explain how molecular movement causes the pressure exerted by a gas and derive and use the relationship pV = 31Nm<c2>, where < c2> is the mean-square speed (a simple model considering one-dimensional collisions and then extending to three dimensions

Kinetic Theory of Gases

The kinetic theory links the microscopic motion of gas molecules to the macroscopic properties

such as pressure, volume and temperature. In this note we derive the relationship

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

where p is pressure, V is volume, N the number of molecules, m the molecular mass,

and \$\langle c^{2}\rangle\$ the mean‑square speed of the molecules.

1. Assumptions of the Simple Model

• The gas consists of a large number \$N\$ of identical, spherical molecules.
• Molecules move in straight lines between perfectly elastic collisions.
• Collisions with the container walls are the only source of pressure.
• There are no intermolecular forces except during collisions.
• The distribution of molecular velocities is isotropic (no preferred direction).

2. One‑Dimensional Collision with a Wall

Consider a cubic container of side length \$L\$ (volume \$V=L^{3}\$). Focus on a single molecule

moving with velocity component \$c_{x}\$ towards the wall perpendicular to the \$x\$‑axis.

1. When the molecule strikes the wall it rebounds with the same speed but opposite direction,

   so the change in momentum is

   \$\Delta p{x}= -m c{x} - (m c{x}) = -2m c{x}.\$

   The magnitude of the impulse is \$2m|c_{x}|\$.

2. The time between successive collisions with the same wall is the time required to travel

   a distance \$2L\$ (to the opposite wall and back):

   \$\Delta t = \frac{2L}{|c_{x}|}.\$

3. The average force exerted on the wall by this molecule is

   \$F{x}= \frac{\Delta p{x}}{\Delta t}= \frac{2m c{x}^{2}}{2L}= \frac{m c{x}^{2}}{L}.\$

4. Pressure is force per unit area. The wall area is \$A = L^{2}\$, so the pressure contribution

   from this molecule is

   \$p{x}= \frac{F{x}}{A}= \frac{m c{x}^{2}}{L^{3}}= \frac{m c{x}^{2}}{V}.\$

3. Summing Over All Molecules

For \$N\$ molecules the total pressure is the sum of the individual contributions:

\$p = \frac{m}{V}\sum{i=1}^{N} c{x,i}^{2}= \frac{Nm}{V}\langle c_{x}^{2}\rangle,\$

where \$\langle c{x}^{2}\rangle\$ is the average of \$c{x}^{2}\$ over all molecules.

4. Extending to Three Dimensions

Because the velocity distribution is isotropic, the mean square speeds in each Cartesian direction are equal:

\$\langle c{x}^{2}\rangle = \langle c{y}^{2}\rangle = \langle c_{z}^{2}\rangle.\$

The total mean‑square speed is the sum of the three components:

\$\langle c^{2}\rangle = \langle c{x}^{2}\rangle + \langle c{y}^{2}\rangle + \langle c{z}^{2}\rangle = 3\langle c{x}^{2}\rangle.\$

Substituting \$\langle c_{x}^{2}\rangle = \langle c^{2}\rangle/3\$ into the pressure expression gives

\$p = \frac{Nm}{V}\frac{\langle c^{2}\rangle}{3} \quad\Longrightarrow\quad pV = \frac{1}{3}Nm\langle c^{2}\rangle.\$

5. Summary of Key \cdot ariables

6. Example Application

Calculate the pressure of \$1.0\times10^{23}\$ molecules of nitrogen (\$m = 4.65\times10^{-26}\,\$kg) in a

container of volume \$0.010\,\$m³ if the mean‑square speed is \$\langle c^{2}\rangle = 6.0\times10^{4}\,\$m²·s⁻².

1. Insert the numbers into \$pV = \frac{1}{3}Nm\langle c^{2}\rangle\$:

   \$pV = \frac{1}{3}(1.0\times10^{23})(4.65\times10^{-26})(6.0\times10^{4})\$

2. Evaluate:

   \$pV = \frac{1}{3}(1.0\times10^{23})(2.79\times10^{-21}) = 9.3\times10^{1}\ \text{Pa·m}^3\$

3. Divide by \$V = 0.010\,\$m³:

   \$p = \frac{9.3\times10^{1}}{0.010} = 9.3\times10^{3}\ \text{Pa}.\$

Thus the gas exerts a pressure of approximately \$9.3\,\$kPa.

7. Conceptual Points to Remember

• The pressure arises from momentum transfer during molecular collisions with the walls.
• Only the component of velocity normal to a wall contributes to the impulse on that wall.
• Isotropy of the velocity distribution allows us to replace \$\langle c_{x}^{2}\rangle\$ by \$\langle c^{2}\rangle/3\$.
• The derived relation connects a macroscopic thermodynamic quantity (pV) to a microscopic average

  (\$\langle c^{2}\rangle\$), forming the basis for further results such as \$p = nk_{B}T\$.