Introduction:
The root mean square velocity (Urms) of a monatomic gas is a measure of the average speed of the gas particles. It is defined as the square root of the average of the squares of the velocities of the gas particles. The relationship between the average kinetic energy of the gas and Urms can be understood using the kinetic theory of gases.

Kinetic Theory of Gases:
The kinetic theory of gases states that gas particles are in constant random motion. The average kinetic energy of the gas particles is directly proportional to the temperature of the gas. The kinetic energy of a gas particle can be related to its velocity using the formula:

KE = (1/2)mv^2

where KE is the kinetic energy, m is the mass of the gas particle, and v is its velocity.

Relation between Average Kinetic Energy and Urms:
1. The average kinetic energy of a gas can be calculated by taking the average of the kinetic energies of all the gas particles. Since the kinetic energy is directly proportional to the square of the velocity, the average kinetic energy can be written as:

Average KE = (1/2) * Average(mv^2)

2. The root mean square velocity (Urms) is the square root of the average of the squares of the velocities. Mathematically, it can be expressed as:

Urms = √(Average(v^2))

3. Substituting the value of v^2 from the kinetic energy equation, we get:

Urms = √(Average(2KE/m))

4. Multiplying and dividing the equation by m, we get:

Urms = √(2Average(KE)/m)

5. From the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

6. Rearranging the equation, we get:

PV = (n/m)RT

7. Since n/m is the number of moles per unit mass, it can be denoted as N, the Avogadro's number.

8. Substituting the value of n/m in the equation for Urms, we get:

Urms = √(3RT/N)

9. As per the kinetic theory of gases, the average kinetic energy is directly proportional to the temperature. Therefore, we can write:

Average KE = (3/2)kT

where k is the Boltzmann constant.

10. Substituting the value of the average kinetic energy in the equation for Urms, we get:

Urms = √(3kT/m)

Conclusion:
The relationship between the average kinetic energy of a gas and the root mean square velocity is given by Urms = √(3kT/m). This equation shows that the Urms is directly proportional to the square root of the temperature and inversely proportional to the square root of the molar mass of the gas. Therefore, the average kinetic energy of a gas is directly related to its Urms.

So easy