Root Mean Square (rms) Speed

Definition

The root mean square speed of the molecules of a gas is the square root of the mean of the squares of the speeds of all the molecules:

$$v_{rms} = \sqrt{\overline{v^{2}}} = \sqrt{\dfrac{v_{1}^{2} + v_{2}^{2} + \cdots + v_{N}^{2}}{N}}$$

It is a kind of average speed at a given temperature, and it is the average that appears naturally in the expression for pressure.

Derivation: rms speed is proportional to the square root of absolute temperature

i. The average pressure of the gas from kinetic theory is

$$P = \dfrac{1}{3}\dfrac{N}{V}m\overline{v^{2}}$$

ii. Hence the mean square speed of the molecules is

$$\overline{v^{2}} = \dfrac{3PV}{Nm}$$

iii. Using the ideal gas equation $PV = nRT$,

$$\overline{v^{2}} = \dfrac{3nRT}{Nm}$$

But the number of moles is $n = \dfrac{N}{N_{A}}$, so

$$\overline{v^{2}} = \dfrac{3NRT}{N_{A}Nm} = \dfrac{3RT}{N_{A}m}$$

Also $mN_{A} = M_{0}$, the molar mass of the gas. Therefore

$$\overline{v^{2}} = \dfrac{3RT}{M_{0}}$$

iv. Taking the square root,

$$v_{rms} = \sqrt{\overline{v^{2}}} = \sqrt{\dfrac{3RT}{M_{0}}}$$

For a given gas $R$ and $M_{0}$ are constants, so

$$v_{rms} \propto \sqrt{T}$$

which is the required result.

Equivalent forms

• Per molecule: dividing numerator and denominator by $N_{A}$, $v_{rms} = \sqrt{\dfrac{3k_{B}T}{m}}$, where $m$ is the mass of one molecule.

• In terms of pressure and density: since $P = \dfrac{1}{3}\rho\overline{v^{2}}$, $v_{rms} = \sqrt{\dfrac{3P}{\rho}}$.

• For comparing two gases, $v_{rms} \propto \sqrt{\dfrac{T}{M_{0}}}$, so

$$\dfrac{v_{1}}{v_{2}} = \sqrt{\dfrac{T_{1}}{M_{1}} \times \dfrac{M_{2}}{T_{2}}}$$

At the same temperature the lighter gas has the greater rms speed; for example, oxygen ($M_{0} = 32$) and sulphur dioxide ($M_{0} = 64$) at S.T.P. give $\dfrac{v_{O_{2}}}{v_{SO_{2}}} = \sqrt{\dfrac{64}{32}} = \sqrt{2}$.

Mean speed and rms speed are different

For a small set of listed speeds, the mean speed is $\bar{v} = \dfrac{v_{1} + v_{2} + \cdots + v_{N}}{N}$, while the rms speed averages the squares. For speeds 2, 3 and 4 m s$^{-1}$, $\bar{v} = 3$ m s$^{-1}$ but $v_{rms} = \sqrt{\dfrac{29}{3}} = 3.109$ m s$^{-1}$. The rms speed is the larger of the two unless all the speeds are equal.

Distribution of speeds of molecules

The molecules of a gas are in continuous random motion and their speeds vary over the whole range from zero to infinity. The function giving the number of molecules as a function of speed is called the distribution of speeds, and for a gas at temperature $T$ it is Maxwell's distribution of molecular speeds.

Diagram in words: a curve of $n_{v}$ (number of molecules per unit range of speed) against speed $v$, starting from zero at $v = 0$, rising to a single peak and falling away gradually at large $v$, with a long tail. The shaded strip of width $dv$ under the curve, of area $n_{v}\,dv$, is the number of molecules having speeds between $v$ and $v + dv$. Once the distribution is known, average values of physical quantities such as $\overline{v^{2}}$ can be calculated from it.

Comparison with the speed of sound

The speed of sound in a gas is $v_{s} = \sqrt{\dfrac{\gamma RT}{M_{0}}}$, where $\gamma = \dfrac{C_{P}}{C_{V}}$ is the adiabatic ratio, whose maximum value is $5/3$ for monatomic gases. Since $\gamma < 3$, the sound wave cannot move faster than the rms speed of the molecules, though the two speeds are of the same order of magnitude. The molecules serve as the medium that transports the sound energy: the speed of sound in H$_2$ is comparable to the rms speed of H$_2$ molecules, and in N$_2$ to that of N$_2$ molecules.

Standard results

$$v_{rms} = \sqrt{\dfrac{3RT}{M_{0}}} = \sqrt{\dfrac{3k_{B}T}{m}} = \sqrt{\dfrac{3P}{\rho}}, \qquad v_{rms} \propto \sqrt{T}, \qquad v_{rms} \propto \dfrac{1}{\sqrt{M_{0}}}$$

$$\overline{v^{2}} = \dfrac{3PV}{Nm} \propto T$$

• Typical values: $v_{rms} = 517$ m s$^{-1}$ for nitrogen and $483$ m s$^{-1}$ for oxygen, both at 300 K.