The Kinetic Theory Of Gases

The mean free path of a molecule is the average distance a molecule travels before colliding with another molecule. According to kinetic theory, the mean free path is inversely proportional to the number density of molecules and the square of the radius of the molecule. The formula for the mean free path is: Thus, the mean free path is inversely proportional to .

The correct option is (A) :

The formula for the r.m.s. speed is given by: where is the Boltzmann constant, is the temperature, and is the mass of the molecules. Since the gas is ideal and the mass of the molecules remains constant, the r.m.s. speed of molecules is proportional to the square root of the temperature. Therefore, we can write the relation: Given that , , and , we can substitute into the formula: Thus, So, the new r.m.s. speed of the molecules is .

The correct option is (C) :

For a polyatomic molecule with 3 translational and 3 rotational degrees of freedom, the contribution to the specific heat is: Each vibrational mode contributes to , and there are 2 vibrational modes: For , we use the relation: Thus, the ratio is:

The correct option is (E) :

Step 1: Recall that the average kinetic energy ( ) of an ideal gas is proportional to the absolute temperature ( ).
If is doubled, the absolute temperature must also double.
Step 2: Convert the initial temperature from Celsius to Kelvin: .
Step 3: The new temperature in Kelvin when doubles is .
Step 4: Convert back to Celsius: , which rounds to .

The mean free path is inversely proportional to the number of molecules per unit volume .
Hence, we have the relation: Let the initial number of molecules per unit volume be and the initial mean free path be .
When the number of molecules per unit volume is increased to , the mean free path becomes: Substituting the values: Thus: Hence, the correct answer is (B).