Chapter 13, Problem 40A

Interpretation Introduction

Interpretation: The factor by which average kinetic energy changes by increasing temperature from $300\text{ K}$ to $900\text{ K}$ is to be evaluated.

Concept introduction:

The relationship between kinetic energy and temperature is as follows:

  $E_k=\frac{3}{2}{k}_{B}T$

Here, $E_k$ is kinetic energy, $k_B$ is Boltzmann’s constant, and $T$ is temperature.

Answer to Problem 40A

The average kinetic energy becomes three times.

Explanation of Solution

The initial temperature is $300\text{ K}$ .

The final temperature is $600\text{ K}$ .

Calculate kinetic energy at $300\text{ K}$ as shown’

  $\begin{array}{c}{E}_1=\frac{3}{2}{k}_{B}T\\ =\frac{3}{2}{k}_B\times 300\end{array}$

Calculate kinetic energy at $600\text{ K}$ as shown’

  $\begin{array}{c}{E}_2=\frac{3}{2}{k}_{B}T\\ =\frac{3}{2}{k}_B\times 900\end{array}$

Calculate the factor by which kinetic energy changes as shown,

  $\begin{array}{c}\frac{E_2}{E_1}=\frac{\frac{3}{2}{k}_B\times 900}{\frac{3}{2}{k}_B\times 300}\\ =\frac{\bcancel{\frac{3}{2}{k}_B}\times 900}{\bcancel{\frac{3}{2}{k}_B}\times 300}\\ =3\end{array}$

The average kinetic energy becomes three times.