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42E

Expert-verifiedFound in: Page 405

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?**

- The average speed of gas molecule in a classical ideal gas is $\sqrt{\frac{8{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{\pi m}}}$.
- The average velocity of gas molecule is zero.

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

**Where, **

**m is the mass of the particle. **

**v is velocity of particle.**

**T is temperature. **

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

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

- 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.

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