The value of $\gamma\left(=\frac{C_{p}}{C_{y}}\right)$ , for hydrogen, helium and another ideal diatomic gas $X$ (whose molecules are not rigid but have an additional vibrational mode), are respectively equal to :

Question

In the given cycle ABCDA, the heat required for an ideal monoatomic gas will be:

Answer 1

To solve this problem, we need to understand the concept of specific heat ratio $\gamma$ which is defined as:

$\gamma = \frac{C_p}{C_v}$

where $C_p$ is the specific heat at constant pressure and $C_v$ is the specific heat at constant volume.

For monatomic gases like helium, $C_p = \frac{5}{2}R$ and $C_v = \frac{3}{2}R$. Therefore,
$\gamma = \frac{C_p}{C_v} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3}$
For diatomic gases with only translational and rotational motion (like hydrogen), $C_p = \frac{7}{2}R$ and $C_v = \frac{5}{2}R$. Therefore,
$\gamma = \frac{C_p}{C_v} = \frac{\frac{7}{2}R}{\frac{5}{2}R} = \frac{7}{5}$
For a diatomic gas with an additional vibrational mode (such as gas X in the question), one vibrational mode contributes an additional $R$ to both $C_v$ and $C_p$. Therefore, $C_p = \frac{9}{2}R$ and $C_v = \frac{7}{2}R$. Thus,
$\gamma = \frac{C_p}{C_v} = \frac{\frac{9}{2}R}{\frac{7}{2}R} = \frac{9}{7}$

Now, based on the calculations, we have the values of $\gamma$ for each gas:

Therefore, the correct answer is: $\frac{7}{5}, \frac{5}{3}, \frac{9}{7}$

Answer 2

To solve this problem, we need to understand the concept of specific heat ratio $\gamma$ which is defined as:

$\gamma = \frac{C_p}{C_v}$

where $C_p$ is the specific heat at constant pressure and $C_v$ is the specific heat at constant volume.

For monatomic gases like helium, $C_p = \frac{5}{2}R$ and $C_v = \frac{3}{2}R$. Therefore,
$\gamma = \frac{C_p}{C_v} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3}$
For diatomic gases with only translational and rotational motion (like hydrogen), $C_p = \frac{7}{2}R$ and $C_v = \frac{5}{2}R$. Therefore,
$\gamma = \frac{C_p}{C_v} = \frac{\frac{7}{2}R}{\frac{5}{2}R} = \frac{7}{5}$
For a diatomic gas with an additional vibrational mode (such as gas X in the question), one vibrational mode contributes an additional $R$ to both $C_v$ and $C_p$. Therefore, $C_p = \frac{9}{2}R$ and $C_v = \frac{7}{2}R$. Thus,
$\gamma = \frac{C_p}{C_v} = \frac{\frac{9}{2}R}{\frac{7}{2}R} = \frac{9}{7}$

Now, based on the calculations, we have the values of $\gamma$ for each gas:

Therefore, the correct answer is: $\frac{7}{5}, \frac{5}{3}, \frac{9}{7}$

List-I	List-II
(A) Heat capacity of body	(I) \( J\,kg^{-1} \)
(B) Specific heat capacity of body	(II) \( J\,K^{-1} \)
(C) Latent heat	(III) \( J\,kg^{-1}K^{-1} \)
(D) Thermal conductivity	(IV) \( J\,m^{-1}K^{-1}s^{-1} \)

The Correct Option is A