Revise Kinetic theory of gases for Cambridge 9702 (15.3)

At its heart, the kinetic theory postulates that gases consist of countless tiny molecules in continuous, random motion. Evidence for this tireless dance comes from phenomena like Brownian motion, where larger particles suspended in a fluid are seen to jiggle erratically due to unseen molecular bombardments. This confirms the existence and constant movement of molecules, forming the bedrock of the theory.

To simplify the complex interactions within a real gas, the kinetic theory uses the concept of an ideal gas. This model relies on several crucial assumptions, which are essential for deriving the gas laws. While no real gas is perfectly ideal, these assumptions provide an excellent approximation under many conditions, especially at high temperatures and low pressures.

The pressure a gas exerts is the result of countless molecular collisions with the container walls. We can derive the pressure equation by considering a single molecule in a cubic box of side length $L$.

1. A molecule with mass $m$ and x-component of velocity $c_x$ collides elastically with a wall. Its momentum changes from $mc_x$ to $-mc_x$. The change in momentum is $\Delta p = 2mc_x$.
2. The time between two consecutive collisions with the same wall is the time taken to travel to the opposite wall and back, a distance of $2L$. So, $\Delta t = 2L/c_x$.
3. The force exerted by the molecule on the wall is the rate of change of momentum: $F = \frac{\Delta p}{\Delta t} = \frac{2mc_x}{2L/c_x} = \frac{mc_x^2}{L}$.
4. For $N$ molecules, the total force on the wall is the sum of the forces from each molecule: $F_{total} = \frac{m}{L}(c_{x1}^2 + c_{x2}^2 + ... + c_{xN}^2)$.
5. We use the mean square speed in the x-direction, $\overline{c_x^2} = \frac{\sum c_{xi}^2}{N}$, so $F_{total} = \frac{Nm\overline{c_x^2}}{L}$.
6. Since motion is random, the average motion is the same in all three directions: $\overline{c_x^2} = \overline{c_y^2} = \overline{c_z^2}$. The total mean square speed is $\overline{c^2} = \overline{c_x^2} + \overline{c_y^2} + \overline{c_z^2} = 3\overline{c_x^2}$. Therefore, $\overline{c_x^2} = \frac{1}{3}\overline{c^2}$.
7. Substituting this into the force equation gives $F_{total} = \frac{Nm}{L}(\frac{1}{3}\overline{c^2}) = \frac{1}{3}\frac{Nm\overline{c^2}}{L}$.
8. Pressure is force per unit area ($A=L^2$): $P = \frac{F_{total}}{A} = \frac{\frac{1}{3}Nm\overline{c^2}/L}{L^2} = \frac{1}{3}\frac{Nm\overline{c^2}}{L^3}$.
9. Since the volume of the cube is $V=L^3$, we arrive at the final equation.

$P = \frac{1}{3}\frac{Nm\overline{c^2}}{V} \quad \text{or} \quad pV = \frac{1}{3}Nm\overline{c^2}$

Here, $N$ is the total number of gas molecules, $m$ is the mass of a single molecule, $V$ is the volume of the gas, and $\overline{c^2}$ represents the mean square speed of the gas molecules. The mean square speed is the average of the squares of the speeds of all the individual molecules.

One of the most profound insights from the kinetic theory is the direct link between the absolute temperature of a gas and the average translational kinetic energy of its molecules. Simply put, a hotter gas means its molecules are, on average, moving faster and therefore possess more kinetic energy.

The average translational kinetic energy ($E_k$) of a single gas molecule is given by:

$E_k = \frac{1}{2}m\overline{c^2}$

By combining the kinetic theory equation ($pV = \frac{1}{3}Nm\overline{c^2}$) with the ideal gas law in terms of molecules ($pV = NkT$), we can establish a direct link between energy and temperature.

$NkT = \frac{1}{3}Nm\overline{c^2} \implies kT = \frac{1}{3}m\overline{c^2}$

Rearranging this gives the crucial relationship:

$\frac{3}{2}kT = \frac{1}{2}m\overline{c^2} = E_k$

This shows that the average translational kinetic energy of a molecule is directly proportional to the absolute temperature. The Boltzmann constant ($k \approx 1.38 \times 10^{-23} \text{ J K}^{-1}$) is the constant of proportionality. Remember, all temperature values used in these calculations MUST be in Kelvin (K).