Question

The molar specific heat at constant pressure of an ideal gas is $\left({\displaystyle \frac{7}{2}}\right)R$. The ratio of specific heat at constant pressure to that at constant volume is:-
A. ${\displaystyle \frac{7}{5}}$
B. ${\displaystyle \frac{8}{7}}$
C. ${\displaystyle \frac{5}{7}}$
D. ${\displaystyle \frac{9}{7}}$

Answer 1

Hint:The ratio of specific heat at constant pressure to the ratio of specific heat at constant pressure is denoted as $\gamma$. In the question, the value of specific heat at constant pressure is given, we will calculate the specific heat at constant volume by using the formula given below. Also, the ratio of specific heats can be calculated by dividing the specific heat at constant pressure to the specific heat at constant volume.

Formula used:
The formula used for calculating the specific heat coefficient at constant volume is given by
$C_p-C_v=R$
$\Rightarrow C_v=C_p+R$
Here, $C_p$ is the coefficient of specific heat at constant pressure, $C_v$ is the coefficient of specific heat at constant volume and $R$ is the gas constant from the equation of state.
The ratio of specific heat at constant pressure to that at constant volume is given below
$\gamma ={\displaystyle \frac{C_p}{C_v}}$
Here, $\gamma$ is the ratio of the specific heats.

Complete step by step answer:
As given in the question,the molar specific heat at constant pressure is, $C_p=\left({\displaystyle \frac{7}{2}}\right)R$. Now, to calculate the specific heat at constant volume can be calculated by using the formula given below;
$C_p-C_v=R$
$\Rightarrow C_v=C_p-R$
$\Rightarrow C_v=\left({\displaystyle \frac{7}{2}}\right)R-R$
$\Rightarrow C_v={\displaystyle \frac{5}{2}}R$
Now, the ratio of specific heat at constant pressure to the specific heat at constant volume can be calculated as given below
$\gamma ={\displaystyle \frac{C_p}{C_v}}$
$\Rightarrow \gamma ={\displaystyle \frac{{\displaystyle \frac{7}{2}}R}{{\displaystyle \frac{5}{2}}R}}$
$\therefore \gamma ={\displaystyle \frac{7}{5}}$
Therefore, the ratio of specific heat at constant pressure to that at constant volume is ${\displaystyle \frac{7}{5}}$.

Hence, option A is the correct option.

Note: Here, $C_v$ is the specific heat at constant volume which means that there are no moving particles in the system. On the other hand, $C_p$ is the specific heat at constant pressure which suggests that volume is changing. Here, in the above question, the value of $C_p$ and $C_v$ is in the form of $R$.