The gas laws, the ideal-gas equation, Dalton's and Graham's laws, the kinetic molecular theory and why real gases deviate — exam-focused for the BIEK / Sindh Board paper. Read it straight through, or open the interactive lecture and play with the gas simulator.
1 — The gaseous state
Of the three states, gases are the simplest to describe because their particles are far apart and move freely.
General properties of gases
- No fixed shape or volume — a gas fills its container completely.
- Highly compressible (large empty space between molecules).
- Very low density compared with solids and liquids.
- Exert pressure equally in all directions (wall collisions).
- Diffuse and mix spontaneously; expand on heating.
A gas's state is fixed by four measurable quantities: pressure (P), volume (V), temperature (T) and amount (n, moles).
2 — Pressure of a gas
- Pressure — force exerted per unit area by gas molecules colliding with the container walls. P = Force / Area.
| Unit | Equivalent |
|---|
| 1 atmosphere (atm) | 760 mmHg = 760 torr |
| 1 atm | 101325 Pa = 101.325 kPa |
| 1 atm | 1.01325 bar |
Measured with a barometer (atmospheric) or a manometer (a gas sample). STP = 0 °C (273 K) and 1 atm; SATP = 25 °C, 1 bar.
3 — Boyle's law
At constant temperature and amount, the volume of a gas is inversely proportional to its pressure.
Boyle's lawV ∝ 1/P ⇒ PV = constant ⇒ P₁V₁ = P₂V₂
A graph of P vs V is a curve (hyperbola); P vs 1/V is a straight line through the origin.
Boyle's law
A gas occupies 2 dm³ at 1 atm. What volume at 4 atm (constant T)?
P₁V₁ = P₂V₂ → (1)(2) = (4)(V₂) → V₂ = 0.5 dm³
4 — Charles's law
At constant pressure and amount, the volume of a gas is directly proportional to its absolute (Kelvin) temperature.
Charles's lawV ∝ T ⇒ V/T = constant ⇒ V₁/T₁ = V₂/T₂ (T in K)
Absolute zero: extrapolating V→0 gives −273.15 °C = 0 K, the lowest possible temperature. Always convert °C to K (K = °C + 273).
Charles's law
A gas is 300 cm³ at 27 °C. Volume at 127 °C (constant P)?
T₁ = 300 K, T₂ = 400 K → V₂ = V₁ T₂/T₁ = 300 × 400/300 = 400 cm³
5 — Gay-Lussac's & Avogadro's laws
- Gay-Lussac's (pressure) law — at constant V, P ∝ T (P₁/T₁ = P₂/T₂).
- Avogadro's law — at constant T and P, V ∝ n. Equal volumes of all gases contain equal numbers of molecules; 1 mole of any gas = 22.4 dm³ at STP.
6 — The ideal-gas equation
Combining all the laws (V ∝ 1/P, V ∝ T, V ∝ n) gives the general (ideal) gas equation:
Ideal gas equationPV = nRT
R = 0.0821 dm³·atm·K⁻¹·mol⁻¹ = 8.314 J·K⁻¹·mol⁻¹
The combined gas law (fixed n) is P₁V₁/T₁ = P₂V₂/T₂.
ideal gas equation
Volume of 2 mol of gas at 27 °C and 1 atm?
V = nRT/P = (2)(0.0821)(300)/1 = 49.3 dm³
7 — Dalton's law of partial pressures
The total pressure of a mixture of non-reacting gases equals the sum of the partial pressures each gas would exert alone.
Dalton's lawP_total = P₁ + P₂ + P₃ + …
partial pressure: Pᵢ = (mole fraction xᵢ) × P_total
Use: correcting the volume of a gas collected over water — subtract the water vapour (aqueous tension): P_dry = P_total − P_water.
8 — Graham's law of diffusion
At the same temperature and pressure, the rate of diffusion (or effusion) of a gas is inversely proportional to the square root of its molar mass (or density).
Graham's lawrate ∝ 1/√M ⇒ r₁/r₂ = √(M₂/M₁)
Graham's law
Compare the diffusion rates of H₂ (M=2) and O₂ (M=32).
r_H₂/r_O₂ = √(32/2) = √16 = 4 — hydrogen diffuses 4× faster
9 — Kinetic Molecular Theory (KMT)
The model that explains gas behaviour. Its postulates for an ideal gas:
- Gases consist of tiny particles in continuous, random, straight-line motion.
- The volume of the molecules is negligible compared with the volume of the container.
- There are no attractive or repulsive forces between molecules.
- Collisions are perfectly elastic — no kinetic energy is lost.
- The average kinetic energy is directly proportional to the absolute temperature.
Kinetic equation & energyPV = ⅓ m N c̄²
average K.E. = (3/2) kT (per molecule)
10 — Temperature & molecular speed
Because average K.E. ∝ T, raising the temperature makes the molecules move faster and strike the walls harder and more often — so pressure rises (at fixed V).
Root-mean-square speedc_rms = √(3RT / M)
At the same T, lighter molecules move faster (this is why H₂ diffuses fastest) — but all gases have the same average kinetic energy.
11 — Real gases & deviation from ideal behaviour
Real gases obey PV = nRT only at low pressure and high temperature. They deviate because two KMT assumptions fail:
- Molecules do have a finite volume (matters at high pressure).
- Molecules do attract each other (matters at low temperature / high pressure).
van der Waals equation(P + an²/V²)(V − nb) = nRT
a corrects for intermolecular attraction; b corrects for molecular volume.
12 — Liquefaction of gases
A gas can be turned to liquid by cooling (reducing K.E.) and compressing (bringing molecules close so attractions act).
- Critical temperature (Tc) — the temperature above which a gas cannot be liquefied by pressure alone, however great. CO₂ Tc = 31 °C.
13 — Worked numericals
combined gas law
2 dm³ at 1 atm, 273 K → volume at 2 atm, 546 K?
V₂ = V₁ (P₁/P₂)(T₂/T₁) = 2 × (1/2) × (546/273) = 2 dm³
molar mass from PV=nRT
0.5 g of a gas occupies 0.4 dm³ at 1 atm, 300 K. Find M.
n = PV/RT = (1×0.4)/(0.0821×300) = 0.01624 mol
M = mass/n = 0.5/0.01624 = ≈ 30.8 g/mol
partial pressure
2 mol N₂ + 3 mol O₂ at total 5 atm. Partial pressure of O₂?
x_O₂ = 3/5 = 0.6 → P_O₂ = 0.6 × 5 = 3 atm
14 — Exam recap
- Properties of gases; pressure & its units; STP.
- Boyle's (PV=k), Charles's (V/T=k), Gay-Lussac's, Avogadro's laws.
- Ideal-gas equation PV = nRT and the combined gas law.
- Dalton's law of partial pressures; Graham's law (rate ∝ 1/√M).
- KMT postulates; K.E. ∝ T; c_rms.
- Real gases, van der Waals, critical temperature.