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The first two problems are from textbook “Thermal Physics” by Charles Kittel and
Herbert Kroemer, 2nd edition.
1 Chapter 2 Problem 5 (12 points)
In this problem, you will need to use Eq. (15) and (17) in Chapter 2 of the textbook. You
will also use the formula for Gaussian integral in part (b).
2 Chapter 2 Problem 3 (8 points)
Read the section on page 24 and 25 of the textbook: Example: Multiplicity for harmonic
oscillators, which will help you finish this problem.
3 Entropy of mixing (10 points)
Consider two crystals. One is made from atom A, the other one is made from atom B.
Initially the two systems are separated. Let us bring the two together, and A and B atoms
now can reside on any lattice sites in the whole system, see the figure below. Assume the
total number of atoms is N , the percentage of A atoms is x, and the percentage of B atoms
is 1− x.
Figure 1: Mixing of two types of atoms
1
(a) What is the multiplicity function of the mixed system? Assume N is large, use the
Stirling formula, logN ! = N logN−N , to find the entropy difference between the final state
and initial state, show it is given by
∆σ = −N [x log x+ (1− x) log(1− x)]. (1)
Does the entropy increase or decrease after mixing?
(b) Now consider a similar situation, but now we mix 3 different kinds of atoms, A, B
and C. Total number of atoms is N , and percentages of A, B, and C atoms are x1, x2 and x3,
respectively (with the constrain that x1 + x2 + x3 = 1). Find the entropy difference between
the final state and initial state in terms of N, x1, x2, and x3.
(c) Do you find any pattern in these exercises? Write down the general formula for
entropy of mixing for the case of m different species of atoms.