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43E

Expert-verified
Found in: Page 405

### Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

# Using the Maxwell speed distribution, determine the most probable speed of a particle of mass m in a gas at temperature THow does this compare with vrms ? Explain.

1. The most probable speed is $\sqrt{\frac{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}$.
2. The most probable speed is $\sqrt{\frac{2}{3}}$ times ${\mathrm{v}}_{\mathrm{rms}}$.
See the step by step solution

## Step 1: Maxwell Probability Distribution.

1. ${\mathbf{P}}\left(v\right){\mathbf{=}}{\left(\frac{m}{2{\mathrm{\pi k}}_{B}T}\right)}^{\frac{\mathbf{3}}{\mathbf{2}}}{\mathbf{4}}{{\mathbf{\pi v}}}^{{\mathbf{2}}}{{\mathbf{e}}}^{\mathbf{-}\frac{{\mathbf{mv}}^{\mathbf{2}}}{\mathbf{2}{\mathbf{k}}_{\mathbf{B}}\mathbf{T}}}$…..(1)

Where,

m is the mass of the particle.

v is velocity of particle.

T is temperature.

kB is Boltzmann constant.

$\begin{array}{rcl}\frac{d\mathrm{P}}{d\mathrm{v}}& =& \frac{\mathrm{d}}{\mathrm{dv}}\left[\sqrt{\frac{2}{\mathrm{\pi }}}{\left(\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}^{\frac{3}{2}}{\mathrm{v}}^{2}{\mathrm{e}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\right]\\ \frac{d\mathrm{P}}{d\mathrm{v}}& =& \sqrt{\frac{2}{\mathrm{\pi }}}{\left(\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}^{\frac{3}{2}}\frac{\mathrm{d}}{\mathrm{dv}}\left[{\mathrm{v}}^{2}{\mathrm{e}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\right]\\ \frac{d\mathrm{P}}{d\mathrm{v}}& =& \sqrt{\frac{2}{\mathrm{\pi }}}{\left(\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}^{\frac{3}{2}}\left[2{\mathrm{ve}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}-\left(2\mathrm{v}\right)\left(\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right){\mathrm{v}}^{2}{\mathrm{e}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\right]\\ \frac{d\mathrm{P}}{d\mathrm{v}}& =& 2\sqrt{\frac{2}{\mathrm{\pi }}}{\left(\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}^{\frac{3}{2}}\left[1-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right]{\mathrm{ve}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\\ 0& =& 2\sqrt{\frac{2}{\mathrm{\pi }}}{\left(\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}^{\frac{3}{2}}\left[1-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right]{\mathrm{ve}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\\ 0& =& 1-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\\ \frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}& =& 1\\ {\mathrm{v}}^{2}& =& \frac{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}\\ \mathrm{v}& =& \sqrt{\frac{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}\end{array}$

Therefore, the most probable speed is $\sqrt{\frac{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}$.

## Step 2: Mathematical Expression of rms Speed.

${\mathrm{v}}_{\mathrm{rms}}=\sqrt{\frac{3{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}$

Rearrange for $\sqrt{\frac{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}$,

$\sqrt{\frac{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}=\frac{{\mathrm{v}}_{\mathrm{rms}}}{\sqrt{3}}$

The mathematical expression for the most probable speed is,

$\mathrm{v}=\sqrt{\frac{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}$

Substitute $\frac{{\mathrm{v}}_{\mathrm{rms}}}{\sqrt{3}}$ for $\sqrt{\frac{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{m}}}$,

$\begin{array}{rcl}\mathrm{v}& =& \sqrt{2}\frac{{\mathrm{v}}_{\mathrm{rms}}}{\sqrt{3}}\\ & =& \sqrt{\frac{2}{3}}{\mathrm{v}}_{\mathrm{rms}}\end{array}$

Therefore, the most probable speed is $\sqrt{\frac{2}{3}}$ times vrms.