Physics H133: Problem Set #15
Here are some hints, suggestions, and comments on the problem set.
Remember to give a good explanation, no longer than
- 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.
- T4T.6: This is a simple plug and chug problem if you know which
equation to use. Which is it?!
- 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,
- 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
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
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?
Your comments and
suggestions are appreciated.
Physics H133: Hints for Problem Set 15.
Last modified: 05:05 pm, May 05, 2012.