Gas laws & kinetic molecular theory
The idea
The ideal gas law PV = nRT compresses Boyle's, Charles's, and Avogadro's separate observations into one equation of state, and kinetic molecular theory (KMT) explains why it works: gas particles are tiny relative to their container, in ceaseless random motion, colliding elastically, with negligible mutual attraction — and their average kinetic energy depends only on absolute temperature. You already used single-variable gas laws; the university move is wielding PV = nRT quantitatively, including the workhorse rearrangement M = mRT/(PV) that turns a vapor's mass, volume, pressure, and temperature into a molar mass.
Because average kinetic energy is fixed by T alone, two different gases at the same temperature must differ in speed: lighter molecules move faster on average (the root of Graham's law of effusion). The widespread misconception is that all molecules at one temperature share one speed — in fact each gas has a broad Maxwell–Boltzmann distribution, and equal temperature means equal average kinetic energy, not equal velocity.
Know where ideality fails: at high pressure, molecular volume stops being negligible, and at low temperature, attractions slow molecules near the walls — both corrected empirically by the van der Waals equation. Under ordinary lab conditions, though, PV = nRT is reliable to about a percent.
Worked example
A 0.612 g sample of a volatile liquid is vaporized, filling a 250.0 mL flask at 100.0 °C and 0.980 atm. Find the molar mass of the compound and decide whether it could be carbon disulfide, CS₂ (76.1 g/mol). Use R = 0.08206 L·atm/(mol·K).
- Convert every quantity to the units R expects: V = 0.2500 L and T = 100.0 + 273 = 373 K; pressure is already in atm.
- Count the moles of vapor with the ideal gas law: n = PV/RT = (0.980 × 0.2500)/(0.08206 × 373) = 0.245/30.61 = 8.00 × 10⁻³ mol.
- Molar mass is mass per mole: M = 0.612 g/0.00800 mol = 76.5 g/mol.
- Compare with the candidate: CS₂ has 12.01 + 2 × 32.06 = 76.1 g/mol, within about 0.5% of the measurement — the identification is consistent.
- Sanity-check the assumptions: at roughly 1 atm and well above the boiling point, the vapor is dilute and ideal behavior is a fair model, so the small deviation is ordinary experimental scatter rather than non-ideality.
Answer. The molar mass is about 76.5 g/mol, consistent with the vapor being CS₂.
Check your understanding
- Why does the ideal gas law need no information about the identity of the gas, and which experimental conditions break that indifference?
- How does KMT explain why lighter gases effuse faster even though all gases at one temperature share the same average kinetic energy?
- What systematic error would you expect in the measured molar mass if some of the liquid failed to vaporize, and in which direction?
- How would you decide, given P, V, T data for a real gas, whether attractions or molecular volume dominate its deviation from ideality?
Build the foundations first
Gas laws & kinetic molecular theory builds on these concepts. If any feel shaky, start there.