Chapter 10, Problem 83QP

Interpretation Introduction

Interpretation:

The root mean square speed values of ${\text{N}}_{\text{2}}$, ${\text{O}}_{\text{2}}$, and ${\text{O}}_{\text{3}}$ found in the stratosphere, with the absolute temperature of –23°C, are to be calculated.

Concept introduction:

Kinetic molecular theory states that the total kinetic energy of a mole of gas is equal to $\frac{3}{2}RT$. Here, R is universal gas constant and T is the absolute temperature. The average kinetic energy of one molecule is $\frac{1}{2}m\overline{u^2}$. Therefore,

$N_A\left(\frac{1}{2}m\overline{u^2}\right)=\frac{3}{2}RT$

$m{u}^2=3RTor{u}^2=3RT/moru={\left(3RT/m\right)}^{1/2}$

From Maxwell’s Speed Distribution Curve, the molecular speed of two gas samples can be compared using the following relation:

$u_{rms}\left(1\right)=\sqrt{\frac{3RT}{{\mu }_1}}$

and,

$u_{rms}\left(2\right)=\sqrt{\frac{3RT}{{\mu }_2}}$

Now,

$\begin{array}{c}\frac{u_{rms}\left(1\right)}{u_{rms}\left(2\right)}=\frac{\sqrt{\frac{3RT}{{\mu }_1}}}{\sqrt{\frac{3RT}{{\mu }_2}}}\\ =\sqrt{\frac{{\mu }_2}{{\mu }_1}}\end{array}$ …… (1)

Here, $u_{rms}\left(1\right)$ and $u_{rms}\left(2\right)$ are the root mean square speeds of each gas compound, respectively, $R$ is the Gas Constant, $T$ is the absolute temperature, and ${\mu }_1\text{ and }{\mu }_2$ are the molar masses of each of the gas compounds, respectively.