Chapter 13, Problem 56P

(a)

To determine

To Calculate:The root mean square speed.

Answer to Problem 56P

The root mean square speed $\text{=461 m/s}$ .

Explanation of Solution

Given:

Temperature, $T=0°\text{C}=0+273\text{K}=273\text{K}$

And, $k\text{=1}{\text{.38×10}}^{\text{-23}}\text{J/K}$

Mass of the oxygen gas, $m\text{=16×}\left(\text{1}{\text{.66×10}}^{\text{-27}}\text{kg}\right)$

Formula used:

The root mean square speed is,

  $v_{rms}=\sqrt{\frac{3kT}{2m}}$

Here,

$k$ is boltzmann constant,

$T$ is temperature in $\text{K}$ ,

  $m$ is mass of molecule.

Calculation:

Now substitute the values in the given formula,

  $\begin{array}{l}{v}_{rms}=\sqrt{\frac{3kT}{2m}}\text{=}\sqrt{\frac{\text{3×}\left(\text{1}{\text{.38×10}}^{\text{-23}}\text{J/K}\right)\text{×}\left(\text{273K}\right)}{\text{2×}\text{16×}\left(\text{1}{\text{.66×10}}^{\text{-27}}\text{kg}\right)}}\\ {\text{v}}_{\text{rms}}\text{=}\text{461}\text{m/s}\\ \end{array}$

Conclusion:

The rms speed is $\text{=461m/s}$ .

(b)

To determine

To calculate:The time per second for the given motion.

Answer to Problem 56P

The time per second for the given motion is $\text{19}\text{round/sec}$ .

Explanation of Solution

Given:

Room’s length, $\text{a}\text{=}\text{7}\text{.0}\text{m}$

Formula used:

  $t=\frac{d}{v_x}=\frac{\sqrt{3}d}{v_{rms}}$

Here,

  $t$ is time,

  $d$ is length,

  $v_{rms}$ is root mean square speed.

Calculation:

Let us consider the particle has no fix direction. So, the velocity of the particle is written as,

  $\begin{array}{l}{v}^2{}_{rms}=v^2{}_x+v^2{}_y+v^2{}_z=3v^2{}_x\\ \sqrt{v_x}=\frac{v_{rms}}{\sqrt{3}}\\ \end{array}$

The time for crossing the room first time,

  $t=\frac{d}{v_x}=\frac{\sqrt{3}d}{v_{rms}}$

So, time taken by the particle for complete one round is,

  $t=2\frac{\sqrt{3}d}{v_{rms}}$

Therefore, the number of times per second when it will move back and forth is given as,

  $\begin{array}{l}\frac{\text{round}\text{trip}}{\text{sec}}\text{=}\frac{1}{t}=\frac{1}{2\frac{\sqrt{3}d}{v_{rms}}}=\frac{v_{rms}}{2\sqrt{3}d}\\ \text{=}\frac{\text{461 m/s}}{\text{2}\sqrt{\text{3}}\text{×}\left(\text{7}\text{.0}\text{m}\right)}\text{=}\text{19}\text{round/sec}\end{array}$

Conclusion:

The round trip per second is $\text{19}\text{round/sec}$ .