Root-Mean-Square Speed | CAIE 9702 Paper 4 Notes

Chapters

Paper 4

Physics9702

Root-Mean-Square Speed

The molecules of a gas do not all move at the same speed.

This chapter uses one defined average speed, because it is the speed that appears in the kinetic energy equation.

What r.m.s. speed means

Square the speed of every molecule. Take the mean of all those squares. That mean is the mean-square speed $\langle c^2 \rangle$ .
Now take the square root of it. The result is the , written $c_{\text{rms}}$ .

Definition

1 mark

What is meant by root-mean-square speed?

Model answer: The square root of the mean of the squares of the molecular speeds.

c_{\text{rms}} = \sqrt{\langle c^2 \rangle}

Symbols

$c_{\text{rms}}$ = root-mean-square speed of the molecules (m s⁻¹)
$\langle c^2 \rangle$ = mean-square speed (m² s⁻²)

A plain average of the velocities would be zero, because the molecules move in all directions equally. Squaring removes the direction, so this average is useful.

Start from the mean translational kinetic-energy equation.
$\tfrac{1}{2}m\langle c^2\rangle=\tfrac{3}{2}kT$
Make $\langle c^2\rangle$ the subject (9702/41/O/N/24 Q3(b)(i)).
$\langle c^2\rangle=\dfrac{3kT}{m}$
Take the square root.
$c_{\mathrm{rms}}=\sqrt{\dfrac{3kT}{m}}$

In that equation $m$ is the mass of one molecule, in kg. If a question gives the mass in u, change it first using the conversion below.
$1\ \mathrm{u}=1.66\times10^{-27}\ \mathrm{kg}$

Worked example

Exam version: the r.m.s. speed of nitrogen

Nitrogen gas is at 210 °C. One nitrogen molecule has a mass of $4.7 \times 10^{-26}$ kg. Find the r.m.s. speed of the molecules. (9702/42/F/M/25 Q3(b)(iii))

Change the temperature to kelvin.
$T=210+273=483\ \mathrm{K}$
Substitute the molecular mass in kg.
$c_{\mathrm{rms}}=\sqrt{\dfrac{3kT}{m}}$
$3kT=2.00\times10^{-20}\ \mathrm{J}$
$c_{\mathrm{rms}}=\sqrt{\dfrac{2.00\times10^{-20}}{4.7\times10^{-26}}}$
State the speed with its unit.
$c_{\mathrm{rms}}=650\ \mathrm{m\,s^{-1}}$

Answer

c_{\text{rms}} = 650\ \text{m s}^{-1}

Common mistake

If the question asks for the mean-square speed, stop at

\langle c^2 \rangle

and give the unit m² s⁻². If it asks for the r.m.s. speed, take the square root and give the unit m s⁻¹. Papers ask for both, in alternate years.

Your turnquick check

The thermodynamic temperature of the same gas becomes four times larger. What happens to its r.m.s. molecular speed?

Show worked answer

It doubles. For a fixed molecular mass:

c_{\mathrm{rms}}\propto\sqrt{T}