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42E

Expert-verified
Found in: Page 405

### Modern Physics

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

# Calculate the average speed of a gas molecule in a classical ideal gas. What is the average velocity of a gas molecule?

1. The average speed of gas molecule in a classical ideal gas is $\sqrt{\frac{8{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{\pi m}}}$.
2. The average velocity of gas molecule is zero.
See the step by step solution

## Step 1: Maxwell Probability Distribution.

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

Where,

m is the mass of the particle.

v is velocity of particle.

T is temperature.

kB is Boltzmann constant.

Average speed

${\mathrm{v}}_{\mathrm{avg}}={\int }_{0}^{\infty }\mathrm{vP}\left(\mathrm{v}\right)\mathrm{dv}$

Substituting expression (1) in (2).

${\mathrm{v}}_{\mathrm{avg}}={\int }_{0}^{\infty }\mathrm{v}{\left(\frac{\mathrm{m}}{2{\mathrm{\pi k}}_{\mathrm{B}}\mathrm{T}}\right)}^{\frac{3}{2}}4{\mathrm{\pi v}}^{2}{\mathrm{e}}^{-\frac{{\mathrm{mv}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\mathrm{dv}$

Let $\mathrm{b}=\frac{1}{2{\mathrm{a}}^{2}}=\frac{\mathrm{m}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}$

$\begin{array}{rcl}{\mathrm{v}}_{\mathrm{avg}}& =& 4\mathrm{\pi }{\left(\frac{\mathrm{b}}{\mathrm{\pi }}\right)}^{\frac{3}{2}}{\int }_{0}^{\infty }{\mathrm{v}}^{3}{\mathrm{e}}^{-{\mathrm{bv}}^{2}}\mathrm{dv}\\ & =& 4\mathrm{\pi }{\left(\frac{\mathrm{b}}{\mathrm{\pi }}\right)}^{\frac{3}{2}}\frac{1}{2{\mathrm{b}}^{2}}\\ & =& \sqrt{\frac{4}{\mathrm{\pi b}}}\\ & =& \sqrt{\frac{8{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{\pi m}}}\end{array}$

## Step 2: Average Velocity.

1. The average velocity of a body is the pace at which it changes position from one location to another. It's a quantity with a vector. The fact that gas molecules move in random directions is well known, and so the gas molecules have velocity in all possible directions. As a result, the vector sum of all velocities equals zero. As a result, a gas molecule's average velocity is zero.