# quiz-4-03-25-10-answers - Quiz 4 for Physics 176 Solutions...

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Quiz 4 for Physics 176: Solutions Professor Greenside Thursday, March 26, 2010 The following equations may be useful: dU = TdS - PdV + μ dN, p s = e - βE s Z , Z = X s e - βE s , E = X s p s E s = - ln( Z ) ∂β . (1) D ( v ) = 4 π m 2 πkT · 3 / 2 v 2 e - mv 2 / (2 kT ) . (2) As usual, the following solutions and discussion are much more detailed than what were necessary to obtain full credit on any given problem. But the details should help you to understand the thermal physics and to solve related problems. 1. (10 points) Consider a small system S in thermodynamic equilibrium with a large reservoir that has a constant temperature T and constant chemical potential μ . (The latter condition corresponds to a constant pressure if the reservoir is an ideal gas.) The small system and the reservoir consist of the same kind of particles, and the small system can exchange particles but not energy with the reservoir, so the number of particles in the small system can vary with time. Using an argument similar to how we derived the Boltzmann factor in class, and clearly stating your assumptions and approximations, derive an expression for the probability for the small system to be in a particular state s that has N ( s ) particles . Your expression will look like an “exponential of something” over a quantity that looks like a partition function. Answer: p ( N = N s ) = e βμN s s e βμN s . (3) To derive this, we can adapt the argument I gave in class for the derivation of the Boltzmann factor. (The derivation I gave makes various assumptions more clear than Schroeder’s argument in his book, although the two approaches are equivalent.) A derivation can also be found on pages 257-258 in Section 7.1 of Schroeder, for the more general case in which a small system exchange particles and energy with a reservoir. (We will discuss Section 7.1 this coming week in lecture.) So assume we have a small system in thermodynamic equilibrium with a large reservoir that has a constant temperature T and constant chemical potential μ . The system and reservoir together form a closed system which means that the total number of particle N + N R = N 0 is conserved: whenever the number of the particles in the system increases, the number of particles in the reservoir decreases, and this typically decreases the multiplicity of the reservoir rapidly since the multiplicity Ω( N ) is generally a rapidly increasing function of N as N increases (see Eq. (2.21) on page 64 and Eq. (2.40) on page 71 of Schroeder). Because the reservoir and system form an isolated equilibrium system, the fundamental postulate of statistical physics (see page 57 of Schroeder) implies that all accessible microstates of the reservoir and system must be equally likely. (This is one of several points that you had to mention to get full credit.) This implies that the probability P ( N = N s ) for the system to be in a single particular state with the number of particles equal to N s must be proportional to the number of accessible microstates of the entire isolated system that have N s particles in the small system. But if we are asking for the

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