Chapter 5, Problem 5.72QP

Interpretation Introduction

Interpretation:

The highest root-mean-square speed has to be predicted

Concept introduction:

Average kinetic energy: Average kinetic energy of the gas molecules is proportional to the temperature of the gas in kelvins.

• Average kinetic energy of a molecule is given by,

$\overline{KE}=\frac{1}{2}m\overline{u^2}=CT$

Where, m is the mas of molecule

u is the speed

C is proportionality constant

T is absolute temperature

According to the equation, the absolute temperature of a gas is a measure of the average kinetic energy of the molecules. Higher the temperature, the molecules are highly energetic. Thus, random molecular motion is referred to as thermal motion.

Root-Mean-Square speed:

One of the results of the kinetic theory of gases is that the total kinetic energy of a mole of any gas equals $\frac{3}{2}RT$. The average kinetic energy of one molecule is $\frac{1}{2}m\overline{u^2}$.

So,

$\begin{array}{l}\overline{KE}=\frac{3}{2}RT\\ N_A\left(\frac{1}{2}m\overline{u^2}\right)=\frac{3}{2}RT\end{array}$

Where,

$N_A$ is Avogadro’s number

m is mass of a single molecule.

As known, $N_{A}m=M$

Where,

M is molar mass.

This equation can be arranged to give,

$\sqrt{\overline{u^2}}=u_{rms}=\sqrt{\frac{3RT}{M}}$

Taking the square root of both sides gives,

$\sqrt{\overline{u^2}}=u_{rms}=\sqrt{\frac{3RT}{M}}-----\left(1\right)$

The unit of each term in the equation (1) is,

Where, R is proportionality constant called gas constant $\left(J/mol.K\right)$

M is molar mass $\left(Kg/mole\right)$

T is temperature $\left(K\right)$