The Timpson Lyceum · University Physics I (2025)

Thermal Properties of Matter - Lecture 18 (University Physics I)

Overview

This lecture launches the study of thermodynamics by describing the macroscopic behavior of gases and connecting it to the microscopic motion of their molecules. We start with the ideal-gas law as an empirical equation of state, rewrite it in several useful forms, and note the more accurate Van der Waals correction. We then build the kinetic-molecular model, deriving the link between temperature and molecular kinetic energy and defining the root-mean-square (rms) speed. Finally we use the equipartition of energy to explain the measured molar heat capacities of gases and solids, and close with the phases of matter and how to read a phase diagram. The payoff is a single picture in which temperature, pressure, and molecular jiggling are all the same story.

Key Concepts

Notation used throughout

• $N$ — the actual number of molecules (a molecule is an atom or a bonded group of atoms).
• $n$ — the number of moles.
• $N_A$ — Avogadro's number, the number of molecules in one mole, $N_A \approx 6.022\times10^{23}\ \text{mol}^{-1}$. A mole is a quantity of a substance, not just a pure number, so $N = nN_A$.
• $M$ — molar mass (kg/mol in this course; periodic tables list g/mol).
• $m$ — mass of one molecule; $m_{\text{total}}$ — mass of the whole gas sample.
• $R$ — universal gas constant, $R = 8.314\ \text{J/(mol}\cdot\text{K)}$.
• $k$ — Boltzmann's constant, $k = R/N_A \approx 1.381\times10^{-23}\ \text{J/K}$.
• Mass relations: $m_{\text{total}} = Nm = nM$.

Equation of state. An equation of state gives the state of a system from its variables. For a solid, volume responds to temperature (thermal expansion) and pressure (compression):

$V = V_0\left(1 + \beta\,\Delta T - \kappa\,\Delta P\right)$

where $\beta$ is the volume thermal-expansion coefficient and $\kappa$ the compressibility (the negative sign reflects that higher pressure shrinks volume).

Ideal-gas law. A gas is ideal when its temperature is high and its molecules are widely spaced (low density / low pressure) — ordinary room air is close to ideal.

• $PV = nRT$ (moles form).
• Same side of the equation $\Rightarrow$ inversely related; opposite sides $\Rightarrow$ directly related. So $V$ rises with $T$ and with $n$, and falls as $P$ rises.
• Constant-$n$ form (the gas amount is fixed): $R$ cancels, giving $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}.$
• Molecule form: since $N = nN_A$ and $k = R/N_A$, $\;PV = NkT$.
• Density form: with $n = m_{\text{total}}/M$ and $\rho = m_{\text{total}}/V$, $\;\displaystyle \rho = \frac{PM}{RT}$.
• Always use absolute temperature in kelvin ($T_K = T_C + 273.15$). The only exception is a change in temperature.

Van der Waals equation. A more accurate equation of state: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT.$

• The $nb$ term subtracts the volume actually occupied by the molecules, so $V - nb$ is the free space; $b$ is larger for physically bigger molecules (size, not mass, matters).
• The $an^2/V^2$ term accounts for weak intermolecular attraction, which slightly lowers the pressure the gas exerts on the walls. The attraction effect scales with density squared because only nearby neighbors pull on a given molecule.
• When $n$ is small compared with $V$, both correction terms vanish and the equation reduces to $PV = nRT$.

Kinetic-molecular model. Treat the gas as many tiny molecules bouncing elastically inside a box; pressure is the average force per area from those impacts. Each wall collision reverses one velocity component, $\Delta p = 2mv$; a molecule re-hits the same wall every $\Delta t = 2L/v$, so it delivers an average force $mv^2/L$. Summing over molecules and using $P = F/A$ leads to the central result: $K_{\text{total}} = \tfrac{3}{2}nRT.$

• Temperature is a measure of (translational) kinetic energy. This counts translation only, not rotation or vibration.
• Average translational KE of one molecule: $\displaystyle \overline{K} = \tfrac{3}{2}kT$.
• At equal temperature, gases have equal average KE; a heavier gas therefore moves slower than a lighter one.

Root-mean-square speed. The representative molecular speed is found by averaging the squares of the speeds, then taking the root — because $v^2$ (not $v$) sets the kinetic energy: $v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}.$ The rms speed is slightly larger than the plain arithmetic mean and is the one that reproduces the correct kinetic energy.

Mean free path and collisions. Modeling molecules as spheres of radius $r$ and tracking one moving through stationary neighbors:

• Average time between collisions and average distance traveled between collisions (the mean free path $\lambda$): $\lambda = \frac{V}{4\pi\sqrt{2}\,r^2 N} = \frac{kT}{4\pi\sqrt{2}\,r^2 P}.$
• Letting all molecules move (rather than just one) only introduces the factor of $\sqrt{2}$.

Heat capacities and equipartition. $C_V$ is the molar heat capacity at constant volume (volume must be fixed to even define a gas's heat capacity). By the equipartition principle, each available degree of freedom (a natural place for energy to go) takes $\tfrac12 R$ per mole:

• Monatomic gas (He, Ar): 3 translational degrees $\Rightarrow C_V = \tfrac{3}{2}R \approx 12.47\ \text{J/(mol}\cdot\text{K)}$. Spheres don't pick up spin from random collisions.
• Diatomic gas (H$_2$, N$_2$, O$_2$): 3 translational + 2 rotational $\Rightarrow C_V = \tfrac{5}{2}R \approx 20.8\ \text{J/(mol}\cdot\text{K)}$. (A diatomic molecule won't spin about its long axis from collisions.)
• Polyatomic gas (CO$_2$, SO$_2$): 3 translational + 3 rotational $\Rightarrow C_V \approx 3R \approx 24.9\ \text{J/(mol}\cdot\text{K)}$.
• Vibration adds further degrees of freedom and explains why the measured values vary among molecules with the same atom count.
• Solids (atoms as masses on springs): 3 kinetic + 3 potential degrees of freedom $\Rightarrow C \approx 3R \approx 25\ \text{J/(mol}\cdot\text{K)}$ for nearly all solids (the Dulong–Petit value).

Phases of matter. The interatomic force is strongly repulsive at very short range and weakly attractive farther out, with a potential-energy minimum at an equilibrium separation. Gases sit far out where attraction is negligible; solids are held near the minimum by bonds; liquids teeter near the minimum and are messy to model.

Phase diagram. A plot of $P$ versus $T$ showing which phase (solid, liquid, gas) is stable, separated by transition lines.

• Triple point: the unique $(P,T)$ where solid, liquid, and gas coexist. Below the triple-point pressure a substance cannot exist as a liquid — e.g. CO$_2$ (triple point $\approx 5$ atm, $\approx 216.55\ \text{K}$) sublimes directly from solid to gas at 1 atm.
• Critical point: beyond it the liquid–gas boundary disappears; the substance changes between gas-like and liquid-like states with no sharp transition.

Worked Example

Final temperature of a compressed gas. An engine compresses a fixed amount of gas to $\tfrac19$ of its volume. The initial state is $T_1 = 27^\circ\text{C} = 300.15\ \text{K}$, $P_1 = 1\ \text{atm}$; the final pressure is $P_2 = 21.7\ \text{atm}$. Find $T_2$.

Since $n$ is constant, use $\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$ and solve for $T_2$. With $V_2 = \tfrac19 V_1$, the volumes enter only as a ratio, so the actual values cancel:

rms speed of oxygen at $27^\circ$C. Treat oxygen as O$_2$, so $M = 0.032\ \text{kg/mol}$, $T = 300\ \text{K}$:

Summary

• The ideal-gas law $PV = nRT$ is an empirical equation of state; it also takes the forms $PV = NkT$, $\rho = PM/RT$, and (at fixed $n$) $P_1V_1/T_1 = P_2V_2/T_2$. Always work in kelvin.
• The Van der Waals equation corrects for finite molecular size ($nb$) and intermolecular attraction ($an^2/V^2$), reducing to the ideal-gas law in the dilute limit.
• The kinetic model gives $K_{\text{total}} = \tfrac32 nRT$ and $\overline{K} = \tfrac32 kT$: temperature is average translational kinetic energy, so at equal $T$ heavier molecules move slower.
• The rms speed $v_{\text{rms}} = \sqrt{3RT/M} = \sqrt{3kT/m}$ is the speed that reproduces the correct kinetic energy; the mean free path $\lambda = kT/(4\pi\sqrt{2}\,r^2 P)$ sets the typical distance between collisions.
• Equipartition ($\tfrac12 R$ per degree of freedom) explains $C_V$: $\tfrac32 R$ for monatomic, $\tfrac52 R$ for diatomic, $\approx 3R$ for polyatomic gases and for solids.
• A phase diagram maps solid/liquid/gas regions in $P$–$T$ space; the triple point is where all three coexist and the critical point is where the liquid–gas distinction vanishes.