Gas in Harmonic Well
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 21
29-8-2016
1539
Gas in Harmonic Well
A classical system of N distinguishable noninteracting particles of mass m is placed in a three-dimensional harmonic well:
a) Find the partition function and the Helmholtz free energy.
b) Regarding V as an external parameter, find the thermodynamic force 
conjugate to this parameter, exerted by the system; find the equation of state and compare it to that of a gas in a container with rigid walls.
c) Find the entropy, internal energy, and total heat capacity at constant volume.
SOLUTION
a) The partition function is given by a standard integral (where the molecules are indistinguishable):
(1)
The Helmholtz free energy F follows directly from the partition function:
(2)
b) We may find the force from F:
(3)
The equation of state is therefore analogous to the gas in a container with rigid walls, where
c) The entropy, energy, and heat capacity all follow in quick succession from F:
(4)
(5)
(6)
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