Gases

Of the three classical states of matter, gases are the simplest to model: low density, complete fluidity, and a uniform pressure that responds to volume and temperature in predictable ways. The PMDC MDCAT 2026 syllabus expects working knowledge of Boyle's law, Charles's law, the ideal gas equation, the kinetic molecular theory (KMT) and the differences between real and ideal gases. Expect 2-3 MCQs from this chapter.

PMC Table of Specifications. Boyle's, Charles's and ideal-gas calculations are the highest-yield topics. Memorise R in both unit systems and the molar volume at STP.

Boyle's Law

At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume:

PV = constant; or P₁V₁ = P₂V₂.

A graph of P vs V is a hyperbola; P vs 1/V is a straight line through the origin. Each curve at a different temperature is an isotherm.

Charles's Law

At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature:

V/T = constant; or V₁/T₁ = V₂/T₂ (T in kelvin).

The volume of any gas decreases by 1/273 of its volume at 0°C for every 1°C drop in temperature. A graph of V vs T (kelvin) is a straight line through the origin; V vs t (°C) extrapolates back to V = 0 at −273.15°C.

The four gas laws — what's held constant, what relates to what
Law	Held constant	Relation	Equation	Discoverer
Boyle's Law	T, n	P ∝ 1/V	PV = const → P₁V₁ = P₂V₂	Robert Boyle (1662)
Charles's Law	P, n	V ∝ T	V/T = const → V₁/T₁ = V₂/T₂	Jacques Charles (1787)
Gay-Lussac's Law	V, n	P ∝ T	P/T = const → P₁/T₁ = P₂/T₂	Joseph Gay-Lussac (1802)
Avogadro's Law	P, T	V ∝ n	V/n = const at same P, T	Amedeo Avogadro (1811)
Ideal Gas Law	—	Combines all four	PV = nRT	Compiled (~1834)
Combined Gas Law	n	For fixed amount	P₁V₁/T₁ = P₂V₂/T₂	—
Dalton's Law of Partial Pressures	V, T	P_total = ∑P_i	P_i = x_i · P_total	John Dalton (1801)
Graham's Law of Effusion	P, T	rate ∝ 1/√M	r₁/r₂ = √(M₂/M₁)	Thomas Graham (1848)

Absolute Zero

Absolute zero is the temperature at which the volume (and pressure) of an ideal gas would extrapolate to zero: −273.15°C = 0 K. It is the lower limit of the kelvin (thermodynamic) temperature scale; molecular motion would cease (in classical terms) and the gas can no longer cool further. Real gases liquefy or solidify long before reaching this temperature.

Mnemonic for the gas laws. "Boyle's Pushes" (P↔V at fixed T); "Charles's Cooks" (V↔T at fixed P). Always convert temperature to kelvin before using either law.

Ideal Gas Equation

Combining Boyle's, Charles's and Avogadro's laws gives the ideal gas equation:

PV = nRT

where P = pressure, V = volume, n = moles, T = absolute temperature, and R = universal gas constant. The equation describes a hypothetical gas that obeys it exactly at all P and T.

General gas equation (combined gas law)

For a fixed amount of gas: P₁V₁/T₁ = P₂V₂/T₂.

Unit of R

The numerical value of R depends on the units used:

R = 8.314 J K⁻¹ mol⁻¹ (SI; P in Pa, V in m³).
R = 0.0821 dm³ atm K⁻¹ mol⁻¹ (P in atm, V in dm³ or L).
R = 62.36 L torr K⁻¹ mol⁻¹ (P in torr/mmHg).
R = 1.987 cal K⁻¹ mol⁻¹.

STP — Standard Temperature and Pressure

The IUPAC defines STP in MDCAT context as T = 273.15 K (0°C) and P = 1 atm (101.325 kPa, 760 mmHg). At STP, one mole of an ideal gas occupies a molar volume of 22.4 L (dm³). This single number short-cuts many MDCAT calculations.

Common trap. The post-1982 IUPAC "standard pressure" is 1 bar (100 kPa), giving a molar volume of 22.7 L. PMDC MDCAT and Pakistani FSc textbooks use the older convention — 1 atm and 22.4 L/mol. Use that unless told otherwise.

Kinetic Molecular Theory (KMT)

KMT is a set of postulates that derive the gas laws from molecular motion:

Gases consist of large numbers of tiny particles in continuous, random motion.
The particle volume is negligible compared to the container volume.
There are no intermolecular forces (attractive or repulsive) between particles.
Collisions are perfectly elastic — total kinetic energy is conserved.
The average kinetic energy of the particles is directly proportional to the absolute temperature — KE_avg = (3/2) k_BT.

Maxwell-Boltzmann distribution

Not all molecules in a gas have the same speed. The Maxwell-Boltzmann distribution gives the fraction of molecules at each speed; raising the temperature shifts the curve to the right and broadens it. Three useful averages:

Most probable speed v_mp = √(2RT/M)
Average speed v_avg = √(8RT/πM)
Root-mean-square speed v_rms = √(3RT/M)

Graham's law of effusion

At constant T and P: rate of effusion ∝ 1/√M. A gas of lower molar mass effuses faster.

Real and Ideal Gas

An ideal gas obeys PV = nRT exactly. Real gases show deviations because molecules do have a finite volume and do attract one another. Deviations are largest at high pressure (molecular volumes matter) and low temperature (attractive forces matter).

van der Waals equation

The most familiar correction is van der Waals's:

(P + an²/V²)(V − nb) = nRT

a corrects for intermolecular attractive forces (raises the effective pressure).
b corrects for the finite volume of the molecules (reduces the free volume).
Polar/large molecules → higher a; bulkier molecules → higher b.

When does a real gas behave most ideally?

At low pressure and high temperature — the molecules are far apart and moving fast, so volume and attraction effects are negligible.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for gases. Read every explanation — the deeper concept lives there.

Q1. The volume of one mole of an ideal gas at STP (0°C, 1 atm) is approximately:

11.2 L
22.4 L
24.0 L
1.0 L

Substituting in PV = nRT with P = 1 atm, T = 273.15 K, n = 1 mol, R = 0.0821 L·atm/(mol·K) gives V = 22.4 L. This is the molar volume at STP — a number you must memorise.

Q2. Boyle's law states that at constant temperature, for a fixed mass of gas:

V ∝ T
P ∝ 1/V
P ∝ V
V ∝ n

Boyle (1662) showed pressure and volume are inversely proportional at constant T (and n). PV is therefore constant, and a P-vs-V plot is a hyperbola.

Q3. A real gas approaches ideal behaviour at:

High pressure and low temperature
High pressure and high temperature
Low pressure and high temperature
Low pressure and low temperature

At low P, molecules are far apart so their finite volume is negligible compared to V. At high T, kinetic energy overwhelms intermolecular attractions. Both KMT assumptions (zero size, no forces) hold best in this regime.

Q4. The numerical value of R in dm³ atm K⁻¹ mol⁻¹ is:

8.314
0.0821
62.36
1.987

R = 0.0821 L·atm K⁻¹ mol⁻¹ when P is in atm and V in litres (= dm³). The SI value 8.314 J K⁻¹ mol⁻¹ applies when P is in Pa and V in m³.

Q5. According to KMT, the average kinetic energy of gas molecules depends only on:

Pressure
Volume
Absolute temperature
Molar mass

KE_avg = (3/2) k_BT, so it depends only on T (in kelvin). Two gases at the same temperature share the same average KE even though heavier molecules move more slowly.

Quick Recap

Boyle's law: PV = const at fixed T, n.
Charles's law: V/T = const at fixed P, n (T in kelvin).
Absolute zero = −273.15°C = 0 K.
Ideal gas equation: PV = nRT.
R = 8.314 J K⁻¹ mol⁻¹ = 0.0821 dm³ atm K⁻¹ mol⁻¹.
STP (PMDC) = 273.15 K and 1 atm; molar volume = 22.4 L.
KMT: random motion, point particles, no forces, elastic collisions, KE ∝ T.
Real gases deviate at high P / low T; van der Waals corrects with constants a (attraction) and b (volume).

Test yourself. Take a timed practice test or browse the topic-wise MCQs to lock these concepts in.