Revise Gas laws for IB Physics (B.3)

To make a gas of $10^{23}$ molecules tractable, physics uses a deliberately simplified picture called the ideal gas model. It keeps the features that matter — many molecules, constant motion — and throws away the complications of molecular size and intermolecular forces. An ideal gas is a theoretical gas that obeys this model exactly. No real gas is perfectly ideal, but under everyday conditions the model is an excellent approximation, and it is the basis of every equation in this topic.

Pressure is where the model earns its keep. When a molecule strikes a wall and rebounds, its momentum reverses, so the wall must have exerted a force on it — and by Newton's third law the molecule pushes back on the wall with an equal and opposite force. One collision is imperceptible, but a gas produces an unimaginable number of collisions every second, and the steady average of all those forces spread over the wall area is exactly what we measure as pressure ($P = F/A$). This single idea explains the gas laws qualitatively: anything that makes collisions more frequent or more forceful raises the pressure.

Long before the molecular picture was accepted, experimenters found the laws by changing one quantity while holding another fixed. Each law describes a fixed mass (fixed amount) of gas and holds one quantity constant. Read them together and they are simply three faces of the same underlying relationship.

At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume: halve the volume and you double the pressure. Molecularly, squeezing the gas into a smaller space means each molecule reaches a wall sooner, so collisions are more frequent and the pressure climbs.

$P \propto \frac{1}{V} \quad \text{(constant } T, n) \;\; \implies \;\; P_1V_1 = P_2V_2$

On a graph, plotting $P$ against $V$ gives a curve (a hyperbola) falling away from both axes. To get a straight-line test of the law, plot $P$ against $1/V$: the result is a straight line through the origin, whose gradient is the constant value of $PV$.

At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature: heat it and it expands, cool it and it contracts. For the pressure to stay constant while the faster molecules push outward, the gas must take up more room, so the volume grows in step with the Kelvin temperature.

$V \propto T \quad \text{(constant } P, n) \;\; \implies \;\; \frac{V_1}{T_1} = \frac{V_2}{T_2}$

A graph of $V$ against absolute temperature $T$ (in kelvin) is a straight line passing through the origin. If instead you plot $V$ against Celsius temperature, you still get a straight line, but extrapolating it backwards to zero volume meets the axis at $-273\,^\circ\text{C}$ — this is one experimental route to absolute zero.

At constant volume, the pressure of a fixed mass of gas is directly proportional to its absolute temperature. In a rigid, sealed container, heating the gas makes the molecules move faster, so they strike the walls both harder and more frequently, and the pressure rises. This is why a sealed can heated in a fire can burst.

$P \propto T \quad \text{(constant } V, n) \;\; \implies \;\; \frac{P_1}{T_1} = \frac{P_2}{T_2}$

A graph of $P$ against absolute temperature is again a straight line through the origin. As with Charles's law, plotting against Celsius and extrapolating back to zero pressure gives an intercept at $-273\,^\circ\text{C}$.

The straight-line proportionalities in Charles's and Gay-Lussac's laws only work because temperature is measured on the absolute, or Kelvin, scale. Absolute zero (0 K, equal to $-273\,^\circ\text{C}$) is the temperature at which molecular kinetic energy is at its minimum — you cannot go lower. Because the scale starts from a true physical zero, doubling the Kelvin temperature genuinely doubles the quantity the gas law depends on, which a Celsius ratio can never do. Converting is the first move in almost every gas-law calculation.

$T(\text{K}) = \theta(^\circ\text{C}) + 273$

The three experimental laws are special cases of one relationship for a fixed mass of gas. Merging them gives the combined gas law, which handles problems where pressure, volume and temperature all change at once. Setting any one of the three quantities constant recovers Boyle, Charles or Gay-Lussac in turn.

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

The combined gas law works for a fixed mass of gas, but it says nothing about HOW MUCH gas there is. Bringing in the amount of substance, $n$ (in moles), gives the full equation of state — the ideal gas equation. It ties all four macroscopic quantities together through a single universal constant $R = 8.31\text{ J K}^{-1}\text{mol}^{-1}$ (from the data booklet), and it is the most powerful single relationship in the topic.

PV = nRT

There is an equivalent form written in terms of the number of molecules, $N$, rather than moles. It uses the Boltzmann constant $k = 1.38 \times 10^{-23}\text{ J K}^{-1}$ and is the natural choice when you are thinking about individual particles. The two forms are identical physics: $n = N/N_A$ and $R = N_A k$, where $N_A = 6.02 \times 10^{23}\text{ mol}^{-1}$ is Avogadro's constant.

PV = NkT

The kinetic model gives temperature a deep physical meaning: it is a measure of the average translational kinetic energy of the molecules. For an ideal gas the average kinetic energy of a single molecule depends only on the absolute temperature — not on the type of gas — through the relation below, where $k$ is the Boltzmann constant. So the average kinetic energy is directly proportional to the absolute temperature: $E_k \propto T$.

$E_k = \frac{3}{2}kT \quad \Longrightarrow \quad E_k \propto T \;\text{(absolute)}$

The ideal gas model rests on two convenient lies: that molecules have no volume of their own, and that they do not attract one another. Real molecules do both, so real gases only obey $PV = nRT$ approximately, and they depart from it most where those assumptions fail. At HIGH pressure the molecules are forced close together, so their own volume is no longer negligible compared with the container. At LOW temperature the molecules move slowly, so the intermolecular attractions that the model ignores become significant, and the gas is heading towards liquefaction. The practical summary: a real gas behaves most like an ideal gas at LOW pressure and HIGH temperature, and least like one at high pressure and low temperature.

This is the showcase for a calculation topic. In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A). Method marks and error-carried-forward (ECF) mean an early arithmetic slip does not have to cost you every mark that follows, but only if your method is written down. Study how each mark below is earned by a specific line.

The kinetic model you have built here underpins the rest of thermal physics: the same molecular collisions explain why a gas does work when it expands, the link $E_k = \frac{3}{2}kT$ feeds into internal energy and the first law of thermodynamics, and the ideal-versus-real distinction reappears whenever a gas is compressed or cooled towards a phase change. Master the two habits of this topic — convert to Kelvin and SI units first, then reach for the right law — and every thermal calculation ahead becomes a variation on a method you already own.