11.3.3 Root-mean-square Speed

According to the kinetic theory, a gas consists of an unimaginably large number of gas particle in random motion. It is impossible to analyse the motion of each and every gas particle. But having large number of particles in random motion also means that the system as a whole can be modelled accurately using statistical methods.

For example, a very useful statistical average is called the mean-square speed $\left\langle {{{c}^{2}}} \right\rangle$ of the gas particles in the gas[1],

$\displaystyle \left\langle {{{c}^{2}}} \right\rangle =\frac{{{{c}_{1}}^{2}+{{c}_{2}}^{2}+{{c}_{3}}^{2}...+{{c}_{N}}^{2}}}{N}$

If we take the square root of $\displaystyle \left\langle {{{c}^{2}}} \right\rangle$ , we obtain c_rms, the root-mean-square speed of the gas particles,

$\displaystyle {{c}_{{rms}}}=\sqrt{{\left\langle {{{c}^{2}}} \right\rangle }}=\sqrt{{\frac{{{{c}_{1}}^{2}+{{c}_{2}}^{2}+{{c}_{3}}^{2}...+{{c}_{N}}^{2}}}{N}}}$

Firstly, let’s be clear that ${{c}_{{rms}}}$ is neither equal to the average velocity $\displaystyle \left\langle v \right\rangle =\frac{{{{v}_{1}}+{{v}_{2}}+{{v}_{3}}...+{{v}_{N}}}}{N}$ nor the average speed $\displaystyle \left\langle c \right\rangle =\frac{{{{c}_{1}}+{{c}_{2}}+{{c}_{3}}...+{{c}_{N}}}}{N}$ . By the way, $\left\langle v \right\rangle$ is expected to be zero since the motion of the gas particles in an ideal gas is assumed to be random in direction.

So what’s so great about ${{c}_{{rms}}}$ ? Well, it is a meaningful statistical average because it is directly related to the KE and pressure of a gas.

For a gas containing atoms of mass m, the average kinetic energy of the atoms can be expressed as

$\begin{aligned}\left\langle {KE} \right\rangle &=\left\langle {\frac{1}{2}m{{c}^{2}}} \right\rangle \\&=\frac{1}{2}m\left\langle {{{c}^{2}}} \right\rangle \text{ or }\frac{1}{2}m{{c}_{{rms}}}^{2}\end{aligned}$

And its pressure can be expressed as

$\displaystyle p=\frac{1}{3}\rho \left\langle {{{c}^{2}}} \right\rangle$

where ρ is the density of the gas. This formula will be discussed in more detail in the following section.

Example

For a mickey-mouse scenario of 5 gas particles moving along the x-axis with velocities

-2 m s^-1, -1 m s^-1, 0 m s^-1, +1 m s^-1 and +2 m s^-1

Calculate

a) mean-square speed $\displaystyle \left\langle {{{c}^{2}}} \right\rangle$ ,

b) root-mean-square speed c_rms,

c) average speed $\left\langle c \right\rangle$

d) average velocity $\left\langle v \right\rangle$

of the gas particles.

Solution

$\left\langle {{{c}^{2}}} \right\rangle =\frac{{{{{2.0}}^{2}}+{{{1.0}}^{2}}+{{{0.0}}^{2}}+{{{1.0}}^{2}}+{{{2.0}}^{2}}}}{5}=2.0\text{ }{{\text{m}}^{2}}\text{ }{{\text{s}}^{{-2}}}$