Animated Solution for Physics - Thermodynamics: If one mole of the polyatomic gas is having two vibrational modes and $\beta$ is the ratio of molar specific heats for polyatomic gas $\left(\beta =\frac{C_p}{C_V}\right)$, then the value of $\beta$ is

Imagine a complex polyatomic molecule floating in a container. It is a dynamic, energetic entity that can store thermal energy in multiple independent ways. In thermodynamics, we call these independent modes of storing energy the degrees of freedom ($f$).

To find the ratio of specific heats, denoted here as $\beta$ (which is identical to the adiabatic index $\gamma$), we must first take a complete inventory of these degrees of freedom. The total degrees of freedom is the sum of translational, rotational, and vibrational modes:

Let's break down the movement of our polyatomic gas:

1. Translation: No matter how complex the molecule is, its center of mass can move freely in three-dimensional space (along the $x$, $y$, and $z$ axes). This gives us exactly $3$ translational degrees of freedom.

2. Rotation: A non-linear polyatomic molecule has a significant moment of inertia about all three spatial axes. Therefore, it can tumble and spin in three independent directions, giving us $3$ rotational degrees of freedom.

3. Vibration: Here is where we must be careful. The problem explicitly states that the gas has two vibrational modes. A vibrational mode acts like a microscopic spring connecting the atoms. According to the Law of Equipartition of Energy, a harmonic oscillator stores energy in two forms: kinetic energy (as the atoms move) and potential energy (stored in the "spring"). Because of this dual nature, each vibrational mode contributes $2$ degrees of freedom.

Adding them all together, we find the total degrees of freedom for our specific gas:

The ratio of molar specific heats at constant pressure ($C_p$) to constant volume ($C_V$) is a fundamental thermodynamic property. From the equipartition theorem, this ratio is elegantly tied to the degrees of freedom by the formula:

Now, we simply substitute our hard-earned value of $f=10$ into the equation:

This clean, elegant result perfectly matches option (b). It is fascinating to note that as a molecule becomes more complex and activates more vibrational modes, its degrees of freedom increase, which in turn causes the ratio of specific heats ($\beta$) to decrease, asymptotically approaching $1$.