeroth to the third), the zeroth law is really a consequence of the second law, and the third law is only important at tem- peratures close to absolute zero. So, for most practical purposes, the two laws that actually matter are the first law and the second law. For an infinitesimal process, the first law of thermodynamics is written d̄Q = dE + d̄W, (6.2) where dE is the change in internal energy of the system, d̄Q the heat absorbed by the system from its surroundings, and d̄W the work done by the system on its surroundings. Note that this particular formulation of the first law is merely a convention. We could equally well write the first law in terms of the heat emitted by the system, or the work done on the system. It does not really matter, as long as we are consistent in our definitions. The second law of thermodynamics implies that d̄Q = T dS , (6.3) for an infinitesimal quasi-static process, where T is the thermodynamic temperature, and dS the change in entropy of the system. Furthermore, for systems in which the only external parameter is the volume (e.g., gases), the work done on the environment is d̄W = p dV, (6.4) 80 6.2. Ideal Gas Equation of State It follows from mathematics that dE = ( ∂E ∂T ) V dT + ( ∂E ∂V ) T dV, (6.15) where the subscript V reminds us that the first partial derivative is taken at constant volume, and the subscript T reminds us that the second partial derivative is taken at constant temperature. The first and second laws of thermodynamics imply that for a quasi-static change of parameters, T dS = dE + p dV. (6.16) [See Equation (6.5).] The ideal gas equation of state, (6.10), can be used to express the pressure in term of the volume and the temperature in the previous expression: dS = 1 T dE + νR V dV. (6.17) Using Equation (6.15), this becomes dS = 1 T ( ∂E ∂T ) V dT + [ 1 T ( ∂E ∂V ) T + νR V ] dV. (6.18) However, dS is the exact differential of a well-defined state function, S . This means that we can consider the entropy to be a function of the temperature and volume. Thus, S = S (T,V), and mathematics immediately yields dS = ( ∂S ∂T ) V dT + ( ∂S ∂V ) T dV. (6.19) The previous expression is valid for all small values of dT and dV , so a comparison with Equa- tion (6.18) gives ( ∂S ∂T ) V = 1 T ( ∂E ∂T ) V , (6.20) ( ∂S ∂V ) T = 1 T ( ∂E ∂V ) T + νR V . (6.21) A well-known property of partial differentials is the equality of second derivatives, irrespective of the order of differentiation, so ∂ 2S ∂V ∂T = ∂ 2S ∂T ∂V . (6.22) This implies that ( ∂ ∂V ) T ( ∂S ∂T ) V = ( ∂ ∂T ) V ( ∂S ∂V ) T . (6.23) The previous expression can be combined with Equations (6.20) and (6.21) to give 1 T ( ∂ 2E ∂V ∂T ) = [ − 1 T 2 ( ∂E ∂V ) T + 1 T ( ∂ 2E ∂T ∂V )] . (6.24) Classical Thermodynamics 81 Because second derivatives are equivalent, irrespective of the order of differentiation, the previous relation reduces to ( ∂E ∂V ) T = 0, (6.25) which implies that the internal energy is independent of the volume for any gas obeying the ideal equation of state. This result was confirmed experimentally by James Joule in the middle of the nineteenth century. 6.3 Specific Heat Suppose that a body absorbs an amount of heat ∆Q, and its temperature consequently rises by ∆T . The usual definition of the heat capacity, or specific heat, of the body is C = ∆Q ∆T . (6.26) If the body consists of ν moles of some substance then the molar specific heat (i.e., the specific heat of one mole of this substance) is defined c = 1 ν ∆Q ∆T . (6.27) In writing the previous expressions, we have tacitly assumed that the specific heat of a body is independent of its temperature. In general, this is not true. We can overcome this problem by only allowing the body in question to absorb a very small amount of heat, so that its temperature only rises slightly, and its specific heat remains approximately constant. In the limit that the amount of absorbed heat becomes infinitesimal, we obtain c = 1 ν d̄Q dT . (6.28) In classical thermodynamics, it is usual to define two molar specific heats. Firstly, the molar specific heat at constant volume, denoted cV = 1 ν ( d̄Q dT ) V , (6.29) and, secondly, the molar specific heat at constant pressure, denoted cp = 1 ν ( d̄Q dT ) p . (6.30) Consider the molar specific heat at constant volume of an ideal gas. Because dV = 0, no work is done by the gas on its surroundings, and the first law of thermodynamics reduces to d̄Q = dE. (6.31) 82 6.3. Specific Heat It follows from Equation (6.29) that cV = 1 ν ( ∂E ∂T ) V . (6.32) Now, for an ideal gas, the internal energy is volume independent. [See Equation (6.25).] Thus, the previous expression implies that the specific heat at constant volume is also volume independent. Because E is a function of T only, we can write dE = ( ∂E ∂T ) V dT. (6.33) The previous two expressions can be combined to give dE = ν cV dT (6.34) for an ideal gas. Let us now consider the molar specific heat at constant pressure of an ideal gas. In general, if the pressure is kept constant then the volume changes, and so the gas does work on its environment. According to the first law of thermodynamics, d̄Q = dE + p dV = ν cV dT + p dV, (6.35) where use has been made of Equation (6.34). The equation of state of an ideal gas, (6.10), im- plies that if the volume changes by dV , the temperature changes by dT , and the pressure remains constant, then p dV = νR dT. (6.36) The previous two equations can be combined to give d̄Q = ν cV dT + νR dT. (6.37) Now, by definition, cp = 1 ν ( d̄Q dT ) p , (6.38) so we obtain cp = cV + R (6.39) for an ideal gas. Note that, at constant volume, all of the heat absorbed by the gas goes into increasing its internal energy, and, hence, its temperature, whereas, at constant pressure, some of the absorbed heat is used to do work on the environment as the volume increases. This means that, in the latter case, less heat is available to increase the temperature of the gas. Thus, we expect the specific heat at constant pressure to exceed that at constant volume, as indicated by the previous formula. The ratio of the two specific heats, cp/cV , is conventionally denoted γ. We have γ ≡ cp cV = 1 + R cV (6.40) Classical Thermodynamics 85 in an adiabatic process (in which no heat is absorbed). [See Equation (6.35).] The ideal gas equation of state, (6.10), can be differentiated, yielding p dV + V dp = νR dT. (6.54) The temperature increment, dT , can be eliminated between the previous two expressions to give 0 = cV R (p dV + V dp) + p dV = (cV R + 1 ) p dV + cV R V dp, (6.55) which reduces to (cV + R) p dV + cV V dp = 0. (6.56) Dividing through by cV p V yields γ dV V + dp p = 0, (6.57) where γ ≡ cp cV = cV + R cV . (6.58) It turns out that cV is a slowly-varying function of temperature in most gases. Consequently, it is usually a good approximation to treat the ratio of specific heats, γ, as a constant, at least over a limited temperature range. If γ is constant then we can integrate Equation (6.57) to give γ ln V + ln p = constant, (6.59) or p V γ = constant. (6.60) This result is known as the adiabatic gas law. It is straightforward to obtain analogous relationships between V and T , and between p and T , during adiabatic expansion or contraction. In fact, because p = νR T/V , the previous formula also implies that T V γ−1 = constant, (6.61) and p 1−γ T γ = constant. (6.62) Equations (6.60)–(6.62) are all completely equivalent. 6.6 Hydrostatic Equilibrium of Atmosphere The gas that we are most familiar with in everyday life is, of course, the Earth’s atmosphere. It turns out that we can use the isothermal and adiabatic gas laws to explain most of the observed features of the atmosphere. Let us, first of all, consider the hydrostatic equilibrium of the atmosphere. Consider a thin vertical slice of the atmosphere, of cross-sectional area A, that starts at height z above ground 86 6.7. Isothermal Atmosphere level, and extends to height z + dz. The upward force exerted on this slice by the gas below it is p(z) A, where p(z) is the pressure at height z. Likewise, the downward force exerted by the gas above the slice is p(z + dz) A. The net upward force is [p(z) − p(z + dz)] A. In equilibrium, this upward force must be balanced by the downward force due to the weight of the slice, which is ρ A dz g, where ρ is the mass density of the gas, and g the acceleration due to gravity. It follows that the force balance condition can be written [p(z) − p(z + dz)] A = ρ A dz g, (6.63) which reduces to dp dz = −ρ g. (6.64) This result is known as the equation of hydrostatic equilibrium for the atmosphere. We can express the mass density of a gas in the following form, ρ = ν µ V , (6.65) where µ is the molecular weight of the gas, and is equal to the mass of one mole of gas particles. For instance, the molecular weight of nitrogen gas is 28 × 10−3 kg. The previous formula for the mass density of a gas, combined with the ideal gas law, p V = νR T , yields ρ = p µ R T . (6.66) It follows that the equation of hydrostatic equilibrium can be rewritten dp p = − µ g R T dz. (6.67) 6.7 Isothermal Atmosphere As a first approximation, let us assume that the temperature of the atmosphere is uniform. In such an isothermal atmosphere, we can directly integrate the previous equation to give p = p0 exp ( − z z0 ) . (6.68) Here, p0 is the pressure at ground level (z = 0), which is generally about 1 bar (10 5 N m−2 in SI units). The quantity z0 = R T µ g (6.69) is known as the isothermal scale-height of the atmosphere. At ground level, the atmospheric temperature is, on average, about 15◦ C, which is 288 K on the absolute scale. The mean molecular weight of air at sea level is 29 × 10−3 kg (i.e., the molecular weight of a gas made up of 78% nitrogen, 21% oxygen, and 1% argon). The mean acceleration due to gravity is 9.81 m s−2 at Classical Thermodynamics 87 ground level. Also, the molar ideal gas constant is 8.314 joules/mole/degree. Combining all of this information, the isothermal scale-height of the atmosphere comes out to be about 8.4 kilometers. We have discovered that, in an isothermal atmosphere, the pressure decreases exponentially with increasing height. Because the temperature is assumed to be constant, and ρ ∝ p/T [see Equation (6.66)], it follows that the density also decreases exponentially with the same scale- height as the pressure. According to Equation (6.68), the pressure, or the density, of the atmo- sphere decreases by a factor 10 every ln10 z0, or 19.3 kilometers, increase in altitude above sea level. Clearly, the effective height of the atmosphere is very small compared to the Earth’s radius, which is about 6, 400 kilometers. In other words, the atmosphere constitutes a relatively thin layer covering the surface of the Earth. Incidentally, this justifies our neglect of the decrease of g with increasing altitude. One of the highest points in the United States of America is the peak of Mount Elbert in Col- orado. This peak lies 14, 432 feet, or about 4.4 kilometers, above sea level. At this altitude, Equa- tion (6.68) predicts that the air pressure should be about 0.6 atmospheres. Surprisingly enough, after a few days acclimatization, people can survive quite comfortably at this sort of pressure. In the highest inhabited regions of the Andes and Tibet, the air pressure falls to about 0.5 atmo- spheres. Humans can just about survive at such pressures. However, people cannot survive for any extended period in air pressures below half an atmosphere. This sets an upper limit on the altitude of permanent human habitation, which is about 19, 000 feet, or 5.8 kilometers, above sea level. Incidentally, this is also the maximum altitude at which a pilot can fly an unpressurized aircraft without requiring additional oxygen. The highest point in the world is, of course, the peak of Mount Everest in Nepal. This peak lies at an altitude of 29, 028 feet, or 8.85 kilometers, above sea level, where we expect the air pressure to be a mere 0.35 atmospheres. This explains why Mount Everest was only conquered after lightweight portable oxygen cylinders were invented. Admittedly, some climbers have subse- quently ascended Mount Everest without the aid of additional oxygen, but this is a very foolhardy venture, because, above 19, 000 feet, the climbers are slowly dying. Commercial airliners fly at a cruising altitude of 32, 000 feet. At this altitude, we expect the air pressure to be only 0.3 atmospheres, which explains why airline cabins are pressurized. In fact, the cabins are only pressurized to 0.85 atmospheres (which accounts for the “popping” of passangers ears during air travel). The reason for this partial pressurization is quite simple. At 32, 000 feet, the pressure difference between the air in the cabin, and hence that outside, is about half an atmosphere. Clearly, the walls of the cabin must be strong enough to support this pressure difference, which means that they must be of a certain thickness, and, hence, that the aircraft must be of a certain weight. If the cabin were fully pressurized then the pressure difference at cruising altitude would increase by about 30%, which means that the cabin walls would have to be much thicker, and, hence, the aircraft would have to be substantially heavier. So, a fully pressurized aircraft would be more comfortable to fly in (because your ears would not “pop”), but it would also be far less economical to operate. 90 6.8. Adiabatic Atmosphere the isothermal scale-height calculated using this temperature. The pressure profile is easily calcu- lated from the adiabatic gas law p 1−γ T γ = constant, or p ∝ T γ/(γ−1). It follows that p = p0 ( 1 − γ − 1 γ z z0 )γ/(γ−1) . (6.77) Consider the limit γ → 1. In this limit, Equation (6.75) yields T independent of height (i.e., the atmosphere becomes isothermal). We can evaluate Equation (6.77) in the limit as γ → 1 using the mathematical identity lt m→0 (1 + m x)1/m ≡ exp(x). (6.78) We obtain p = p0 exp ( − z z0 ) , (6.79) which, not surprisingly, is the predicted pressure variation in an isothermal atmosphere. In reality, the ratio of specific heats of the atmosphere is not unity, but is about 1.4 (i.e., the ratio for diatomic gases), which implies that in the real atmosphere p = p0 ( 1 − z 3.5 z0 )3.5 . (6.80) In fact, this formula gives very similar results to the isothermal formula, Equation (6.79), for heights below one scale-height (i.e., z < z0). For heights above one scale-height, the isothermal formula tends to predict too high a pressure. (See Figure 6.1.) So, in an adiabatic atmosphere, the pressure falls off more quickly with altitude than in an isothermal atmosphere, but this effect is only noticeable at pressures significantly below one atmosphere. In fact, the isothermal formula is a fairly good approximation below altitudes of about 10 kilometers. Because ρ ∝ p/T , the variation of density with height is ρ = ρ0 ( 1 − γ − 1 γ z z0 )1/(γ−1) = ρ0 ( 1 − z 3.5 z0 )2.5 , (6.81) where ρ0 is the density at ground level. Thus, the density falls off more rapidly with altitude than the temperature, but less rapidly than the pressure. Note that an adiabatic atmosphere has a sharp upper boundary. Above height z1 = [γ/(γ−1)] z0, the temperature, pressure, and density are all zero. In other words, there is no atmosphere. For real air, with γ = 1.4, the upper boundary of an adiabatic atmosphere lies at height z1  3.5 z0  29.4 kilometers above sea level. This behavior is quite different to that of an isothermal atmosphere, which has a diffuse upper boundary. In reality, there is no sharp upper boundary to the atmosphere. The adiabatic gas law does not apply above about 20 kilometers (i.e., in the stratosphere) because, at these altitudes, the air is no longer strongly mixed. Thus, in the stratosphere, the pressure falls off exponentially with increasing height. In conclusion, we have demonstrated that the temperature of the lower atmosphere should decrease approximately linearly with increasing height above ground level, while the pressure Classical Thermodynamics 91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p/ p 0 0 1 2 3 4 5 z/z0 Figure 6.1: The solid curve shows the variation of pressure (normalized to the pressure at ground level) with altitude (normalized to the isothermal scale-height at ground level) in an isothermal atmosphere. The dashed curve shows the variation of pressure with altitude in an adiabatic atmo- sphere. should decrease far more rapidly than the temperature, and the density should decrease at some intermediate rate. We have also shown that the lapse rate of the temperature should be about 10◦ C per kilometer in dry air, but somewhat less than this in wet air. In fact, all of these predictions are, more or less, correct. It is amazing that such accurate predictions can be obtained from the two simple laws—p V = constant for an isothermal gas, and p V γ = constant for an adiabatic gas. 6.9 Internal Energy Equation (6.5) can be rearranged to give dE = T dS − p dV. (6.82) This equation shows how the internal energy, E, depends on independent variations of the entropy, S , and the volume, V . If S and V are considered to be the two independent parameters that specify the system then E = E(S ,V). (6.83) It immediately follows that dE = ( ∂E ∂S ) V dS + ( ∂E ∂V ) S dV. (6.84) 92 6.10. Enthalpy Now, Equations (6.82) and (6.84) must be equivalent for all possible values of dS and dV . Hence, we deduce that ( ∂E ∂S ) V = T, (6.85) ( ∂E ∂V ) S = −p. (6.86) We also know that ∂ 2E ∂V ∂S = ∂ 2E ∂S ∂V , (6.87) or ( ∂ ∂V ) S ( ∂E ∂S ) V = ( ∂ ∂S ) V ( ∂E ∂V ) S . (6.88) In fact, this is a necessary condition for dE in Equation (6.84) to be an exact differential. (See Section 4.5.) It follows from Equations (6.85) and (6.86) that( ∂T ∂V ) S = − ( ∂p ∂S ) V . (6.89) 6.10 Enthalpy The analysis in the previous section is based on the premise that S and V are the two independent parameters that specify the system. Suppose, however, that we choose S and p to be the two independent parameters. Because p dV = d(p V) − V dp, (6.90) we can rewrite Equation (6.82) in the form dH = T dS + V dp, (6.91) where H = E + p V (6.92) is termed the enthalpy. The name is derived from the Greek enthalpein, which means to “to warm in.” The analysis now proceeds in an analogous manner to that in the preceding section. First, we write H = H(S , p), (6.93) which implies that dH = ( ∂H ∂S ) p dS + ( ∂H ∂p ) S dp. (6.94) Classical Thermodynamics 95 Thus, it follows from Equations (6.115) and (6.116) that − ( ∂S ∂p ) T = ( ∂V ∂T ) p . (6.119) Equations (6.89), (6.99), (6.109), and (6.119) are known collectively as Maxwell relations. 6.13 General Relation Between Specific Heats Consider a general homogeneous substance (not necessarily a gas) whose volume, V , is the only relevant external parameter. Let us find the general relationship between this substance’s molar specific heat at constant volume, cV , and its molar specific heat at constant pressure, cp. The heat capacity at constant volume is given by CV = ( d̄Q dT ) V = T ( ∂S ∂T ) V . (6.120) Likewise, the heat capacity at constant pressure is written Cp = ( d̄Q dT ) p = T ( ∂S ∂T ) p . (6.121) Experimentally, the parameters that are most easily controlled are the temperature, T , and the pressure, p. Let us consider these as the independent variables. Thus, S = S (T, p), which implies that d̄Q = T dS = T  ( ∂S ∂T ) p dT + ( ∂S ∂p ) T dp  (6.122) in an infinitesimal quasi-static process in which an amount of heat d̄Q is absorbed. It follows from Equation (6.121) that d̄Q = T dS = Cp dT + T ( ∂S ∂p ) T dp. (6.123) Suppose that p = p(T,V). The previous equation can be written d̄Q = T dS = Cp dT + T ( ∂S ∂p ) T [( ∂p ∂T ) V dT + ( ∂p ∂V ) T dV ] . (6.124) At constant volume, dV = 0. Hence, Equation (6.120) gives CV = T ( ∂S ∂T ) V = Cp + T ( ∂S ∂p ) T ( ∂p ∂T ) V . (6.125) This is the general relationship between CV and Cp. Unfortunately, it contains quantities on the right-hand side that are not readily measurable. 96 6.13. General Relation Between Specific Heats Consider (∂S/∂p)T . According to the Maxwell relation (6.119),( ∂S ∂p ) T = − ( ∂V ∂T ) p . (6.126) Now, the quantity αV ≡ 1 V ( ∂V ∂T ) p , (6.127) which is known as the volume coefficient of expansion, is easily measured experimentally. Hence, it is convenient to make the substitution ( ∂S ∂p ) T = −V αV (6.128) in Equation (6.125). Consider the quantity (∂p/∂T )V . Writing V = V(T, p), we obtain dV = ( ∂V ∂T ) p dT + ( ∂V ∂p ) T dp. (6.129) At constant volume, dV = 0, so we obtain( ∂p ∂T ) V = − ( ∂V ∂T ) p / ( ∂V ∂p ) T . (6.130) The (usually positive) quantity κT ≡ − 1 V ( ∂V ∂p ) T , (6.131) which is known as the isothermal compressibility, is easily measured experimentally. Hence, it is convenient to make the substitution ( ∂p ∂T ) V = αV κT (6.132) in Equation (6.125). It follows that Cp −CV = V T α 2 V κT , (6.133) and cp − cV = v T α 2 V κT , (6.134) where v = V/ν is the molar volume. As an example, consider an ideal gas, for which p V = νR T. (6.135) Classical Thermodynamics 97 At constant p, we have p dV = νR dT. (6.136) Hence, ( dV dT ) p = νR p = V T , (6.137) and the expansion coefficient defined in Equation (6.127) becomes αV = 1 T . (6.138) At constant T , we have p dV + V dp = 0. (6.139) Hence, ( ∂V ∂p ) T = −V p , (6.140) and the compressibility defined in Equation (6.131) becomes κT = 1 p . (6.141) Finally, the molar volume of an ideal gas is v = V ν = R T p . (6.142) Hence, Equations (6.134), (6.138), (6.141), and (6.142) yield cp − cV = R, (6.143) which is identical to Equation (6.39). 6.14 Free Expansion of Gas Consider a rigid container that is thermally insulated. The container is divided into two compart- ments separated by a valve that is initially closed. One compartment, of volume V1, contains the gas under investigation. The other compartment is empty. The initial temperature of the system is T1. The valve is now opened, and the gas is free to expand so as to fill the entire container, whose volume is V2. What is the temperature, T2, of the gas after the final equilibrium state has been reached? Because the system consisting of the gas and the container is adiabatically insulated, no heat flows into the system: that is, Q = 0. (6.144)