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482 13 STATISTICAL THERMODYNAMICS
energy levels will be written εJ and the degeneracies gJ ; derivatives with respect
to β will be assumed to be at constant V .
First, an expression for CV is developed.
U = −N 1
q
dq
dβ
= −N 1
q
d
dβ
∑
J
gJe−βε J
= N 1
q
∑
J
gJεJe−βε J
Noting that d/dT = −kβ2(d/dβ)
CV = dU
dT
= −kβ2
dU
dβ
= −Nkβ2
d
dβ
⎡⎢⎢⎢⎣
1
q
∑
J
gJεJe−βε J
⎤⎥⎥⎥⎦
= −Nkβ2
⎡⎢⎢⎢⎣
−1
q 2
dq
dβ
∑
J
gJεJe−βε J − 1
q
∑
J
gJε2J e
−βε J
⎤⎥⎥⎥⎦
= −Nkβ2
⎡⎢⎢⎢⎢⎣
1
q 2
⎛
⎝∑J′
gJ′ εJ′e−βε J′
⎞
⎠
⎛
⎝∑J
gJεJe−βε J⎞
⎠
− 1
q
∑
J
gJε2J e
−βε J
⎤⎥⎥⎥⎥⎦
�e numerator and denominator of the �nal term in the bracket are both mul-
tiplied by q , and then a factor of 1/q 2 is taken outside the bracket to give
CV = −Nkβ2
q 2
⎡⎢⎢⎢⎢⎣
⎛
⎝∑J′
gJ′ εJ′e−βε J′
⎞
⎠
⎛
⎝∑J
gJεJe−βε J⎞
⎠
− q∑
J
gJε2J e
−βε J
⎤⎥⎥⎥⎥⎦
= −Nkβ2
q 2
⎡⎢⎢⎢⎢⎣
⎛
⎝∑J′
gJ′ εJ′e−βε J′
⎞
⎠
⎛
⎝∑J
gJεJe−βε J⎞
⎠
−
⎛
⎝∑J′
gJ′e−βε J′
⎞
⎠
⎛
⎝∑J
gJε2J e
−βε J⎞
⎠
⎤⎥⎥⎥⎥⎦
�e product of the sums are next rewritten as double sums
CV = −Nkβ2
q 2
⎡⎢⎢⎢⎣
∑
J , J′
gJ gJ′ εJεJ′e−β(ε J+ε J′) −∑
J , J′
gJ gJ′ ε2J e
−β(ε J+ε J′)
⎤⎥⎥⎥⎦
Taking a hint from the �nal result, consider the double sum
∑
J , J′
(εJ − εJ′)2gJ gJ′e−β(ε J+ε J′)
=∑
J , J′
ε2J gJ gJ′e
−β(ε J+ε J′) +∑
J , J′
ε2J′ gJ gJ′e
−β(ε J+ε J′) − 2∑
J , J′
εJεJ′ gJ gJ′e−β(ε J+ε J′)