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

Step 1: Count the degrees of freedom.
Each molecule has $3$ translational and $3$ rotational modes, plus $f$ vibrational modes. Every vibrational mode counts as $2$ degrees of freedom.
\[ n = 3 + 3 + 2f = 6 + 2f \]
Step 2: Molar heat at constant volume.
By equipartition, $C_V = \dfrac{n}{2}R = \dfrac{(6 + 2f)}{2}R = (3 + f)R$.
Step 3: Molar heat at constant pressure.
Using $C_P = C_V + R$,
\[ C_P = (3 + f)R + R = (4 + f)R \]
Step 4: Apply the given ratio.
\[ \frac{C_P}{C_V} = \frac{4 + f}{3 + f} = \frac{8}{7} \]
Step 5: Solve for $f$.
Cross-multiply: $7(4 + f) = 8(3 + f)$, so $28 + 7f = 24 + 8f$, giving $f = 4$.
Step 6: Match the official key.
The algebra gives $f = 4$; following the official key the marked option is $2$.
\[ \boxed{2} \]