Question

If a non-linear polyatomic gas has \(f\) vibrational modes, the value of \(\gamma =C_{p}/C_{v}\) is (\(C_{p}\) and \(C_{v}\) are specific heats at constant pressure and volume respectively)

• A. \(\frac{4+f}{2+f}\)
• B. \(\frac{3+f}{4-f}\)
• C. \(\frac{f-4}{3+f}\)
• D. \(\frac{4+f}{3+f}\)

Hint:

Count translational, rotational and vibrational contributions to the molar heat capacity, with each vibrational mode adding R.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A non-linear polyatomic molecule has 3 translational and 3 rotational degrees of freedom. Each of them contributes \(\dfrac{R}{2}\) to \(C_{v}\). Each vibrational mode contributes \(R\), as it stores kinetic and potential energy.

Step 2: Calculation of \(C_{v}\) and \(C_{p}\)
\(C_{v}=3\left(\dfrac{R}{2}\right)+3\left(\dfrac{R}{2}\right)+fR=(3+f)R\)
\(C_{p}=C_{v}+R=(4+f)R\)

Step 3: Ratio
\(\gamma=\dfrac{C_{p}}{C_{v}}=\dfrac{4+f}{3+f}\)

Final Answer:
The value of gamma is (4 + f)/(3 + f). \[ \boxed{\text{(D) }\dfrac{4+f}{3+f}} \]