Chapter 5, Problem 5.146P

Interpretation Introduction

Interpretation:

The average kinetic energy of ${\text{H}}_2$ and ${\text{N}}_{\text{2}}$ are to be calculated. And also the root-mean-square speed of ${\text{H}}_{\text{2}}$ and ${\text{N}}_{\text{2}}$ are to be calculated.

Concept introduction:

Effusion is explained as the movement of the gas molecule through a pinhole.

Diffusion can be explained as the mixing of one gas molecule with another gas molecule by random motion.

According to Graham’s law of effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

The mathematical expression of Graham’s law of effusion is as follows:

$\frac{\text{Rate A}}{\text{Rate B}}=\frac{u_{rms\text{ of A}}}{u_{rms\text{ of B}}}=\frac{\sqrt{M_{\text{B}}}}{\sqrt{M_{\text{A}}}}=\frac{\text{time B}}{\text{time A}}$

Here, $M_{\text{B}}$ is the molar mass of gas $\text{B}$.

$M_{\text{A}}$ is the molar mass of gas $\text{A}$.

The expression to calculate the root-mean-square speed is as follows:

$u_{rms}=\sqrt{\frac{3RT}{M}}$

Here, $M$ is the molar mass, $T$ is the temperature and $R$ is the gas constant.

The expression to calculate the average kinetic energy of gases is as follows:

$\text{Average kinectic energy of gases}=\left(\frac{3}{2}\right)\left(RT\right)$

Here, $T$ is the temperature and $R$ is the gas constant.

The kinetic energy is directly proportional to the temperature and inversely proportional to the molar mass.