# Physics H133: Problem Set #15

Here are some hints, suggestions, and comments on the problem set.

## Two-Minute Problems

Remember to give a good explanation, no longer than two sentences.

1. T3S.1: It seems like your friend heard about Eq. (T1.6), but didn't properly read the caveats about when not to use it.
2. T4T.6: This is a simple plug and chug problem if you know which equation to use. Which is it?!
3. T4T.7: Watch out: read all offered statements carefully! Note the difference between "micro" and "macro".

## Chapter T Problems

• T4B.4: Plug and chug!
• T4S.2: (a) For 2 atoms, how many springs (oscillators) does the Einstein solid have? List all the possible oscillator excitations with two excitation quanta total. (b) Plug and chug.
• T4S.4: (a) Form the macropartition table, listing all the possibilities of distributing 9 excitation quanta over the two subsystems. When done, compute the probabilities for each macropartition (they should be normalized tp 100%=1.0). Should you expect the energy to be equally divided between the solids? If yes, why? If not, why not?
• T4S.8: (a) Show that this follows from the problem description up to this point. (b) Remember to make the correction u->q as described in the Erratum page for this unit. Then follow the hint and just keep going until you understand the pattern (by induction). (c) Draw one or two examples of patterns and how you can rearrange them in a way that the pattern remains the same but it would correspond to a different drawing. To see the point, the patterns should have more than one marble in at least one of the "drawers" separated by matchsticks. (This corresponds to more than one quantum of excitation energy for at least one of the oscillators.) For example, you can draw the two balls in the 4th drawer of the shown pattern in 2 different sequences, either the left ball first or the right ball first. And you can interchange the order of the drawers without changing the pattern (1 triply occupied drawer, one doubly occupied drawer, three singly occupied drawers, and one empty drawer in the figure shown in the statement). [In real life you should replace "occupied drawer" by "excited oscillator".] We should count only drawings leading to different patterns. Follow a similar procedure as in (b) to count the number of possibilities to rearrange the q marbles and the M-1 matchsticks. (d) How is the number M of oscillators in the Einstein solid related to the number N of molecules?
• T4R.1: This is a lot like T4S.8. Part a) is very simple; don't overthink! b) Here we want to know how many combinations there are of N items of which there are only two kinds. So imagine drawing N marbles from a bag, where n are black, representing excited states, and N-n are white, representing ground states. The sequence will define a microstate. Counting all possibilities (N for the first, times N-1 for the second, and so on) overcounts, so we need to divide by the ways to arrange n black marbles and then N-n white marbles ==> the same type calculation! c) Just make the table from the formula in b). The partition 2:6 should have Omega_A = 190, Omega_B = 38760, Omega_AB = 736,440, and 9.6% probability.
• T3R.1: If the air moves quickly, there is no time for heat transfer. Assume an ideal gas. Should it be monatomic or diatomic, considering what air is mostly made of?