Chapter 13, Problem 52P

To determine

To Calculate: The change in root mean square speed.

Answer to Problem 52P

If the pressure is doubled then the root mean speed is changed by the factor of $1.41$ .

Explanation of Solution

Given info:

• Pressure, ${\text{P}}_{\text{1}}\text{=P}\text{and}{\text{P}}_{\text{2}}\text{=}\text{2P}$ .
• Volume is constants.

Formula used:

The root mean square speed of the gas molecules is,

  ${\text{v}}_{\text{rms}}\text{=}\sqrt{\frac{\text{3kT}}{\text{m}}}$

Here, $k$ is the Boltzmann constant,

  $T$ is the temperature of the gas,

  $m$ is mass of the gas.

Calculation:

The rms value for the first case,

  ${\left({\text{v}}_{\text{rms}}\right)}_{\text{1}}\text{=}\sqrt{\frac{{\text{3kT}}_{\text{1}}}{\text{m}}}$ .........Equation (1)

And, also for the second case,

  ${\left({\text{v}}_{\text{rms}}\right)}_{\text{2}}\text{=}\sqrt{\frac{{\text{3kT}}_{\text{2}}}{\text{m}}}$ .........Equation (2)

When volume and amount of the gas molecules are constant then the pressure is directly varying with the temperature such that,

  $\begin{array}{l}P=\frac{nRT}{V}=\left(\text{constant}\right)T\\ P=\left(\text{constant}\right)T\end{array}$

Therefore, if pressure is doubled then the temperature is also doubled.

So, the rms value for the first case,

  ${\left({\text{v}}_{\text{rms}}\right)}_{\text{1}}\text{=}\sqrt{\frac{{\text{3kT}}_{\text{1}}}{\text{m}}}\text{=}\sqrt{\left(\text{constant}\right){\text{×T}}_{\text{1}}}$

The rms speed for the second case,

  $\begin{array}{l}{\left({\text{v}}_{\text{rms}}\right)}_{\text{2}}\text{=}\sqrt{\frac{{\text{3kT}}_{\text{2}}}{\text{m}}}\text{=}\sqrt{\left(\text{constant}\right){\text{×T}}_{\text{2}}}\\ \text{=}\sqrt{\left(\text{constant}\right){\text{×2T}}_{\text{1}}}\left(\because {\text{T}}_{\text{2}}{\text{=2T}}_{\text{1}}\right)\\ {\left({\text{v}}_{\text{rms}}\right)}_{\text{2}}\text{=}\sqrt{\text{2}}{\left({\text{v}}_{\text{rms}}\right)}_{\text{1}}\text{=1}\text{.41}{\left({\text{v}}_{\text{rms}}\right)}_{\text{1}}\end{array}$

Conclusion:

The rms speed is multiplied by the factor of $1.41$ .