Simple Models of Matter and Boltzmann Factor

Gas Laws

When describing an ideal gas, the following assumptions are made:

• Gas particles take up negligible volume
• All collisions between particles and the walls of the container and each other are completely elastic
• There are negligible forces between particles except during collisions
• All particles of a particular gas are identical
• The internal energy of the gas is entirely kinetic
• Newton’s laws of motion apply
• Gravitational, electrostatic and Van de Waals forces can be ignored
• The motion of all molecules is random
• All molecules travel in straight lines. IS: The top three assumptions in bold are mentioned explicitly in the specification.

Boyle’s Law states that for a constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure:

$$pV=\text{constant}p\propto \frac{1}{V}$$

Avogadro’s Law states that for a constant pressure and temperature, the volume of a gas is directly proportional to the number of moles:

$$V\propto n$$

Temperature is a measure of the average kinetic energy of the particles in a substance. Due to the random nature of collisions between particles in a gas, there is a distribution of speeds, and therefore kinetic energies of particles. The range of speeds are represented by the Maxwell Boltzmann distribution.

Charles’ Law states that for a constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature:

$$V\propto T$$

For $T$ in Kelvin (K).

By combining these laws, we get: $PV\propto Tn⟹PV=nRT$ Where $R$ is the ideal gas constant, with a value of $8.314\text{J}{\text{mol}}^{-1}{\text{K}}^{-1}$.

The Boltzmann Constant is similar to the gas constant, $R$, but only for one particle. It is equal to the gas constant divided by $N_a$, the Avogadro constant:

$$k=\frac{R}{N_a}=1.38\times 10^{-23}\text{J}{\text{K}}^{-1}$$

By combining $N=n{N}_a$ with $k=\frac{R}{N_a}$, we get: $nR=Nk$, so we can rewrite the ideal gas equation as:

$$pV=NkT$$

At a constant temperature, the distribution of speeds will remain the same, as well as the average speed. The root mean square speed is:

$$c_{rms}=\sqrt{\frac{c_1^2+c_2^2+\cdots +c_N^2}{N}}=\sqrt{\overline{(c^2)}}$$ $$pV=\frac{1}{3}Nm{c_{rms}}^2$$

Temperature of a gas is proportional to the average kinetic energy of its particles:

$$\begin{array}{rl}\mathrm{Average}\mathrm{KE}& =\frac{3}{2}kT\\ \mathrm{Total}\mathrm{KE}& =\frac{3}{2}nRT\end{array}$$

For monatomic particles (and one dimension), a simple estimation is often used:

$$E_k\approx kT$$

Gas Law Derivation

NIS: Consider a particle of mass $m$ moving at velocity $v$ in a cubic box with side length $x$.

• Assume that the particle is moving only perpendicular to a given wall. In general, one third of the total kinetic energy is in each perpendicular direction¹.
• When the particle collides with a wall, it applies an impulse of $2mv$.
• For a given wall, this happens every $\frac{2x}{v}$ seconds, so there are $\frac{v}{2x}$ collisions with a given wall per second.
• The force exerted on this wall is equal to the rate of change of impulse, so $F=\frac{1}{3}\times 2mv\times \frac{v}{2x}=\frac{m{v}^2}{3x}$
• The pressure is equal to force per unit area, so $p=\frac{m{v}^2}{3x}÷x^2=\frac{m{v}^2}{3x^3}$
• Rearranging, we find $p{x}^3=\frac{1}{3}m{v}^2$
• $x^3$ is just the volume, so $pV=\frac{1}{3}m{v}^2$ for a single particle
• Considering $N$ particles each with mean squared speed $\overline{c^2}$, we get $pV=\frac{1}{3}Nm\overline{c^2}$.

Equating $pV=\frac{1}{3}Nm\overline{c^2}=NkT$, we find:

• $\frac{1}{3}m\overline{c^2}=kT$ (cancelling $N$ from both sides)
• $\frac{2}{3}(\frac{1}{2}m\overline{c^2})=kT$ (pulling out a factor of $\frac{1}{2}$ for KE)
• $\text{KE}=\frac{3}{2}kT$

Specific Heat Capacity

Specific heat capacity is a measure of the energy needed to raise the temperature of $1\text{kg}$ of a substance by $1°\text{C}$. The equation for SHC is:

$$\Delta E=mc\Delta \theta$$

Brownian Motion

Particles in a liquid or gas undergo random motion because fast free-moving molecules move around and collide with each other. The mean free path is the average distance a particle travels before colliding (the size of a ‘step’). If a particles takes $N$ steps, on average its end point is $\sqrt{N}$ steps away from its starting point.

Boltzmann Factor

For many physical processes, a certain amount of energy is needed for the process to take place. This is the activation energy E; for example, in chemistry, reagents are often heated so they have enough kinetic energy to react.

Other physical processes requiring an activation energy are:

• Changes of state
• Thermionic emission, where electrons are emitted from a heated metal surface
• Ionisation, where electrons are removed from an atom
• Conduction in semiconductors
• Viscous flow
• Nuclear fusion IS: The first five of these processes are mentioned explicitly in the specification.

The Boltzmann factor, $f$, is defined as:

$$f=e^{-\frac{E}{kT}}=\exp \left(-\frac{E}{kT}\right)$$

The graph below shows how the value of $f$ varies as $T$ increases, asymptotically approaching $f=1$.

Many physical processes require energies between $15kT$ and $30kT$, for example water evaporation requires around $18kT$ at freezing. The graph below shows varying energy regimes for physical and chemical changes.

Footnotes

1. Footnote on kinetic energies in perpendicular directions Kinetic energies in perpendicular directions add (by the Pythagorean theorem), so on average a third of the total kinetic energy is in any given direction. ↩