Kinetic Theory

A real gas behaves like an ideal gas at low pressure and high temperature.

As per the kinetic theory of gases, there are two main assumptions made explaining the deviation of real gases from ideal gas behavior:
1. Compared to the volume of the vessel, the volume of the gas particle is negligible. But, in the case of a real gas, the volume of every individual gas particle is very significant.
2. There is no interaction between the gaseous particles. However, in a real gas, there are forces of attraction between the molecules.

From the ideal gas equation we know, PV=nRT. Thus, if the pressure of the gas is very high, or the temperature is very low, there is a substantial deviation from the ideal gas equation. Thus, a real gas obtains ideal gas behavior at very low pressure and high temperature.

Also Read: Behavior of Gas Molecules

Sound Velocity and Internal Energy: Explanation

The correct option is (D): and .

The formulas provided for the sound velocity and internal energy are:

where ,

and

Explanation:

Let's break down the formulas and the effect of the number of degrees of freedom ( ):

1. Sound Velocity Equation:

The formula for sound velocity is:

Here, is the sound velocity, is the gas constant, is the temperature, and is the molar mass.

As (the number of degrees of freedom) increases, decreases because:

Since appears in the numerator of the sound velocity equation, a decrease in leads to a decrease in the sound velocity .

2. Internal Energy Equation:

The formula for internal energy is:

Here, is a constant and represents the number of degrees of freedom. As increases, the internal energy increases because is directly proportional to .

Conclusion:

Thus, when increases:

• decreases, which results in a decrease in the sound velocity ( ).
• The internal energy increases because it is directly proportional to .

Therefore, the correct option (D) is confirmed: and .