Question

FIGURE EX20.10 is a histogram showing the speeds of the molecules in a very small gas. What are

(a) the most probable speed,

(b) the average speed, and

(c) the rms speed?

Short answer

(a) The most probable speed is$\mathbf{4}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

(b) The average speed is$\mathbf{4}\mathbf{.}\mathbf{6}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}\mathbf{}$

(c) The rms speed is$\mathbf{4}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}\mathbf{}$

Step-by-step solution

Given information and Formula used

Given : FIGURE EX20.10 is a histogram showing the speeds of the molecules in a very small gas.

Theory used :

The average speed can be calculated using the following formula:

$v_{avg}=\frac{v_1{N}_1+v_2{N}_2+v_3{N}_3+v_4{N}_4}{N_1+N_2+N_3+N_4}$

We may calculate the rms speed using the following formula:

$v_{rms}=\sqrt{{(v²)}_{avg}}$

Calculating the most probable speed 

a) The most likely speed is $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$, based on the fact that$N$is the largest at the given velocity.

Calculating the average speed 

b) The average speed can be calculated using the following formula:

$v_{avg}=\frac{v_1{N}_1+v_2{N}_2+v_3{N}_3+v_4{N}_4}{N_1+N_2+N_3+N_4}$

Substituting values based on the histogram provided:

$v_{avg}=\frac{2·2+4·4+6·3+8·1}{2+4+3+1}=\mathbf{4}\mathbf{.}\mathbf{6}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

Calculating the rms speed 

c) We may calculate the rms speed using the following formula:

$$\begin{gathered} v_{rms}=\sqrt{{(v²)}_{avg}} \\ =\sqrt{\frac{{v^2}_1{N}_1+{v^2}_2{N}_2+{v^2}_3{N}_3+{v^2}_4{N}_4}{N_1+N_2+N_3+N_4}} \end{gathered}$$

Substituting values based on the histogram provided:

$$\begin{gathered} =\sqrt{\frac{2^2·2+4^2·4+6^2·3+8^2·1}{2+4+3+1}} \\ =\sqrt{\frac{244}{10}} \\ =\mathbf{4}\mathbf{.}\mathbf{94}\mathit{m}\mathbf{/}\mathit{s} \end{gathered}$$