IMAT Chemistry · States of Matter and Gas Laws

The Ideal Gas Equation and Mixtures of Gases

PV = nRT joins all the gas laws in one equation; in a mixture each gas adds its own partial pressure, set by its share of the moles.

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Why this matters

The 2025 paper asked for the ideal gas equation itself, and the 2024 paper asked for the partial pressure of one gas in a mixture of three. The 2018 paper asked which changes raise the total kinetic energy of a fixed sample of gas, which is a reading of PV = nRT.

Concept 1 of 4: The ideal gas equation PV = nRT

Boyle, Charles, Gay-Lussac and Avogadro each describe one slice of the same behaviour: pressure times volume grows with the number of moles and with the kelvin temperature. Putting them together with one constant, R, gives a single equation that holds for any ideal gas.

Definition

Ideal gas equation: $PV = nRT$ . Every gas law on the previous page is this equation with two of the quantities held constant.
The gas constant $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$ when $P$ is in Pa and $V$ in m³. With $P$ in atm and $V$ in litres, $R = 0.0821\ \text{L atm mol}^{-1}\text{K}^{-1}$ .
$T$ is always in kelvin. Volume conversions: $1\ \text{L} = 1\ \text{dm}^3 = 10^{-3}\ \text{m}^3$ .
With $n = m/M$ , the equation gives a molar mass: $M = mRT/(PV)$ .
The total translational kinetic energy of the molecules of an ideal gas is $\tfrac{3}{2}nRT = \tfrac{3}{2}PV$ . For a fixed sample, $T$ and the kinetic energy rise whenever $PV$ rises.

Ideal gas equation

PV = nRT

$P$ pressure, in Pa (or atm with the other R)
$V$ volume, in m³ (or L with the other R)
$n$ amount of gas, in mol
$R$ gas constant, 8.314 J/(mol K) or 0.0821 L atm/(mol K)
$T$ absolute temperature, in K

Worked example

2.0 mol of nitrogen is kept in a container of volume 0.050 m³ at 300 K. What is its pressure? (

R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}

)

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Litres with R = 8.314 gives an answer 1000 times wrong

R = 8.314

needs pressure in Pa and volume in m³. Putting in litres gives a pressure 1000 times too large. With litres and atmospheres, use

R = 0.0821

Adding gas at constant pressure and volume cools the gas

From

T = PV/(nR)

, if more moles go into the container while

P

and

V

stay fixed, the temperature must fall, and so does the kinetic energy of the molecules that were there at the start. At fixed

n

, raising either

P

V

raises

T

and the kinetic energy.

Concept 2 of 4: Dalton's law of partial pressures

In a mixture of ideal gases the molecules ignore one another, so each gas hits the walls exactly as it would if it were alone. The total pressure is the sum of these separate pushes. Since pressure depends on the number of molecules, each gas's share of the pressure equals its share of the moles.

Definition

Dalton's law: the total pressure of a mixture of gases is the sum of the partial pressures of the gases: $P_\text{total} = p_A + p_B + \dots$
The partial pressure of a gas is the pressure it would exert alone in the same volume at the same temperature.
The mole fraction $x_A = n_A / n_\text{total}$ , and $p_A = x_A\,P_\text{total}$ . Moles, not masses, decide the share.
Example: dry air is about 21% oxygen by moles, so at 100 kPa the partial pressure of oxygen is about 21 kPa.
A gas collected over water is mixed with water vapour: subtract the vapour pressure of water to get the pressure of the dry gas.

Partial pressure

p_A = x_A\,P_{\text{total}} = \frac{n_A}{n_{\text{total}}}\,P_{\text{total}}

$p_A$ partial pressure of gas A
$x_A$ mole fraction of A
$n_A, n_\text{total}$ moles of A and total moles of gas

Worked example

A diver's cylinder contains 16 g of oxygen (

M = 32

) and 14 g of helium (

M = 4

) at a total pressure of 8.0 atm. Find the partial pressure of each gas.

Practice this conceptself-check · 3 quick reps

Partial pressures follow moles, not masses

In a mixture of 1 g of hydrogen and 1 g of helium, hydrogen exerts two thirds of the pressure, because it has twice as many moles. Always convert masses to moles before sharing out the pressure.

Concept 3 of 4: Graham's law: lighter gases diffuse and effuse faster

At the same temperature every gas molecule has the same average kinetic energy,

\tfrac{1}{2}mv^2

. A lighter molecule must therefore move faster, and so it spreads out (diffuses) or escapes through a tiny hole (effuses) faster. Speed goes as one over the square root of the mass.

Definition

Diffusion: the spreading of a gas through another. Effusion: the escape of a gas through a tiny hole.
Graham's law: at the same temperature and pressure, the rate of effusion is inversely proportional to the square root of the molar mass.
A gas 4 times heavier effuses at half the rate; one 16 times heavier at a quarter of the rate.
Classic demonstration: ammonia ( $M = 17$ ) and hydrogen chloride ( $M = 36.5$ ) released at the two ends of a tube meet nearer the hydrogen chloride end, forming a white ring of ammonium chloride.

Graham's law

\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}

$\text{rate}_1, \text{rate}_2$ rates of effusion of gases 1 and 2
$M_1, M_2$ their molar masses

Worked example

How many times faster does hydrogen (

M = 2

) effuse than oxygen (

M = 32

) under the same conditions?

Practice this conceptself-check · 3 quick reps

Rate goes with the square root of the molar mass

A gas 4 times heavier is 2 times slower, not 4 times slower. Leaving out the square root is the commonest wrong answer.

Concept 4 of 4: Real gases and the assumptions of the ideal gas model

The ideal gas model pretends molecules are points that never attract one another. That is close to true when the molecules are far apart and fast. Squeeze a gas hard, or cool it, and the molecules come close and slow down: their size and their attractions start to matter, and the gas stops obeying PV = nRT exactly.

Definition

The kinetic model of an ideal gas assumes:

the molecules move randomly in straight lines between collisions;
their own volume is negligible compared with the volume of the container;
there are no forces between molecules except during collisions;
collisions are perfectly elastic.

Real gases come closest to ideal at high temperature and low pressure, and deviate most at high pressure and low temperature (near condensation). Small, non-polar molecules such as helium and hydrogen are the most nearly ideal.

Idea	Ideal gas model	Real gas
Volume of the molecules	Negligible	Small but real; matters at high pressure
Forces between molecules	None	Weak attractions; matter at low temperature
Can it be liquefied?	Never	Yes, by cooling and compressing
Closest to ideal	Any gas, at any conditions	Helium and hydrogen, at high temperature and low pressure

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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 (3)

The ideal gas equation PV = nRT
Ideal gas equation
$PV = nRT$
Dalton's law of partial pressures
Partial pressure
$p_A = x_A\,P_{\text{total}} = \frac{n_A}{n_{\text{total}}}\,P_{\text{total}}$
Graham's law: lighter gases diffuse and effuse faster
Graham's law
$\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

Reference tables (1)

Real gases and the assumptions of the ideal gas model4 rows

Idea	Ideal gas model	Real gas
Volume of the molecules	Negligible	Small but real; matters at high pressure
Forces between molecules	None	Weak attractions; matter at low temperature
Can it be liquefied?	Never	Yes, by cooling and compressing
Closest to ideal	Any gas, at any conditions	Helium and hydrogen, at high temperature and low pressure

Watch out for (4)

Litres with R = 8.314 gives an answer 1000 times wrong
→ The ideal gas equation PV = nRT
Adding gas at constant pressure and volume cools the gas
→ The ideal gas equation PV = nRT
Partial pressures follow moles, not masses
→ Dalton's law of partial pressures
Rate goes with the square root of the molar mass
→ Graham's law: lighter gases diffuse and effuse faster

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