Difficulty: Intermediate | Prerequisites: Ideal gas laws (see companion notes), mole concept, basic algebra.
This set of notes picks up where the ideal gas laws leave off. It covers how gases behave in mixtures (Dalton's law, Henry's law), what is happening at the particle level (kinetic molecular theory), how molecular speed and mass relate (Graham's law), and where the ideal model breaks down (real gases, Van der Waals equation). You should already be comfortable with PV = nRT and the individual gas laws before working through this material.
In a mixture of gases, each gas contributes its own partial pressure as though the others were not there (Dalton's law), and gas solubility in a liquid rises with the partial pressure of that gas above it (Henry's law). Kinetic molecular theory explains macroscopic gas behaviour through particle motion: lighter molecules move faster, heavier ones slower (Graham's law). Real gases deviate from ideal predictions at high pressure and low temperature because their particles do have volume and do attract one another; the Van der Waals equation corrects for both.
Partial pressure
The pressure that one gas in a mixture would exert if it alone occupied the entire container at the same temperature. Think of it as each gas's individual contribution to the total push on the walls.
Mole fraction (X)
The ratio of moles of one component to the total moles in the mixture. A dimensionless number between 0 and 1.
Dalton's law of partial pressures
The total pressure of a gas mixture equals the sum of the partial pressures of each component: P_total = P₁ + P₂ + P₃ + ...
Henry's law
The concentration of a dissolved gas is directly proportional to the partial pressure of that gas above the solution: [A] = k_H × P_A.
Vapour pressure
The pressure exerted by evaporated particles above the surface of a liquid at equilibrium between evaporation and condensation.
Kinetic molecular theory (KMT)
A model that explains gas behaviour by treating gases as large numbers of tiny particles in constant, random motion with perfectly elastic collisions.
Boltzmann constant (k_B)
1.38 x 10⁻²³ J/K. Relates the average kinetic energy of a single particle to absolute temperature.
Root-mean-square (RMS) speed
A measure of the average speed of gas particles: u_rms = √(3RT/M). Heavier molecules have lower RMS speeds at the same temperature.
Diffusion
The spreading of gas molecules from a region of high concentration to low concentration through air or another medium. In simple terms, it is why you can smell perfume from across a room.
Effusion
The escape of gas particles through a tiny opening from one compartment to another under pressure.
Graham's law
Under the same conditions of temperature and pressure, the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass: r₁/r₂ = √(M₂/M₁).
Real gas
An actual gas whose particles have finite volume and experience intermolecular attractions, causing it to deviate from ideal behaviour at high pressure or low temperature.
Van der Waals equation
A modified version of the ideal gas law that corrects for intermolecular attractions (constant a) and molecular volume (constant b): (P + n²a/V²)(V - nb) = nRT.
When two or more non-reacting gases share a container, each gas behaves independently.
Each gas exerts the same pressure it would if it were the only gas in the container.
The total pressure equals the sum of all partial pressures: P_total = P₁ + P₂ + P₃ + ...
The partial pressure of any one gas equals its mole fraction times the total pressure: P₁ = X₁ × P_total, where X₁ = n₁ / n_total.
Vapour pressure is the pressure exerted by evaporated particles above the surface of a liquid once equilibrium is reached between evaporation and condensation.
Henry's law states that the solubility (concentration) of a gas in a liquid is directly proportional to the partial pressure of that gas above the liquid: [A] = k_H × P_A.
k_H (Henry's constant) is specific to each gas and solvent combination.
Equivalently: [A]₁/P₁ = [A]₂/P₂ = k_H.
Higher partial pressure above the liquid means more gas dissolves.
Gas particles have negligible volume compared with the volume of the container.
There are no intermolecular attractions or repulsions between gas particles.
Particles are in continuous, random motion, colliding with one another and the container walls.
All collisions are perfectly elastic: total kinetic energy and momentum are conserved.
The average kinetic energy of the particles depends only on the absolute temperature, not on the identity or mass of the gas.
Average kinetic energy per particle: KE_avg = ½mv² = (3/2)k_B T, where k_B = 1.38 x 10⁻²³ J/K.
At a given temperature, all gases have the same average kinetic energy regardless of molar mass.
Because KE = ½mv², lighter molecules must move faster to have the same kinetic energy as heavier ones.
Root-mean-square speed: u_rms = √(3RT/M), where R = 8.314 J·K⁻¹·mol⁻¹ and M is molar mass in kg/mol.
Raising temperature increases molecular speed. Increasing molar mass decreases it.
A curve showing the distribution of molecular speeds in a gas sample at a given temperature.
At higher temperatures the curve flattens and shifts to the right (more molecules at higher speeds).
At lower temperatures the curve is taller and narrower (molecules cluster around a lower average speed).
Speed is measured in m/s.
Diffusion: movement of gas molecules from high concentration to low concentration through air or another medium. Heavier gases diffuse more slowly.
Effusion: flow of gas particles through a tiny opening under pressure from one compartment to another. Heavier gases effuse more slowly.
Graham's law applies to both: r₁/r₂ = √(M₂/M₁).
A gas with four times the molar mass of another travels at half the rate of the lighter gas (since √4 = 2).
At moderate pressure, intermolecular attractions pull particles together, so the observed volume is less than ideal gas law predicts.
At very high pressure, the particles themselves take up a significant fraction of the space, so the observed volume is greater than predicted.
As a gas approaches its condensation pressure at a given temperature, attractions become strong enough to liquefy it.
As temperature drops, particles slow down and intermolecular attractions become more significant.
Near the condensation point, the gas occupies less volume than the ideal model predicts (attractions compress it).
At very low temperature, the gas cannot be compressed to zero volume as the ideal law implies, because the particles themselves occupy space.
The closer a gas is to its boiling or condensation point, the less ideally it behaves.
(P + n²a/V²)(V - nb) = nRT
a and b are experimentally determined constants, different for each gas.
a corrects for attractive forces. Larger, more polarisable molecules and polar molecules (e.g. HCl, NH₃) have larger a values. Small, nonpolar molecules have smaller a values.
b corrects for molecular volume. Larger molecules have larger b values.
Memory aid: a = attractive forces, b = big particles.
You typically do not need to memorise this equation for the exam, but you do need to understand what a and b correct for and how real gases deviate.
Law / Relationship | Formula | Notes |
|---|---|---|
Dalton's law (total) | P_total = P₁ + P₂ + P₃ + ... | Sum of all partial pressures |
Dalton's law (component) | P₁ = X₁ × P_total | X = mole fraction |
Henry's law | [A] = k_H × P_A | k_H is gas-specific |
Henry's law (comparative) | [A]₁/P₁ = [A]₂/P₂ = k_H | Useful for two-state problems |
Average kinetic energy | KE_avg = (3/2)k_B T | k_B = 1.38 x 10⁻²³ J/K |
Root-mean-square speed | u_rms = √(3RT/M) | R = 8.314 J·K⁻¹·mol⁻¹, M in kg/mol |
Graham's law | r₁/r₂ = √(M₂/M₁) | Applies to both diffusion and effusion |
Van der Waals equation | (P + n²a/V²)(V - nb) = nRT | a = attraction correction, b = volume correction |
Dalton's law is central to scuba diving: at depth, the partial pressure of each gas in the breathing mix increases, which is why nitrogen narcosis and oxygen toxicity become concerns. Henry's law explains why a fizzy drink goes flat when you open the cap: lowering the CO₂ partial pressure above the liquid causes dissolved gas to come out of solution. Graham's law is used in uranium enrichment, where uranium hexafluoride gas containing the lighter U-235 isotope effuses slightly faster through a barrier than the heavier U-238 version. The Van der Waals equation matters in industrial gas storage, where gases are kept at high pressures and the ideal gas law alone would give dangerously inaccurate volume or pressure estimates.
Students often think heavier gases have more kinetic energy at the same temperature. They do not. All gases at the same temperature have the same average kinetic energy. Heavier molecules simply move more slowly to achieve it.
A common error is to assume real gases always occupy less volume than predicted. At moderately high pressure they do (attractions dominate), but at very high pressure they occupy more (molecular volume dominates).
Students confuse diffusion and effusion. Diffusion is spreading through a medium (open space). Effusion is escape through a tiny hole. Graham's law applies to both, but the physical setup is different.
Some students apply the 22.4 L molar volume when working with gas mixtures at non-STP conditions. The 22.4 L figure is valid only for a single ideal gas at STP.
⚠️ Be able to calculate partial pressures from mole fractions and vice versa. This is a staple exam question.
⚠️ Know the five assumptions of kinetic molecular theory. They often appear as true/false or "which of the following" questions.
⚠️ Understand that KE_avg depends only on temperature, not on molar mass. Speed does depend on molar mass.
⚠️ Graham's law problems typically give two gases and ask which diffuses faster, or ask you to find the ratio of rates. Practise setting up the ratio correctly (heavier mass goes under the square root on the numerator side).
⚠️ For Van der Waals, you likely will not need the equation itself, but you must know what a and b correct for and under what conditions real gases deviate from ideal behaviour (high P, low T).
True or false: At a given temperature, heavier gas molecules have higher average kinetic energy than lighter ones. (False. Average KE depends only on temperature, not mass.)
Fill in the blank: The total pressure of a gas mixture equals the ______ of the partial pressures. (sum)
True or false: Real gases behave most ideally at low temperature and high pressure. (False. They behave most ideally at high temperature and low pressure.)
Fill in the blank: In Graham's law, rate of diffusion is inversely proportional to the square root of ______. (molar mass)
True or false: In the Van der Waals equation, the constant b corrects for intermolecular attractions. (False. b corrects for molecular volume. a corrects for attractions.)
Q: A mixture contains 2.0 mol of N₂ and 3.0 mol of O₂ at a total pressure of 5.0 atm. What is the partial pressure of N₂?
A: Mole fraction of N₂ = 2.0 / (2.0 + 3.0) = 0.40. Partial pressure = 0.40 x 5.0 = 2.0 atm.
Q: Gas A has a molar mass of 4 g/mol and gas B has a molar mass of 16 g/mol. How much faster does gas A diffuse than gas B?
A: r_A / r_B = √(M_B / M_A) = √(16/4) = √4 = 2. Gas A diffuses twice as fast.
Q: Under what conditions do real gases deviate most from ideal behaviour?
A: At high pressure and low temperature. Under these conditions, intermolecular forces and molecular volume become significant.
Q: In the Van der Waals equation, which constant would be larger for NH₃ than for He, and why?
A: The constant a would be larger for NH₃ because NH₃ is polar and has stronger intermolecular attractions. The constant b would also be larger because NH₃ molecules are physically bigger.
Q: Calculate the RMS speed of O₂ (M = 0.032 kg/mol) at 300 K.
A: u_rms = √(3RT/M) = √(3 x 8.314 x 300 / 0.032) = √(233,569) ≈ 483 m/s.
Dalton's law of partial pressures is essential for equilibrium calculations involving gaseous reactions (Kp expressions). Henry's law connects to solubility and intermolecular forces, topics covered elsewhere in general chemistry. Kinetic molecular theory underpins the derivation of the ideal gas law itself, so understanding the assumptions here deepens your grasp of why the gas laws work. Real gas behaviour ties directly into intermolecular forces (London dispersion, dipole-dipole, hydrogen bonding) and phase changes, which are covered in the liquids and solids unit.
Dalton's law, partial pressure, mole fraction, Henry's law, gas solubility, vapour pressure, kinetic molecular theory, KMT, Boltzmann constant, average kinetic energy, root-mean-square speed, RMS speed, Maxwell-Boltzmann distribution, molecular speed distribution, Graham's law, diffusion, effusion, real gases, ideal vs real gas, Van der Waals equation, intermolecular forces in gases, gas phase deviations, CHEM 101, general chemistry gases, gas mixtures