Average Velocity of a Gaseous Molecule and Temperature

Introduction:
The average velocity of a gaseous molecule refers to the average speed at which the molecules of a gas move. It is directly influenced by temperature, as temperature affects the kinetic energy of the gas molecules. When the temperature of a gas is increased, the average velocity of its molecules also increases.

Explanation:
When the temperature of a gas is doubled, the average velocity of its molecules also doubles. This can be explained using the kinetic theory of gases and the relationship between temperature and kinetic energy.

Kinetic Theory of Gases:
The kinetic theory of gases states that gas molecules are in constant motion and have kinetic energy. The kinetic energy of a molecule is directly proportional to its temperature.

Relationship between Temperature and Kinetic Energy:
The relationship between temperature and kinetic energy can be described by the equation:

KE = (3/2) * k * T

Where:
- KE is the average kinetic energy of the gas molecules
- k is the Boltzmann constant (1.38 x 10^-23 J/K)
- T is the temperature in Kelvin

Relationship between Kinetic Energy and Average Velocity:
The average kinetic energy of gas molecules is related to their average velocity through the equation:

KE = (1/2) * m * v^2

Where:
- KE is the average kinetic energy of the gas molecules
- m is the mass of a gas molecule
- v is the average velocity of the gas molecules

Derivation:
By equating the two equations for kinetic energy, we can determine the relationship between average velocity and temperature.

(3/2) * k * T = (1/2) * m * v^2

Simplifying the equation:

v^2 = (3 * k * T) / m

Taking the square root of both sides:

v = √((3 * k * T) / m)

Effect of Doubling Temperature:
When the temperature is doubled, the equation for average velocity becomes:

v' = √((3 * k * 2T) / m)

Simplifying the equation:

v' = √((6 * k * T) / m)

Comparing the two equations:

v' / v = (√((6 * k * T) / m)) / (√((3 * k * T) / m))

v' / v = √((6 * k * T) / m) * √(m / (3 * k * T))

v' / v = √(6 / 3)

v' / v = √2

Therefore, the average velocity of a gaseous molecule increases by a factor of √2 (approximately 1.4) when the temperature is doubled.

Conclusion:
When the temperature of a gas is doubled, the average velocity of its molecules increases by a factor of approximately 1.4. This can be explained by the relationship between temperature and kinetic energy, as well as the relationship between kinetic energy and average velocity.