PHYS 333
Winter 2009
Assignment 3
Due: 5:00 pm Friday January 23
Most of these problems are from Schroeder’s book.
1. Schroeder, problems 2.18/22/30
(a) Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any
large values of
N
and
q
, is approximately
Ω(
N, q
)
'
±
q
+
N
q
²
q
±
q
+
N
N
²
N
p
2
πq
(
q
+
N
)
/N
The square root in the denominator is merely large, and can often be neglected. However,
it what follows, it is needed.
(b)
2.22.
Consider two Einstein solids, each with
N
oscillators, in thermal contact with each
other. Suppose that the total number of energy units in the combined system is exactly
2
N
. How many diﬀerent macrostates (that is, possible values for the total energy in the
ﬁrst solid) are there for this combined system?
(c) Use the result of part (a)
(2.18)
to ﬁnd an approximate expression for the total number
of microstates for the combined system. (Hint: treat the combined system as a single
Einstein solid. Do
not
throw away factors of “large” numbers, since you will eventually
be dividing two “very large” numbers that are nearly equal.)
Ans:
2
4
N
/
√
8
πN
.
(d) The most likely macrostate for this system is, of course, the one in which the energy
is shared equally between the two solids. Use the result of part (a)
(2.18)
to ﬁnd an
approximate expression for the multiplicity of this macrostate.
Ans:
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 Winter '09
 Harris
 Physics, Thermodynamics, Energy, Entropy, Black hole

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