The Kinetic Theory Of Gases

For a diatomic gas: Since , we get: For a monatomic gas: Using , we write: Thus, the correct answer is .

To solve this problem, we need to understand the concept of the degrees of freedom in a diatomic gas molecule. The degrees of freedom determine the energy distribution among translation, rotation, and vibration in gas molecules. For a diatomic molecule at room temperature or NTP (Normal Temperature and Pressure), the vibrational modes are not usually active due to lower energy levels. Hence, they exhibit five degrees of freedom: three translational and two rotational.

The formula for average kinetic energy per molecule due to degrees of freedom is given by:

where is the degrees of freedom, is the Boltzmann constant, and is the temperature.

1. At NTP, for a diatomic gas, . Therefore, .
2. At a high temperature, vibrational modes are excited, adding 2 more degrees of freedom, making . Hence, .
3. The ratio of kinetic energies is:

This shows that the ratio of kinetic energy of a diatomic gas molecule at high temperature to that at NTP is .

The rms velocity of a gas molecule is given by the formula: where is the Boltzmann constant, is the temperature, and is the mass of the molecule. Thus, the rms velocity is inversely proportional to the square root of the mass: So, the correct answer is .