Thermodynamics and an Introduction to Thermostatistics 5, Exercises of Chemistry

Download Thermodynamics and an Introduction to Thermostatistics 5 and more Exercises Chemistry in PDF only on Docsity! Group Problems #1 Reciation 3.091 Fall 2003 From McQuarrie: Highly Advised 3­18 3­24 3­26 4­8 4­12 Advised 3­4 3­10 3­12 4­2 4­6 5­4 5­9 Problem 1 A system consists of N non­interacting atoms. Each atom may be in two states, a low energy state with energy, E = 0 and an ‘excited’ state, with energy E. a) How many atoms are in the excited state? b) What is the total energy U of this system as a function of N , E, k (the Boltzmann constant) and T? Problem 2 Consider a system of N distinguishable non­interacting spins in a magnetic field H. Each spin has a magnetic moment of size µ, and each can point either parallel or anti­parallel to the field. The magnetic moment is given by niµ where ni = +/ − 1. Note that since the system is made of non­interacting particles, the total energy of the system does not depend on the arrangements of the spins, i.e. the energy is constant. � � � � � � � � (a) Determine the internal energy of this system as a function of β, H, and N by employing an ensemble characterized by these variables. (b) Determine the entropy of this system as a function of β, H, and N. (c) Determine the behavior of the energy and entropy for this system as T 0. → Problem 3 (a) For the system described in the previous problem, derive the average total magnetization, � N �M = � µni � i=1 as a function of β, H , and N. (b) Similarly determine � (δM)2 � , where δM = M − M and compare your result with the susceptibility � ∂ �M � ∂H β,N (c) Derive the behavior of �M and �(δM)2� in the limit T 0.� → (Note: M denotes the average, so M = M ) Problem 4 Consider the system studied in Problems 2 and 3 above. Use an ensemble in which the total magnetization is fixed, and determine the magnetic field over temperature, βH, as a function of the natural variables for that ensemble. Show that in the limit of large N , the result obtained in this way is equivlent to that obtained in Problem 3. Problem 5 Consider a one­component gas of non­interacting classical structureless particles of mass m at a temperature T. (a) Calculate exactly the grand canonical partition function, Ξ, for this system as a function of volume, V, temperature, and chemical potential, µ. Your result should look like Ξ = exp (zV ) where z is a function of T and µ. (b) From the result of part (a), determine the pressure, p, as a function of T and the average particle density, ρ.