Question

An ideal gas is made of polyatomic molecules. Each molecule has three translational, three rotational and \(f\) number of vibrational modes. If the ratio of heat capacities \[ \frac{C_P}{C_V}=\frac{8}{7} \] then the value of \(f\) is:

• A. 1
• B. 4
• C. 3
• D. 2

Hint:

For every vibrational mode, two degrees of freedom are added. Always remember: \[ C_P=C_V+R \] for an ideal gas.

Answer 1

The Correct Option is D

Concept: For a polyatomic gas, \[ C_V=\frac{nR}{2} \] where \(n\) is the total number of active degrees of freedom. Each vibrational mode contributes two degrees of freedom.

Step 1: Calculate total degrees of freedom
Translational degrees of freedom \[ =3 \] Rotational degrees of freedom \[ =3 \] Vibrational contribution \[ =2f \] Therefore, \[ n=3+3+2f \] \[ n=6+2f \]

Step 2: Write expressions for heat capacities
\[ C_V=\frac{(6+2f)R}{2} \] \[ C_V=(3+f)R \] Also, \[ C_P=C_V+R \] \[ C_P=(4+f)R \]

Step 3: Use the given ratio
Given, \[ \frac{C_P}{C_V} = \frac{8}{7} \] Substituting, \[ \frac{4+f}{3+f} = \frac{8}{7} \] \[ 7(4+f)=8(3+f) \] \[ 28+7f=24+8f \] \[ f=4 \] Since each vibrational mode contributes two degrees of freedom, the number of vibrational modes is \[ \boxed{2} \] Therefore, \[ \boxed{\text{Option (D)}} \]