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Physics Question on Thermodynamics

According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of constant γ=CPCV\gamma = \frac{C_P}{C_V} is (CPC_P where CVC_V are the specific heat capacities of the gas at constant pressure and constant volume, respectively):

A

4+3γγ−1\frac{4 + 3\gamma}{\gamma - 1}

B

3+4γγ−1\frac{3 + 4\gamma}{\gamma -1}

C

4−3γγ−1\frac{4 - 3\gamma}{\gamma -1}

D

3−4γγ−1\frac{3 - 4\gamma}{\gamma - 1}

Answer

4−3γγ−1\frac{4 - 3\gamma}{\gamma -1}

Explanation

Solution

For a polyatomic gas with 3 translational, 3 rotational, and ff vibrational modes:

Internal energy (U) =32kBT+32kBT+fkBT=(3+f)kBT= \frac{3}{2}k_BT + \frac{3}{2}k_BT + fk_BT = (3 + f)k_BT

Cv=(3+f)RC_v = (3 + f)R

Cp=(4+f)RC_p = (4 + f)R

γ=CpCv=4+f3+f\gamma = \frac{C_p}{C_v} = \frac{4 + f}{3 + f}

3γ+fγ=4+f3\gamma + f\gamma = 4 + f

f(γ−1)=4−3γf(\gamma - 1) = 4 - 3\gamma

f=4−3γγ−1f = \frac{4 - 3\gamma}{\gamma - 1}