Question

Match the$\frac{C_p}{C_v}$ ratio for ideal gases with the different types of molecules:

Column 1	Column 2
Monoatomic	$\frac{7}{5}$
Diatomic rigid molecules	$\frac{9}{7}$
Diatomic non rigid molecules	$\frac{4}{3}$
Triatomic rigid molecules	$\frac{5}{3}$

Answer 1

Part A:

1. For the monoatomic gas the degree of freedom is$3$.
2. As we know,
   $$\begin{gathered} \Rightarrow \gamma =\frac{C_p}{C_v} \\ \Rightarrow \gamma =1+\frac{2}{f} \end{gathered}$$
   (where $f=$the degree of freedom, $C_p=$Specific heat capacity at constant pressure, $C_v=$Specific heat capacity at constant volume )
3. The value of$\frac{C_p}{C_v}$ is,
   $\begin{array}{rcl}\gamma & =& 1+\frac{2}{f}\\ & =& 1+\frac{2}{3}\\ & =& \frac{5}{3}\\ & & \end{array}$

Hence, (A) - (IV)

Part B:

1. For the diatomic rigid molecules, the degree of freedom is$5$.
2. As we know,
   $$\begin{gathered} \Rightarrow \gamma =\frac{C_p}{C_v} \\ \Rightarrow \gamma =1+\frac{2}{f} \end{gathered}$$
   (where $f=$the degree of freedom, $C_p=$Specific heat capacity at constant pressure, $C_v=$Specific heat capacity at constant volume )
3. The value of$\frac{C_p}{C_v}$ is,
   $\begin{array}{rcl}\gamma & =& 1+\frac{2}{f}\\ & =& 1+\frac{2}{5}\\ & =& \frac{7}{5}\\ & & \end{array}$

Hence, (B) - (I)

Part C:

1. For the diatomic non-rigid molecules, the degree of freedom is$7$.
2. As we know,
   $$\begin{gathered} \Rightarrow \gamma =\frac{C_p}{C_v} \\ \Rightarrow \gamma =1+\frac{2}{f} \end{gathered}$$
   (where $f=$the degree of freedom, $C_p=$Specific heat capacity at constant pressure, $C_v=$Specific heat capacity at constant volume )
3. The value of$\frac{C_p}{C_v}$ is,
   $\begin{array}{rcl}\gamma & =& 1+\frac{2}{f}\\ & =& 1+\frac{2}{7}\\ & =& \frac{9}{7}\\ & & \end{array}$

Hence, (C) - (II)

Part D:

1. For the triatomic rigid molecules, the degree of freedom is$6$.
2. As we know,
   $$\begin{gathered} \Rightarrow \gamma =\frac{C_p}{C_v} \\ \Rightarrow \gamma =1+\frac{2}{f} \end{gathered}$$
   (where $f=$the degree of freedom, $C_p=$Specific heat capacity at constant pressure, $C_v=$Specific heat capacity at constant volume )
3. The value of$\frac{C_p}{C_v}$ is,
   $\begin{array}{rcl}\gamma & =& 1+\frac{2}{f}\\ & =& 1+\frac{2}{6}\\ & =& \frac{4}{3}\\ & & \end{array}$

Thus, (D) - (III)

Hence, the correct answer is A-IV, B-I, C-2, D-III