Problems and Solutions on Thermodynamics and Statistical Mechanics

Statiationl Phyaca 193

Thus, rotation begins to contribute to the specific heat at T = 75 K, and
vibration does so at T = 6250 K.
When TI = 25 K, only the translational motion contributes to C, then

3 5

2 2

Cv = -R = 12.5 J/K, C, = -R = 20.8 J/K

When Tz = 250 K, only translation and 'rotation contribute to C, then

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2

Cv = iR == 20.8 J/K, C, = -R = 29.1 J/K.

When T3 = 2500 K, the result is the same as for T2 = 250 K.

When T4 = 10000 K, vibration also contributes to C, then

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C - -R = 29.1 J/K, C - -R = 37.4 J/K.

v- 2 p- 2

2051

Derive an expression for the vibrational specific heat of a diatomic gas

as a function of temperature. (Let hwo/k = 0). For full credit start with

an expression for the vibrational partition function, evaluate it, and use the

result to calculate Cvib.

Describe the high and low T limits of Cvib.

( wi3 CO fMitL)

Solution:

The vibrational energy levels of a diatomic gas are

E" = hwo(w + 1/2)] u = 0,1,2, *...

The partition function is

where z = Phwo. The free energy of 1 mole of the gas is