MHT-CET Physics · Kinetic Theory of Gases

The Gas Laws and the Ideal Gas Equation

For a fixed amount of an ideal gas PV/T stays constant, which contains Boyle's law (PV constant at fixed T), Charles's law (V ∝ T at fixed P) and Gay-Lussac's law (P ∝ T at fixed V); written for n moles it is PV = nRT, or PV = NkT for N molecules.

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Why this matters

19 PYQs, none HARD. Eleven apply one gas law — the percentage change in pressure for a percentage change in volume, a tyre warming up, a balloon rising. Eight use PV = nRT or PV = NkT to compare the number of molecules in two jars, find a density or molecular mass, or count the moles that leak out. Two cards.

Concept 1 of 2: One Law at a Time

Hold one quantity fixed and the other two are tied. At constant temperature PV is constant, so a 5% smaller volume needs pressure multiplied by 20/19 — 5.26% more, not 5%. At constant volume P ∝ T in kelvin, so a 2.5% rise in pressure for a 4 K rise means the gas started at 160 K. Combining them, P₁V₁/T₁ = P₂V₂/T₂. A real gas behaves most like an ideal one at low pressure and high temperature, where the molecules are far apart and fast. The force a gas exerts on a closed container's wall is its pressure times the area, so it goes as T to the first power.

Definition

Boyle (T const): $P_1V_1 = P_2V_2$ — V down 5% ⇒ P up 5.26%; P down 20% ⇒ V up 25%.
Gay-Lussac (V const): $\dfrac{P_1}{T_1} = \dfrac{P_2}{T_2}$ — 270 kPa at 27 °C ⇒ 279 kPa at 37 °C.
Combined: $\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$ — 500 m³ at 27 °C, 1 atm ⇒ 900 m³ at −3 °C, 0.5 atm.
Ideal behaviour: low pressure, high temperature. Force on a closed container's wall ∝ $T^1$ .

Combined gas law

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

Worked example

A closed vessel holds gas at 100 °C. Its pressure rises by 4%. By how much has the temperature risen?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 22 April Shift II · Q38Moderate

Example 1 · Kinetic Theory of Gases · Gas Laws and Ideal Gas Equation

When the pressure of the gas contained in a closed vessel is increased by

2.5\%

, the temperature of the gas increases by 4 K . The initial temperature of the gas is

Equal percentages in Boyle's law

PV is constant, not P + V. Reducing V by 5% multiplies P by 1/0.95 — an increase of 5.26%. The options include the plain 5%.

Using degrees Celsius in P ∝ T

27 °C to 37 °C is 300 K to 310 K, a 3.3% rise — not 37/27. Convert before taking the ratio.

Drill 10 more on one law at a time

Concept 2 of 2: PV = nRT and Counting Molecules

The ideal gas equation counts the gas: n = PV/RT moles, or N = PV/kT molecules. Comparing two jars, the ratio of molecule numbers is the ratio of PV/T. With mass m and molar mass M, n = m/M, so density is ρ = PM/RT: at the same pressure and temperature a heavier gas is denser, and samples of equal mass, volume and pressure have temperatures in the ratio of their molar masses. Gas that leaks from a rigid vessel at constant temperature takes with it (P − P′)V/RT moles.

Definition

$PV = nRT = NkT$ , $n = \dfrac{m}{M}$ .
Two jars: $\dfrac{N_1}{N_2} = \dfrac{P_1V_1/T_1}{P_2V_2/T_2}$ (P, V, T against P, V/4, 2T ⇒ 4 : 1).
Density: $\rho = \dfrac{PM}{RT}$ ( $\rho \propto P/T$ ); $\dfrac{M_A}{M_B} = \dfrac{\rho_A P_B}{\rho_B P_A}$ .
Same m, V, P: $T \propto M$ (O₂ : H₂ = 16 : 1).
Leak at constant T: $\Delta n = \dfrac{V}{RT}(P - P')$ .

Ideal gas equation

PV = nRT = NkT, \qquad \rho = \frac{PM}{RT}

Worked example

A gas has density ρ₀ at P₀ and T₀. What is its density at 2P₀ and 4T₀?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 11th May Shift 1 · Q24Moderate

Example 2 · Kinetic Theory of Gases · Gas Laws and Ideal Gas Equation

Two vessels separately contain two ideal gases A and B at the same temperature, pressure of A being twice that of B. Under such conditions, the density of A is found to be 1.5 times the density of B. The ratio of molecular weights of A and B is

Leaving T out of a molecule count

The number of molecules is PV/kT. Two jars at different temperatures cannot be compared by PV alone.

Drill 7 more on pv = nrt and counting molecules

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

One Law at a Time
Combined gas law
$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$
PV = nRT and Counting Molecules
Ideal gas equation
$PV = nRT = NkT, \qquad \rho = \frac{PM}{RT}$

Watch out for (3)

Equal percentages in Boyle's law
→ One Law at a Time
Using degrees Celsius in P ∝ T
→ One Law at a Time
Leaving T out of a molecule count
→ PV = nRT and Counting Molecules

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