The ideal gas equation

What you'll learn

What each symbol means in $p V = n R T$ .
How to convert common laboratory units into the required SI units.
How to rearrange and use the ideal gas equation in calculations.
How gas measurements can be used to find the $M_r$ of a volatile liquid.

The big idea

Gases are very spread out compared with solids and liquids, so their behaviour depends strongly on pressure, volume, temperature and amount of substance.

The ideal gas equation links all four:

p V = n R T

It is one of the most unit-sensitive equations in A-Level Chemistry. Most mistakes are not chemistry mistakes — they are usually unit conversion mistakes.

The variables you need

Amount of substance, $n$

The amount of substance tells you how many particles you have, measured in moles, mol.

Definition

Amount of substance

The amount of substance, $n$ , is measured in mol. One mole contains the Avogadro constant number of particles.

You have already met:

n=mMn=\frac{m}{M}

where $m$ is mass and $M$ is molar mass. In gas questions, you may calculate $n$ using the ideal gas equation, then connect it to mass or molar mass.

Pressure, $p$

Pressure is the force exerted per unit area. In a gas, pressure comes from gas particles colliding with the walls of the container.

For this specification, pressure must be in pascals, Pa, when using $p V = n R T$ .

Volume, $V$

Volume is the space occupied by the gas.

For $p V = n R T$ , volume must be in cubic metres, m³ — not cm³ or dm³.

Temperature, $T$

Temperature must be in kelvin, K. Kelvin temperature is also called absolute temperature.

Definition

Kelvin temperature

The kelvin scale starts at absolute zero, so it must be used in gas calculations. Convert from degrees Celsius using $T=θ+273T=\theta+273$ , where $θ\theta$ is the temperature in °C.

The gas constant, $R$

$R$ is the gas constant. Its value connects the units on both sides of the equation.

A common value is:

mol−1R=8.31\ \text{J K}^{-1}\text{ mol}^{-1}

In AQA questions, you are not expected to recall the value of $R$ ; it will be given.

The diagram below summarises the symbols and the unit conversions you should think about before substituting numbers into the equation.

Ideal gas equation symbols and SI unit conversions

Key Idea

SI units are not optional

For $p V = n R T$ in this specification, use pressure in Pa, volume in m³, amount in mol and temperature in K. If one value is in the wrong unit, the final answer will usually be wrong by a factor of 1000 or more.

The equation is:

p V = n R T

where:

$p$ = pressure in Pa
$V$ = volume in m³
$n$ = amount of gas in mol
$R$ = gas constant, usually given as $mol−18.31\ \text{J K}^{-1}\text{ mol}^{-1}$
$T$ = temperature in K

Definition

Ideal gas

An ideal gas is a model gas whose particles have negligible volume and no intermolecular forces. Real gases often behave close to ideal gases at relatively low pressure and high temperature.

A useful unit check is that pressure multiplied by volume has units of energy:

m3=J\text{Pa}\ \text{m}^3=\text{J}

So the left-hand side, $p V$ , has units of J. The right-hand side also has units of J because:

mol−1×K=J\text{mol}\times \text{J K}^{-1}\text{ mol}^{-1}\times \text{K}=\text{J}

This is why the units of $R$ only work properly when you use the correct units for $p$ , $V$ and $T$ .

Converting units before using the equation

Here are the most common conversions:

kPa to Pa: multiply by 1000
cm³ to m³: multiply by $10^{-6}$
dm³ to m³: multiply by $10^{-3}$
°C to K: add 273

Common Mistake

Using laboratory units directly

Do not substitute kPa, cm³, dm³ or °C directly into $p V = n R T$ . Convert first, then calculate.

Example

Preparing values for the equation

A gas has a pressure of 98.0 kPa, a volume of 250 cm³ and a temperature of 21 °C. Convert these values for use in $p V = n R T$ .

Convert pressure into Pa: $Pa98.0\ \text{kPa}=98.0\times 1000=9.80\times 10^4\ \text{Pa}$ .
Convert volume into m³: $m3250\ \text{cm}^3=250\times 10^{-6}=2.50\times 10^{-4}\ \text{m}^3$ .
Convert temperature into K: $KT=21+273=294\ \text{K}$ .

Rearranging $p V = n R T$

You may need to calculate any one of the variables. Rearrange before substituting numbers.

For amount of gas:

n=pVRTn=\frac{pV}{RT}

For pressure:

p=nRTVp=\frac{nRT}{V}

For volume:

V=nRTpV=\frac{nRT}{p}

For temperature:

T=pVnRT=\frac{pV}{nR}

Tip

Rearrange first

Rearrange the equation symbolically before putting numbers in. This reduces calculator errors and makes your working much easier to follow.

Example

Calculating amount of gas

A gas syringe contains 200 cm³ of gas at 100 kPa and 25 °C. Calculate the amount of gas in mol. Use $mol−1R=8.31\ \text{J K}^{-1}\text{ mol}^{-1}$ .

Convert the measurements into SI units: $Pap=100\times 1000=1.00\times 10^5\ \text{Pa}$ , $m3V=200\times 10^{-6}=2.00\times 10^{-4}\ \text{m}^3$ , and $KT=25+273=298\ \text{K}$ .
Rearrange the equation to make $n$ the subject: $n=pVRTn=\frac{pV}{RT}$ .
Substitute and calculate:

n=(1.00×105)(2.00×10−4)(8.31)(298)n=\frac{(1.00\times 10^5)(2.00\times 10^{-4})}{(8.31)(298)}

moln=8.08\times 10^{-3}\ \text{mol}

Finding the $M_r$ of a volatile liquid

A volatile liquid is a liquid that evaporates easily. If you vaporise a known mass of the liquid and measure the gas volume, pressure and temperature, you can calculate the amount of vapour using $p V = n R T$ .

Then use:

M=mnM=\frac{m}{n}

If mass is in g and amount is in mol, $M$ is in g mol⁻¹. The numerical value is the relative molecular mass, $M_r$ , which has no units.

Example

Finding the Mr of a volatile liquid

A 0.342 g sample of a volatile liquid is vaporised. The vapour occupies 100 cm³ at 101 kPa and 95 °C. Calculate the $M_r$ of the liquid. Use $mol−1R=8.31\ \text{J K}^{-1}\text{ mol}^{-1}$ .

Convert the gas measurements into SI units: $Pap=101\times 1000=1.01\times 10^5\ \text{Pa}$ , $m3V=100\times 10^{-6}=1.00\times 10^{-4}\ \text{m}^3$ , and $KT=95+273=368\ \text{K}$ .
Calculate the amount of vapour:

n=pVRT=(1.01×105)(1.00×10−4)(8.31)(368)n=\frac{pV}{RT} =\frac{(1.01\times 10^5)(1.00\times 10^{-4})}{(8.31)(368)}

moln=3.30\times 10^{-3}\ \text{mol}

Use the mass and amount to find the molar mass:

mol−1M=\frac{m}{n} =\frac{0.342\ \text{g}}{3.30\times 10^{-3}\ \text{mol}} =104\ \text{g mol}^{-1}

State the relative molecular mass: $Mr≈104M_r\approx 104$ .

Common Mistake

When the ideal model is weakest

Real gases deviate most from ideal behaviour at high pressure and low temperature, because particle volume and intermolecular forces become more important. In A-Level calculations, assume ideal behaviour unless the question tells you otherwise.

Quick sense checks

At around room temperature and pressure, 1 mol of gas occupies about 24 dm³. This is not a replacement for $p V = n R T$ , but it is a useful check.

For example, if you calculate that 200 cm³ of gas contains several moles, something has gone badly wrong: 200 cm³ is only 0.200 dm³, much smaller than 24 dm³.

Tip

Spotting impossible answers

For gas samples measured in a syringe, amounts are often small, such as $10^{-3}$ to $10^{-2}$ mol. Very large answers usually mean the volume was left in cm³ or the pressure was left in kPa.

Exam technique

In the exam

Convert all values before substitution: kPa to Pa, cm³ or dm³ to m³, and °C to K.
Rearrange $p V = n R T$ first, then substitute numbers with units shown in your working.
For $M_r$ questions, calculate $n$ from the gas data, then use $M=mnM=\frac{m}{n}$ and give $M_r$ with no units.

Self review

Check yourself

Why must temperature be converted to kelvin before using the ideal gas equation?
What values of $p$ , $V$ and $T$ would you substitute for 150 cm³ of gas at 99.0 kPa and 20 °C?
How could you use the mass of a vaporised liquid and its gas measurements to find its $M_r$ ?

What you'll learn

What each symbol means in $p V = n R T$ .
How to convert common laboratory units into the required SI units.
How to rearrange and use the ideal gas equation in calculations.
How gas measurements can be used to find the $M_r$ of a volatile liquid.

The big idea

Gases are very spread out compared with solids and liquids, so their behaviour depends strongly on pressure, volume, temperature and amount of substance.

The ideal gas equation links all four:

p V = n R T

It is one of the most unit-sensitive equations in A-Level Chemistry. Most mistakes are not chemistry mistakes — they are usually unit conversion mistakes.