Thermodynamics of Ideal Gases: Understanding State Functions and Energy Changes - Prof. Mi, Study notes of Physical Chemistry

Download Thermodynamics of Ideal Gases: Understanding State Functions and Energy Changes - Prof. Mi and more Study notes Physical Chemistry in PDF only on Docsity! Chem 476 Spring 2009 Statistical Thermodynamics and Kinetics Lecture Notes 1 M. D. Barnes Overview of Concepts and Constructs from (nonstatistical) Thermodynamics ü Introduction Thermodynamics - the physics of ensembles of particles - is a strange and funny thing, and our experience this spring semester will be quite different (in terms of concepts and mathematical procedures) than our experience with quantum chemistry last fall. On the surface, many aspects of thermodynamics seem intuitive and simple: A child understands that when you place a 'hot thing' in "contact" with a 'cold thing', the cold thing becomes warmer, and the hot thing gets cooler. We intuitively grasp ideas of direction of spontaneous physical change. For example, we know that a gas sample doesn't spontaneously compress, and a rubber ball placed on a table won't spontaneously start bouncing if we heat up the table. Thermodynamics deals with macro- scopic things we see, perceive, and experience every day, but quantifying the behavior of ensemble systems - and understanding the connection to microscopic properties of the material that makes up the ensemble is not a trivial task at all. The apparent disconnection between Thermodynamics and the quantum mechanics of atoms and molecules derives in part because of its historical development preceding the microscopic theory of electrons, atoms, and molecules. As we will see, much of the "fine-grained" quantum behavior of atoms or molecules is invisible in the limit when a large number of quantum states are populated; from our experience last fall, this should strike you as very strange: If the ensemble is made up from molecules with unique quantum properties (distinct electronic, translational, vibrational, and rotational states), how come the changes in thermo- dynamic properties don't seem to care about that? This is a very strange thing - and as we go through the course, we will unders- tand why this happens. As a result, the basic constructs of thermodynamics ( Internal/Ensemble Energy, U; Enthalpy, H; the Entropy, S; Helmholtz Free Energy, A; and the Gibbs Free Energy, G) require no reference to the identity (or properties) of the atomic/molecular species that make up the ensemble. NonStatistical Thermodynamics expresses differential changes in these thermodynamic "State Func- tions" in terms of any two "State Variables" (# of particles, N, Temperature, T, Volume, V, or Pressure, P). Here, I (very) briefly summarize the basics of nonstatistical thermodynamics, that will serve as a framework for our discussion of the statistical mechanics of ensembles. As we will see, the energy configuration of the ensemble (that is, finding the probability that an atom or molecule occupies a particular quantum state with energy ei ) provides a complete description of the thermody- namic state functions (U, H, S, A, G), not just the differential or integrated changes offered by nonstatistical thermodynamics. Thus, the correspondence between the statistical and nonstatistical models is an important one, thus I offer a brief (and necessar- ily somewhat superficial) overview of the basic concepts and constructs of nonstatistical thermodynamics here. ü The "system", state variables and equations of state ü System and State Variables The state of our thermodynamic system (closed collection of particles) is defined by the following State Variables: N Ø # particles in the system T Ø Absolute temperature V Ø Volume (m3) P Ø Pressure (Force/area; kg m-1 s-2) ü equations of state and Ideal Gas Law The equation of state of our thermodynamic system relates one state variable in terms of the other 3 (that is, we can indepen- dently specify 3 of the 4 state variables). eg: P = f (N, T, V) or V = f ( N, T, P), etc. If we know the equation of state (deduced empirically for ideal gases), then we know how one of the state variables changes as we change the other 3. The IDEAL GAS LAW , derived empirically in the late 1700's, provides a simple equation of state to use as a basis for develop- ing relations for thermodynamic state functions (U, H, S, A, G). It reads: P = N k TV = n R T V where k is Boltzmann's constant (k = 1.38 x 10-23 J/K), and the other (perhaps more familiar form) n is the number of moles, and R is the Gas Constant (R= 8.314 J/(mol K) - it's just k multiplied by Avogadro's number. There are two basic assumptions of the ideal gas law: (1) The particles in the ensemble DO NOT interact with each other (2) The particles are considered as point masses (they have no volume) There are lots of different equations of state that have been developed to account more accurately for particle interactions and finite molecular volume. We'll deal with some of these later in the homework. One of the most interesting things that we will do in this semester (in my view) is to show how equations of state for any species - and at any level of detail in terms of molecular interactions/volume - can be derived from statistical thermodynamics. 2 Chem476_Spr09_NSthermo_rev.01.nb Ukinetic = N k T 2 + k T 2 + k T 2 = 3 2 N k T If there are no other degrees of freedom (e.g. electronic, vibrational, rotational) available to the particles, then Ukinetic = U, the total ensemble energy of the system. ü The First Law of Thermodynamics The last result tells us that for a mono-atomic ideal gas (ignoring electronic energy), the ensemble energy U Ø f (N, T). The energy scales linearly with number of particles, and linearly with the temperature. Thus, if we keep N fixed and we keep the volume fixed, we can (obviously) increase or decrease the ensemble energy by transfer- ring energy in the form of HEAT to or from the system. We would express this change in energy in a differential form as: dU Hconstant N, VL = d q where d q represents a differential amount of heat transferred to/from (+/-) the system. The distinction in notation between 'the bar' and 'not the bar' is important. Differential quantities expressed without the bar are referred to as exact differentials, meaning that the integral of such an exact diffrerential between an initital and final value is independent of the path taken to get from the initial value to the final value. A familiar example is potential energy in a 1D gravitational problem. The gravitational potential energy of an object in such a problem is V(h) = m g h; m is the mass, g is the acceleration due to gravity, and h is ther vertical distance. The above picture shows a fellow trying to get to the top of a tower. He can get there by a variety of different ways (e.g., a ladder, or very long spiral staircase). His change in potential energy (DV = mg (h 1 - h 0) is the same whether he takes the ladder or the stairs; BUT the amount of physical effort (work) that he has to expend to get there clearly depends on the path. Exact differentials have the useful property (among others...) ‡ U1 U2 „U = U2 - U1 = DU However, inexact differentials as indicated with the bar (like d q ), must be integrated along a specific path, and the integrated quantities are therefore path-dependent. ‡ path d q = q where "q" is the sum total (integral over path) of the heat transferred. Of course, we can change the ensemble energy of the system by using part of energy stored in the gas sample to do work for us (internal combustion engines do this all the time!). So, we could imagine thermally isolating our system so no heat can be transferred in our out of the system (such a thermodynamic system is called adiabatic), and we would write the differential change in U as Chem476_Spr09_NSthermo_rev.01.nb 5 Of course, we can change the ensemble energy of the system by using part of energy stored in the gas sample to do work for us (internal combustion engines do this all the time!). So, we could imagine thermally isolating our system so no heat can be transferred in our out of the system (such a thermodynamic system is called adiabatic), and we would write the differential change in U as dU Iconstant N, d q Ø 0M = d w where d w is the inexact differential corresponding to the mechanical work done by/on the system. If our system does work by expansion under adiabatic conditions (no heat transferred), the ensemble energy must decrease (we lost energy in order to perform the work). Thus, the sign convention for expansion/compression work - otherwise known as "P·V" work - is as follows: dU Iconstant N, d q Ø 0M = - p „ V Combining the two expressions for dU in terms of d w and d q, we arrive at the First Law of Thermodynamics in differential form: dU Hconstant NL = d q - p „ V Thus, the first law tells us how to compute changes in ensemble energy by adding/subtracting heat (thermal energy) and/or expansion/compression work. ü Entropy; The 2nd Law of Thermodynamics, Direction of Spontaneous Change ü nonstatistical mechanical definition of entropy and direction of spontaneous change An interesting issue arises directly from considering the functional form of the ensemble energy we derived for a monatomic ideal gas (three modes of translation only) and the first law. Consider the two systems shown in the picture below: The two cylinders contain the same number of particles at the same temperature T, but with different volumes VA and VB, thus the pressures are different ( PA and PB). If we assume the ideal gas law as the equation of state, the two systems have the same energy (recall, U is only a function of N and T for an ideal gas). But, clearly these two systems are different, thus ensemble energy, U, is apparently not a unique way of defining the thermody- namic state of the system (there are an infinite number of different ideal gas systems we can imagine that have the same energy!). The question is then, how do we describe/quantify this difference and why is it important? Since the two systems have the same energy, there is no energetic driving force to inter-convert from ( PA ó PB). Intuitively, we know that there is a direction of spontaneous change: Gas samples don't spontaneously (i.e. without applying an external force, thereby doing work) compress - they expand. So, if we consider the two systems in the figure, the system on the left (VA, PA) can spontaneously expand to PB at constant T - provided that it is in contact with a thermal heat reservoir that provides a quantity of heat, q, necessary to compensate for the work done by the system in expanding from PA,VA Ø PB, VB. 6 Chem476_Spr09_NSthermo_rev.01.nb Since the two systems have the same energy, there is no energetic driving force to inter-convert from ( PA ó PB). Intuitively, we know that there is a direction of spontaneous change: Gas samples don't spontaneously (i.e. without applying an external force, thereby doing work) compress - they expand. So, if we consider the two systems in the figure, the system on the left (VA, PA) can spontaneously expand to PB at constant T - provided that it is in contact with a thermal heat reservoir that provides a quantity of heat, q, necessary to compensate for the work done by the system in expanding from PA,VA Ø PB, VB. ü Microscopic reversibility But, that amount of heat/work involved in the transformation depends on the path! We consider two possible paths for the transformation PA,VA Ø PB, VB (at constant N and T): reversible and irreversible. The distinction between the two words is subtle: a reversible change implies an infinitesimal change in one (or more) state variables so that the system could be restored exactly to it's original state (not just the original values of the state variables!). First, we consider an example of an irreversible transformation: Imagine that the system defined by the state variables, N, T, PA,VA is abruptly expanded against and external pressure PB. The work that is done by the system is trivial to calculate: „wirrev = - Pext „V wirrev = - PB ‡ VA VB „V = - PB HVB - VAL If VB - VA > 0, the quantity of work is negative (Internal energy decreases, because the system DID work on surroundings), thus the quantity of heat that must be absorbed by the system to maintain constant temperature is qirrev = - wirrev = + PB HVB - VAL or, if we wish qirrev = N k T VB HVB - VAL Now consider a reversible transformation. Microscopic reversibility implies that the transformation from N, T, PA, VAØN, T, PB, VB occurs via an infinite number of infinitesimally small external pressure steps, where the external pressure is equal to the internal pressure: Pext = N k T V wrev = - N k T ‡ VA VB „V V = - N k T LogB VB VA F Clearly the amount of work done (and heat transferred) is different for the reversible and irreversible transformations. (Prove to yourself!) The definition of Entropy, S, from nonstatistical thermodynamics is (in differential form) „S = „qrev T which, interestingly, is an exact differential (French Engineer Sadi Carnot invented this construct in the early 1800's. For proof that the entropy is an exact differential - and therefore a state function - see section in text on the Carnot Cycle. ) As you learned in general chemistry, the direction of spontaneous change is associated with a POSITIVE change in entropy Chem476_Spr09_NSthermo_rev.01.nb 7 In differential form, it becomes „G = - S „ T + V „ p ü Maxwell' s relations and partial derivatives The four differential expressions for U, H, A, and G, allow us to calculate changes in any thermodynamic quantity as we change T, V, p, S (Constant N). In summary, they are (again) „ U = T „ S - p „ V „ H = T „ S + V „ p „ A = -S „ T - p „ V „ G = - S „ T + V „ p Each of the above differential expressions allows to relate partial derivatives of a particular variable „ U = T „ S - p „ V = ∂U ∂S V „ S + ∂U ∂V S „ V The latter part is just a general expression for the differential of a function of 2 variables (in this case S and T). Thus T = ∂U ∂S V and p = - ∂U ∂V S likewise, from the differential expression for „ H T = ∂H ∂S p and V = ∂H ∂p S likewise, from the differential expression for „ A - S = ∂A ∂T V and - p = ∂A ∂V T and finally, from the differential expression for „ G - S = ∂G ∂T p and V = ∂G ∂p T Because each of the 4 differentials („ U, „ H, „ A, and „ G) are exact, the following cross-partial derivatives are equal (as a property of exact differentials) Consider a generic function f that depends on two variables x, y. The differential for f (x,y) is „ f Hx, yL = ∂f ∂x y „ x + ∂f ∂y x „ y if „ f is exact, then the following equivalence holds: 10 Chem476_Spr09_NSthermo_rev.01.nb ∂ ∂y ∂f ∂x y x = ∂ ∂x ∂f ∂y x y From the 4 differential expressions for U, H, A, and G, we get a set of 4 equivalent partial derivatives: „ U = T „ S - p „ V and ∂U ∂S V „ S + ∂U ∂V S „ V ∂U ∂S V = T and ∂U ∂V S = - p therefore ∂T ∂V S = - ∂p ∂S V this constitutes one of 4 different Maxwell relations that allow us to express one (potentially obscure) partial derivative in terms of another (hopefully less obscure :-) partial derivative. Here they are (you should be able to derive these on your own): From dH „ H = T „ S + V „ p ∂T ∂p S = ∂V ∂S p From dA „ A = -S „ T - p „ V ∂S ∂V T = ∂p ∂T V and from dG „ G = -S „ T + V „ p - ∂S ∂p T = ∂V ∂T p the latter two are the most useful since they relate to partial derivatives of state variables in terms of another via the choice of equation of state. ü Chemical Potential and Chemical Equilibrium The final thing we need to consider here, is how the various thermodynamic state functions vary with changing number of particles (holding all other state variables constant). This of course is important in many chemical processes such as mixing, reaction, etc., and this quantity is called the chemical potential, µ. The chemical potential of species "x" is defined as Chem476_Spr09_NSthermo_rev.01.nb 11 µ = ∂ Hany state function, U, H, A, GL ∂Nx V, T, P, all other particles Ny ≠ Nx That is, the chemical potential of species "x" can be expressed as the derivative of any state function - U, H, A, or G - with respect to the number of particles of "x" holding all other state variables constant. The condition for chemical equilibrium - a critically important construct in chemistry - between species "x" and "y" - is that their chemical potentials are equal. i.e.: µx = µy The idea is that chemical potential plays a role analogous to electric potential, thermal potential, gravitational potential, etc. in that the potential difference defines the direction of (charge, heat, mass) flow. If we place two systems in contact with each other so that particles of x and y can move between the two systems, the chemical potential difference between the two systems tells us the direction that the particles will move in response to the chemical potential gradient. 12 Chem476_Spr09_NSthermo_rev.01.nb