Question

Molecules of an ideal gas are known to have three translational degrees of freedom. The gas is maintained at a temperature of T. The total internal energy, U of a mole of this gas, and the value of $\gamma = (\frac{C_p}{C_v})$ are given respectively, by

• A. $\frac{U}{}=\frac{5}{2}RT$ and $\gamma = \frac{6}{5}$
• B. $\frac{U}{RT}=5$ and $\gamma = \frac{7}{5}$
• C. $\frac{U}{}=\frac{5}{2}RT$ and $\gamma = \frac{7}{5}$
• D. $U = 5RT$ and $\gamma = \frac{6}{5}$

Answer 1

Answer: (C)

$\frac{U}{}=\frac{5}{2}RT$ and $\gamma = \frac{7}{5}$

Answer 2

Assuming the ideal gas is diatomic (as implied by the options), it has $f=5$ degrees of freedom (3 translational + 2 rotational).

Internal energy per mole $U = \frac{f}{2}RT = \frac{5}{2}RT$.

Molar specific heat at constant volume $C_v = \frac{f}{2}R = \frac{5}{2}R$.

Molar specific heat at constant pressure $C_p = C_v + R = \frac{5}{2}R + R = \frac{7}{2}R$.

Ratio of specific heats $\gamma = \frac{C_p}{C_v} = \frac{7/2 R}{5/2 R} = \frac{7}{5}$.