How Molecules Cause Pressure | CAIE 9702 Paper 4 Notes

Chapters

Paper 4

Physics9702

How Molecules Cause Pressure

A gas pushes on the walls of its container because its molecules keep hitting the walls.

Say that in the words the mark scheme wants. Then turn the same idea into an equation.

Explain pressure in words

The molecules collide with the walls of the container.
The momentum of a molecule changes during each collision with a wall.
That change of momentum is caused by a force from the wall on the molecule.
By Newton's third law, the molecule exerts an equal and opposite force on the wall.
A great many molecules hit the wall each second. Their forces add up over the area of the wall, and pressure is that force divided by the area.
Any three of those five statements score the three marks (9702/42/O/N/24 Q4(b)).

From one molecule to the whole gas

Swipe left or right to see the whole diagram →

The starting picture for the kinetic model: one molecule of mass m and speed c, bouncing straight between two opposite walls of a cube of side l. Each hit on the shaded wall changes the momentum of the molecule by 2mc, and the hits repeat every 2l/c seconds.

Take one molecule of mass $m$ moving at speed $c$ straight at one wall of a cube of side $l$ . Its momentum reverses during an elastic collision.
$\Delta p=(-mc)-(+mc)=-2mc$
The size of the momentum change is $2mc$ .
Between two hits on the same wall the molecule crosses the cube and comes back, a distance $2l$ at speed $c$ .
$\Delta t=\dfrac{2l}{c}$
Force is the rate of change of momentum.
$F=\dfrac{2mc}{2l/c}=\dfrac{mc^2}{l}$
The wall has area $l^2$ , so the pressure from this one molecule is found from force divided by area.
$p=\dfrac{mc^2/l}{l^2}=\dfrac{mc^2}{l^3}$
A real gas has $N$ molecules, and they do not all have the same speed. Replace $c^2$ by the mean of the squares, $\langle c^2 \rangle$ , and multiply by $N$ .
$p=\dfrac{Nm\langle c^2\rangle}{l^3}$
The molecules move equally in three directions, so only one third of the motion is towards this pair of walls. Divide by 3. Finally $l^3$ is the volume $V$ .

pV = \tfrac{1}{3}Nm\langle c^2 \rangle

Symbols

$p$ = pressure of the gas (Pa)
$V$ = volume of the gas (m³)
$N$ = number of molecules (no unit)
$m$ = mass of one molecule (kg)
$\langle c^2 \rangle$ = mean-square speed: the mean of the squares of the molecular speeds (m² s⁻²)

Definition

1 mark

What is meant by mean-square speed?

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

Those two symbol meanings are asked directly, in exactly those words (9702/42/M/J/25 Q4(b)(i)).
This equation is printed on the Paper 4 formula sheet, so you do not have to memorise it. You do have to know what each symbol means, and be able to explain the steps above.
A second form. The quantity $Nm$ is the total mass of the gas, so $Nm/V$ is the density $\rho$ . Dividing the equation by $V$ gives the density form below. Use it when the question gives density instead of a molecule count.
$p=\tfrac{1}{3}\rho\langle c^2\rangle$

Common mistake

\langle c^2 \rangle

is the mean of the squares of the speeds. It is not the square of the mean speed. Square each speed first, then take the mean. In the derivation, the mark most often lost is the reason for the

\tfrac{1}{3}

: say that the motion is shared equally between three directions.

Your turnquick check

For the same gas in the same volume, the r.m.s. molecular speed doubles. What happens to the pressure?

Show worked answer

The pressure becomes four times larger. When $N$ , $m$ and $V$ are fixed:

p\propto c_{\mathrm{rms}}^2