II. Microstates and Fntropy In the figure at right is shown tro systems, iabeled \( A \) and B, that each can hold thermal epery. A and B each love 8 binary degrees of freedom (Le., each DoF can either be empty or hold a packet of energy). There are cight available packets of encrgy. The figure shows one way that the energy packrts can be distributed. A) How many wayx, \( \mathrm{W} \), are there to discribute all 8 packets to system A? If we knew the energy was distributed with 8 packets in A and 0 packets in B, what would be the entropy, \( S \), of that state? Explain. B) Suppose we started out with 6 packets in \( A \) and 2 packets in \( B \), as shown in the figure. 1. How many ways are there to put those 6 packets in \( A\left(W_{A}\right) \) ? 2. How many ways are there to put those 2 packets in \( B\left(W_{B}\right) \) ? 3. What are the entropies (in terms of \( k_{B} \) ) for the two subsystems ( \( S_{A} \) and \( S_{B} \) )? C) What are the total number of ways, \( W_{A B} \), that the full system can be arranged as described in part (B)? And what is the total entropy, \( S_{A B} \), of the system? Explain. (Remember, entropy is an ertensive property!) D) Suppose we start out in a configuration with 6 total packets of energy, with 4 in subsystem A and 2 in subsystem B. If one packet of energy moves from A to B (so that we ead up with 3 packets in each subaystem), calculate (in terms of \( \mathrm{k}_{B} \) ) 1. the change in entropy of suloystem \( A, \Delta S_{A} \) 2. the change in entropy of subsystem \( \mathrm{B}, \Delta S_{B} \) 3. the change in entropy of the whole system, \( \Delta S_{A B} \) E) Repeat part (D) for the case of a packet moving from B to \( A \) (instead of \( A \) to \( B \) ), ending with 5 packets in \( A \) and 1 in \( B \). F) Discuss the results of parts (D) and (B). Consider things like possibility and probability of occurance.
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Call placement of a particle in box A "heads" and placement in box B "tails." Given one particle, there are two ways of arranging it, H or T. For two particles, there are four ways of arranging them, {HH, HT, TH, TT}. We can treat the microstates by considering each particle in order. For example, {H T H H} means the first particle is in box A, the second in box B, the third in box A, and the fourth in box A. (a) List and count the ways of arranging three particles. Now consider four particles. What is the general formula for the number of arrangements versus the number of particles? (ANS. 2N) (b) How many arrangements correspond to having two particles in box A and one in box B? What is the probability of {2H, 1T}? (ANS. 3/8) (c) How many arrangements correspond to {2H, 2T}? {3H, 2T}? {4H, 2T}? {3H, 3T}? (ANS. N!/[(N – m)!m!]) (d) List the macrostates and corresponding number of microstates for an eight-particle, two-box system. What portion of all microstates are parts of either 5:3, 4:4, or 3:5 macrostates? (ANS. 71%) (e) What is the change of entropy in going from a 5:3 macrostate to a 4:4 macrostate? (ANS. 3.08E-24 J/K) (f) Use Stirling's approximation to estimate the change of entropy in going from a distribution of 50.1% of 6.022E23 in box A to a distribution of 50.001%, and from 50.001% to 50.000%. (ANS. 1.2E18 J/K)
Sri K.
Entropy is defined in terms of microstates. We often illustrate how this works using a small number of discrete objects like coins or dice. When rolling three dice, 1+2+6 and 1+6+2 and 3+2+4 are different microstates, even though each adds up to 9. There is only one microstate that adds to 3, but there are 25 that add to 9. That is why you are much more likely to roll a 9 than a 3. With coin-like objects that have two choices, counting the number of possible microstates is illustrated by Pascal's triangle. The branch of mathematics that deals with counting such things is called combinatorics. In any system of more than a handful of atoms, the number of microstates becomes larger than the number of protons in the universe. So we typically calculate changes in entropy, with an eye toward whether it is positive or negative. But in this problem, we'll count microstates. You flip 6 coins. How many microstates are there? 64 What is the number of heads that has the highest number of microstates? If two or more have the same number, give the one with the lowest number of heads. 3 How many microstates are there that have exactly one head? 6 How many times more likely is it that you get the most likely number of heads than that you get one head?
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Three isolated boxes A, B and C each have single-particle energy levels 0, ̵, 2̵, 3̵, 4̵, . . .. Boxes A and B each contain three particles both with total energy 3̵, whilst box C contains two particles with total energy 2̵. The particles are distinguishable and do not interact with each other. (a) Tabulate each distribution and determine the total number of microstates ̵A, ̵B and ̵C accessible to each box separately and show that the total number of mi- crostates accessible to them jointly is, ̵ = 300. (b) Boxes B and C are now put into thermal contact so that energy (but not particles) can be exchanged between them. Show, by creating a table as above, that the new number of microstates accessible to the system is, ̵* = 1260. (c) Calculate the change in entropy of the system resulting from the thermal contact. (d) Box A is now put in thermal contact with B and C. What is the new average energy of each box?
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