16.1 Pressure Exerted by Gas Molecules

Overview

The pressure a gas exerts on the walls of its container is a macroscopic consequence of microscopic molecular motion. Kinetic Molecular Theory (KMT) provides a quantitative link between molecular behaviour and measurable gas pressure.

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Assumptions of the Kinetic Theory (Ideal Gas)

The derivation rests on the following idealising assumptions:

1. A gas consists of a very large number $N$ of identical molecules, each of mass $m$.
2. Molecules are in continuous, random motion.
3. The volume of the molecules themselves is negligible compared with the volume of the container.
4. Intermolecular forces are negligible except during collisions.
5. All collisions (molecule–molecule and molecule–wall) are perfectly elastic.
6. The duration of a collision is negligible compared with the time between collisions.
7. Molecular motion is isotropic — no preferred direction.

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How Molecular Motion Causes Pressure

When a molecule strikes a wall and rebounds elastically, it undergoes a change in momentum. By Newton's second law:

$F=\frac{\Delta p}{\Delta t}$

The continuous bombardment of the wall by a vast number of molecules produces a steady average force. Pressure is this average force per unit area:

$P=\frac{F}{A}$

Key points:

• A single collision transfers momentum $2m{v}_x$ to the wall (for a molecule moving with speed $v_x$ perpendicular to the wall).
• The rate of such collisions per unit area determines the pressure.
• Higher molecular speeds → greater momentum transfer → higher pressure.

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Derivation of $pV=\frac{1}{3}Nm⟨v^2⟩$

Step 1 — Force from one molecule

Consider a cubic box of side $L$ containing $N$ molecules. For a single molecule moving with velocity component $v_x$ along the x-axis:

• Momentum change per collision with the right wall: $\Delta p=2m{v}_x$
• Time between successive collisions with the same wall: $\Delta t=\frac{2L}{v_x}$
• Force on wall from this molecule: $f=\frac{2m{v}_x}{2L/v_x}=\frac{m{v}_x^2}{L}$

Step 2 — Total force from all molecules

Summing over all $N$ molecules:

$F_x=\frac{m}{L}{\sum }_{i=1}^N{v}_{xi}^2=\frac{mN}{L}⟨v_x^2⟩$

Step 3 — Isotropy gives the factor $\frac{1}{3}$

Because molecular motion is isotropic (random, no preferred direction):

$⟨v_x^2⟩=⟨v_y^2⟩=⟨v_z^2⟩=\frac{⟨v^2⟩}{3}$

where $⟨v^2⟩=⟨v_x^2⟩+⟨v_y^2⟩+⟨v_z^2⟩$ is the mean square speed.

Step 4 — Pressure

Pressure on the wall of area $A=L^2$:

$P=\frac{F_x}{L^2}=\frac{mN⟨v_x^2⟩}{L^3}=\frac{mN}{V}\cdot \frac{⟨v^2⟩}{3}$

$\boxed{P=\frac{1}{3}\frac{Nm}{V}⟨v^2⟩}$

Multiplying both sides by $V$:

$\boxed{PV=\frac{1}{3}Nm⟨v^2⟩}$

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Alternative Forms

In terms of density

Since mass density $\rho =\frac{Nm}{V}$:

$\boxed{P=\frac{1}{3}\rho ⟨v^2⟩}$

In terms of average translational kinetic energy

Rewriting $\frac{1}{2}m⟨v^2⟩$ as the average KE per molecule:

$P=\frac{2}{3}\frac{N}{V}⟨\frac{1}{2}m{v}^2⟩=\frac{2}{3}{N}_0⟨KE⟩$

where $N_0=N/V$ is the number density.

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Linking Pressure to Temperature

Comparing $PV=\frac{1}{3}Nm⟨v^2⟩$ with the ideal gas equation $PV=NkT$:

$NkT=\frac{2N}{3}⟨\frac{1}{2}m{v}^2⟩$

$\boxed{⟨\frac{1}{2}m{v}^2⟩=\frac{3}{2}kT}$

where $k=1.38\times 10^{-23}{\text{ J K}}^{-1}$ is the Boltzmann constant.

This is the fundamental result of kinetic theory: absolute temperature is directly proportional to the average translational kinetic energy per molecule.

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Summary Table