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PHYS 258  Thermal Physics:
Assignment 7
Due Monday March 7
Student ID:
Full
Name:
Score:_______/16___
1. [5] Temperature and Heat Capacity of an Einstein Solid.
In class we found that in the high temperature limit of an Einstein solid, the thermal
energy is related to the temperature via
U
=
NkT
, (where N is the number of oscillators) .
What the corresponding result is in the low temperature limit. In Assignment 5, Problem
1, you found that, when
q << N
, the multiplicity of a large Einstein solid is given by
q
q
eN
q
N
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≈
Ω
)
,
(
(a) [2] Using this result, find a formula for the temperature of an Einstein solid as a
function of its energy. Use
U
=
q
ε
, where
is one quantum of energy for an oscillator.
(b) [2] Using your result from part (a), solve for the energy,
U
, as a function of the
temperature. What is the limit of
U
as
T
→
0? What is the average number of quanta of
energy in each oscillator (
q/N
) at temperature
T
? Observe that under the assumptions
stated, this number is always much less than 1 and so
kT
<<
.
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View Full Document(c) [1] Using your result from part (b), calculate the heat capacity of an Einstein solid in
the low temperature limit. Sketch
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 Winter '11
 YuriLeonenko
 Physics, Energy, Heat

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