State the ideal gas equation pV = nRT and use it in calculations including determination of molar mass, explain the assumptions of the kinetic theory, and account for the deviation of real gases from ideal behaviour at high pressure and low temperature

What this dot point is asking

SEAB wants you to apply the ideal gas equation $pV = nRT$ (especially to find a molar mass from vaporised-liquid data), state the assumptions behind the kinetic theory of an ideal gas, and explain in molecular terms why real gases deviate at high pressure and low temperature. The molar-mass calculation is a guaranteed Paper 2 skill.

The answer

The ideal gas equation

where $p$ is pressure (Pa), $V$ is volume (m cubed), $n$ is amount (mol), $R = 8.31$ J per mol per K, and $T$ is temperature (K). The single most common error is unit handling, so always convert before substituting:

• pressure: kPa $\times 10^3$ to Pa
• volume: cm cubed $\times 10^{-6}$, or dm cubed $\times 10^{-3}$, to m cubed
• temperature: degrees Celsius $+ 273$ to K

Finding molar mass

Since $n = m / M_r$, substitute to get a working form:

This is how an unknown volatile liquid is identified: vaporise a known mass, measure the volume it occupies at known $T$ and $p$, and solve for $M_r$.

The kinetic theory assumptions

An ideal gas is a model with these assumptions:

1. Gas molecules are in constant, random motion.
2. The molecules have negligible volume compared with the volume of the container.
3. There are no intermolecular forces between molecules.
4. Collisions are perfectly elastic (no kinetic energy is lost).
5. The average kinetic energy is proportional to the absolute temperature.

Why real gases deviate

Two of the assumptions fail for real gases, and the deviation grows under two conditions:

At high pressure, molecules are forced close together. Their own volume is no longer negligible compared with the container, so the real volume available is smaller than assumed, and the measured $pV/nRT$ rises above 1.

At low temperature (and moderate pressure), molecules move slowly and intermolecular attractions become significant. Attractions pull molecules inward, so they strike the walls less forcefully and the measured pressure is below the ideal value, making $pV/nRT$ dip below 1.

A gas behaves most ideally at high temperature and low pressure, where molecules are far apart and fast moving, so both volume and force effects are negligible.

Which gases are most ideal

Small, light, weakly interacting gases (He, $\text{H}_2$, Ne) are closest to ideal. Larger, more polarisable, or polar molecules ($\text{CO}_2$, $\text{NH}_3$, $\text{SO}_2$) deviate more because of stronger intermolecular forces and greater molecular volume.

Examples in context

Example 1. Verifying a gas law in the lab. A JC practical asks students to confirm $pV = nRT$ by vaporising a measured mass of propanone in a gas syringe in a water bath. Plugging the measured volume, temperature, and atmospheric pressure into $M_r = mRT/(pV)$ should return a value near $58$ g per mol. A value much higher usually means some liquid failed to vaporise; a value much lower usually means a leak in the syringe.

Example 2. Predicting deviation in industry. In the Haber process, nitrogen and hydrogen are compressed to about $200$ atmospheres. At such high pressure the gases deviate significantly from ideal behaviour, and engineers use real-gas corrections (such as the van der Waals equation, beyond H2 syllabus detail) rather than $pV = nRT$ when designing the reactor. This connects the abstract deviation idea to a context met later in equilibrium.

Try this

Q1. Calculate the volume occupied by $0.50$ mol of an ideal gas at $300$ K and $120$ kPa. [2 marks]

• Cue. $V = nRT/p = (0.50 \times 8.31 \times 300)/120000 = 1.04 \times 10^{-2}$ m cubed (about $10.4$ dm cubed).

Q2. State two assumptions of the kinetic theory of an ideal gas that fail for a real gas. [2 marks]

• Cue. Negligible molecular volume; no intermolecular forces.

Q3. Explain why a fixed mass of gas approaches ideal behaviour as temperature is raised at constant pressure. [3 marks]

• Cue. Higher $T$ means faster molecules and larger volume, so molecules spend less time close together; intermolecular forces and molecular volume become negligible relative to the total volume.