# What kinds of speeds can be found from a Maxwell-Boltzmann distribution?

Dec 23, 2017

See below.

#### Explanation:

Concerning the velocity distribution for a Maxwellian gas:

Most probable speed

• The most probable speed corresponds to the maximum of the velocity distribution, where the slope is zero. One solves the equation

$\frac{\mathrm{db} a r \left(f\right) \left(\nu\right)}{\mathrm{dn} u} = \sqrt{\frac{2}{\pi}} {\left(\frac{m}{k T}\right)}^{\frac{3}{2}} \left[2 \nu + \left(\frac{- m \nu}{k T}\right) {\nu}^{2}\right] {e}^{\left(- m {\nu}^{2}\right) / \left(2 k T\right)} = 0$

where $\overline{f} \left(\nu\right)$ is the Maxwell velocity distribution (probability distribution for a molecule's velocity) as a function of velocity $\nu$.

From this, the most probable speed, denoted ${\nu}_{\text{m.p.}}$ emerges as:

$\textcolor{b l u e}{{\nu}_{\text{m.p.}} = \sqrt{\frac{2 k T}{m}}}$

Mean speed

• An average or mean speed $< \nu >$ is computed by weighting the speed $\nu$ with its probability of occurrence $\overline{f} \left(\nu\right) \mathrm{dn} u$ and then integrating:

$< \nu > = {\int}_{0}^{\infty} \nu \overline{f} \left(\nu\right) \mathrm{dn} u = {\int}_{0}^{\infty} {e}^{\left(- m {\nu}^{2}\right) / \left(2 k T\right)} \sqrt{\frac{2}{\pi}} {\left(\frac{m}{k T}\right)}^{\frac{3}{2}} {\nu}^{3} \mathrm{dn} u$

$\implies \textcolor{b l u e}{< \nu > = \sqrt{\frac{8}{\pi}} \sqrt{\frac{k T}{m}}}$

Root mean square speed

• A calculation of $< {\nu}^{2} >$ proceeds as:

$< {\nu}^{2} > = {\int}_{0}^{\infty} {\nu}^{2} \overline{f} \left(\nu\right) \mathrm{dn} u = 3 \frac{k T}{m}$

$\implies \frac{1}{2} m < {\nu}^{2} > = \frac{3}{2} k T$

$\implies \textcolor{b l u e}{{\nu}_{\text{r.m.s}} = \sqrt{\frac{3 k T}{m}}}$

Note:

• The mean speed $< \nu >$ is 13% larger than ${\nu}_{\text{m.p.}}$ and ${\nu}_{\text{r.m.s}}$ is 22% larger.

• The common proportionality to $\sqrt{k T / m}$ has two immediate implications: higher temperature implies higher speed, and larger mass implies lower speed.

**The equivalent expressions in terms of the universal/ideal gas constant $R$ are given in the figure above.