Question

Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational 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 $\gamma \left(\frac{C_p}{C_v}\right)$ are given respectively as:

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

Answer 1

The correct option is C $U=\frac{5}{2}RT\text{and}\gamma =\frac{7}{5}$

Total degree of freedom of the gas is given as
$f=3+2=5$
Total energy, $U=\frac{fRT}{2}=\frac{5RT}{2}$
$\text{And},\gamma =\frac{C_P}{C_v}=1+\frac{2}{f}=1+\frac{2}{5}=\frac{7}{5}$
Hence, $\left(D\right)$ is the correct option.