Question

An ideal gas has molecules with $5$ degrees of freedom. The ratio of specific heats at constant pressure $\left( {{C_p}} \right)$ and at constant volume $\left( {{C_v}} \right)$ is
A. $\dfrac{5}{2}$
B. $\dfrac{7}{5}$
C. $6$
D. $\dfrac{7}{2}$

Answer 1

Hint: An ideal gas has molecules with $5$ degrees of freedom is a diatomic ideal gas. By using the specific heat at constant volume formula for diatomic ideal gas, the relation can by obtained. And with help of the relation between the specific heat at constant pressure and specific heat constant volume relation, the ratio between these two can be derived.

Useful formula:
The specific heat at constant volume for diatomic ideal gas,
${C_v} = \dfrac{5}{2} \times R$
Where, ${C_v}$ is the specific heat at constant volume and $R$ is the gas constant.
The relation between specific heat at constant pressure and specific heat constant volume is,
${C_p} = {C_v} + R$
Where, ${C_v}$ is the specific heat at constant volume and ${C_p}$ is the specific heat at constant pressure.

Complete step by step solution:
Since, the given ideal gas has molecules of $5$ degrees of freedom. Hence, the given ideal gas is diatomic ideal gas.
The specific heat at constant volume for diatomic ideal gas,
${C_v} = \dfrac{5}{2} \times R\;................................\left( 1 \right)$

The relation between specific heat at constant pressure and specific heat constant volume is,
${C_p} = {C_v} + R\;.............................\left( 2 \right)$
By using the equation (1) in equation (2), we get
${C_p} = \left( {\dfrac{5}{2} \times R} \right) + R \\\ {C_p} = \dfrac{{5R}}{2} + R \\\ {C_p} = \dfrac{{5R + 2R}}{2} \\\ {C_p} = \dfrac{{7R}}{2} \\\$

Hence, the ratio between specific heats at constant pressure $\left( {{C_p}} \right)$ and at constant volume $\left( {{C_v}} \right)$ is given by,
$\dfrac{{{C_p}}}{{{C_v}}} = \dfrac{{\left( {\dfrac{{7R}}{2}} \right)}}{{\left( {\dfrac{{5R}}{2}} \right)}} \\\ \dfrac{{{C_p}}}{{{C_v}}} = \dfrac{{7R}}{2} \times \dfrac{2}{{5R}} \\\ \dfrac{{{C_p}}}{{{C_v}}} = \dfrac{7}{5} \\\$

Hence, the option (B) is correct.

Note: As like as the diatomic ideal gas, there are two other gases also exists. They are monoatomic gas and poly atomic gas. They are different based on the number of degrees of freedom of the molecules present in it. Also, the specific at constant volume gets vary for above mentioned different gases.