The root mean square velocity of one mole of a monoatomic...

The root mean square velocity of one mole of a monoatomic gas having molar mass $M$ is $u_{\mathrm{r}, \mathrm{m} . \mathrm{s}} .$ The relation between the average kinetic energy $(E)$ of the gas and $u_{\text {r.m. }}$ is

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases is a framework that describes the behavior of gases in terms of the motion of individual molecules. It establishes relationships between macroscopic properties like pressure and temperature and microscopic quantities like molecular speeds and kinetic energy, providing insight into how energy is stored and transferred in a gas.

Monoatomic Gas

A monoatomic gas consists of single-atom molecules, which simplifies the analysis of its behavior under the kinetic theory of gases. Since monoatomic gases have only translational degrees of freedom, the average kinetic energy per atom directly correlates with the temperature, making it easier to derive relationships such as that between kinetic energy and rms velocity.

Equipartition Theorem

The equipartition theorem is a principle in statistical mechanics that states energy is equally distributed among all degrees of freedom of a system. For a monoatomic gas, this theorem implies that the average kinetic energy per particle is proportional to the temperature, allowing a clear relation between molecular speeds and thermal energy.

Kinetic Energy

Kinetic energy refers to the energy that a particle possesses due to its motion. In the context of gases, it is typically expressed as one-half the mass of a particle multiplied by the square of its speed, linking the microscopic motion of particles to macroscopic thermal energy.

Root Mean Square Velocity

The root mean square (rms) velocity is a statistical measure that represents the square root of the average of the squares of the velocities of gas particles. It provides a meaningful average speed that is directly related to the temperature of the gas in kinetic theory.

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