As adiabatic vessel contains `n_(1) = 3` mole of diatomic gas. Moment of inertia of each molecule is `I = 2.76 xx 10^(-46) kg m^(2)` and root-mean-square angular velocity is `omega_(0) = 5 xx 10^(12) rad//s`. Another adiabatic vessel contains `n_(2) = 5` mole of a monatomic gas at a temperature `470 K`. Assume gases to be ideal, calculate root-mean-square angular velocity of diatomic molecules when the two vessels are connected by a thin tube of negligible volume. Boltzmann constant `k = 1.38 xx 10^(-23) J//:"molecule"`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have two adiabatic vessels containing different gases. We need to find the root-mean-square (RMS) angular velocity of diatomic gas molecules when the two vessels are connected. We will use the law of equipartition of energy to find the final temperature of the system and then use that to find the RMS angular velocity.

### Step 2: Use the law of equipartition of energy
According to the law of equipartition of energy, each degree of freedom contributes \(\frac{1}{2} kT\) to the energy of the system.

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