1. a fair coin. Consider the sequences of N = 4 coin flips HHHH, HHHT , HHTT , HTTT , and TTTT . These sequences have distinct com- positions with nH = 0, 1, . . . , 4 heads and nT = 0, 1, . . . , 4 tails. Make a table for (a)-(c): (a) What is the probability of each sequence? (b) How many per- mutations are there for each composition? (c) What is the total probability for all sequences with each composition?
Now, let’s again suppose the coin was flipped four times, but instead (d) write all possible results of the four flips and group them by composition (i.e.., number of heads, H, and tails, T ). Let’s define a ratio, A: the number of these configurations (or microstates) associated with a particular composition divided by the number of configurations with a 1:1 mixture of H:T . (d) Give the value of A for each composition, and (e) sketch (approximate) histograms that you would would expect for A as the number of coin flips is increased. What do you expect after doing 1 mol of flips?
2. Consider an experiment in which a gas expands. Let’s use the lattice model shown in Figure 1. Initially the N gas particles occupy a volume with M sites. Assume particles can simultaneously occupy any given site and that the particles are distinguishable.
(i) What is the multiplicity of the initial state? Explain your answer. When the wall is removed, what is the multiplicity?
(ii) What is the entropy of the initial state, when the wall is removed, and the entropy change? Do these answers match your intuition? Justify your answers.
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Figure 1: (a) An example of a possible microstate in the initial configuration where the wall keeps all the particles to the left, and (b) a possible microstate in the final configuration after the wall has been removed.
3. Let’s consider a similar experiment (Figure 2) where we have two boxes, each with M sites and N indistinguishable particles. We put the boxes/lattices together with a wall between them as in the first experiment. But, in this case, we have N particles on both sides of the chamber.
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Figure 2: An experiment where (a) initially equal number of particles are on the two halves of a chamber with a wall between them. (b) The particles mix spontaneously when the wall is removed.
(i) Using the multiplicity from Problem 2, what is the multiplicity of the composite system before and after removing the wall? Ex- plain your answer. What is the entropy before and after? What is the entropy change?
(ii) Let’s find the entropy change another way: Use the fact that entropy is extensive to get the entropy of the final state and the entropy change when the wall has been removed. Justify your answer, which should be different from problem 3 part (i).
(iii) Given the number N of identical particles on each side of the wall before and after the experiment, what do you expect the entropy change to be? Explain the contradiction between the result from (ii) and intuition.
(iv) We can resolve this paradox: Recalculate each entropy and the entropy change using the fact that the particles are indistin- guishable when calculating the multiplicity. Show that ∆S agrees with your intuition.
4. Now we can refine the model in Problems 2 and 3 by adding steric inter- actions between the particles. That is, particles exclude volume and cannot occupy the same site. The volume of the system is V = a3M and the in- ternal energy does not depend on the configuration of particles. The ratio φ = N/M is the concentration of particles.
(i) Evaluate the entropy S of the system as a function of N and
M . Apply Stirling’s approximation and rewrite this entropy as a function of φ and M . Keep only the terms of the order of M (the largest ones in the case M » 1; that is, discard terms that are of order ln M ).
(ii) Taylor expand the logarithms up to terms O(φ3) and use
V = a3M to write S(V ).
(iii) Evaluate the free energy F of the system for given N and M
at temperature T .
(iv) From (iii) find the pressure of this lattice gas. Comment on the leading term (i.e., Do you recognize the largest contribution?).
