Ideal Gas Law

The ideal gas law, \( PV = nRT \), is the single most useful equation in introductory chemistry. It relates the four properties that describe any gas under everyday conditions: pressure, volume, amount of substance, and temperature. Once you know any three, you can calculate the fourth. The ‘ideal’ part means the law treats gas molecules as having no volume and no intermolecular forces — approximations that work well at low pressures and high temperatures, and break down only at extremes. Most exam questions and most real-world chemistry calculations sit firmly in the regime where the ideal gas law works fine.

The Equation

The ideal gas law in standard form:

$$ PV = nRT $$

Each symbol has a specific meaning:

• P = pressure of the gas (in atmospheres, kilopascals, or pounds per square inch)
• V = volume of the container (in liters or cubic meters)
• n = amount of gas, in moles
• R = universal gas constant, 0.0821 L·atm / (mol·K) or 8.314 J / (mol·K)
• T = absolute temperature in kelvin (NOT Celsius — this is a common error)

The trickiest part is unit consistency. If you use atmospheres for pressure, use R = 0.0821 L·atm/(mol·K). If you use pascals, use R = 8.314 J/(mol·K) and volume in m³. Mix units and the answer comes out wrong by orders of magnitude.

How the Ideal Gas Law Combines Simpler Laws

The ideal gas law combines four earlier discoveries from the 17th-19th centuries, each of which describes one relationship at a time while holding the others constant.

Earlier law	Discovered by	What it says (with the rest held constant)
Boyle’s law	Robert Boyle, 1662	PV = constant (at constant T, n)
Charles’s law	Jacques Charles, 1787	V/T = constant (at constant P, n)
Gay-Lussac’s law	Joseph Gay-Lussac, 1802	P/T = constant (at constant V, n)
Avogadro’s law	Amedeo Avogadro, 1811	V/n = constant (at constant T, P)

Multiply them together and you get \( PV = nRT \). Émile Clapeyron formalized the combined version in 1834.

Worked Examples

Example 1. How many moles of gas occupy 22.4 L at 1.00 atm and 273 K (standard temperature and pressure, STP)?

\( n = PV/(RT) = (1.00)(22.4) / (0.0821 \times 273) \approx 1.00 \) mol.

This is the famous ‘one mole of any ideal gas occupies 22.4 L at STP’ fact — a direct consequence of the ideal gas law.

Example 2. A 5.00 L container holds 2.50 mol of nitrogen gas at 300 K. What is the pressure?

\( P = nRT/V = (2.50)(0.0821)(300)/5.00 \approx 12.3 \) atm.

Example 3. A balloon contains 1.50 mol of helium at 1.20 atm and 290 K. What is its volume?

\( V = nRT/P = (1.50)(0.0821)(290)/1.20 \approx 29.7 \) L.

Variations You’ll See

The ideal gas law gets rewritten in several useful forms depending on what’s being measured.

Molar mass form

Substituting \( n = m/M \) (mass divided by molar mass):

$$ PV = \dfrac{m}{M} RT \;\;\Rightarrow\;\; M = \dfrac{mRT}{PV} $$

This is useful for determining the molar mass of an unknown gas by measuring its density at known P, V, T.

Density form

Density \( \rho = m/V \), so:

$$ \rho = \dfrac{PM}{RT} $$

At constant temperature and pressure, the densities of two gases are proportional to their molar masses. This is why helium balloons float — He’s molar mass (4) is much less than air’s (~29).

Combined gas law

For a fixed amount of gas (\( n \) constant), the relation between two states becomes:

$$ \dfrac{P_1 V_1}{T_1} = \dfrac{P_2 V_2}{T_2} $$

This is the workhorse formula for ‘before and after’ gas problems where some properties change.

When the Ideal Gas Law Breaks Down

The ideal gas law assumes two things that are not always true:

• Gas molecules have no volume. Real molecules have finite size. At high pressures, the molecules occupy a non-trivial fraction of the container volume.
• No intermolecular forces. Real molecules attract each other slightly (van der Waals forces). At low temperatures, these attractions become significant and cause real gases to deviate from ideal behavior.

For most everyday situations — air at room temperature and atmospheric pressure — these deviations are small (a few percent). For high-pressure industrial processes or low-temperature cryogenics, you need a more accurate equation of state, such as the van der Waals equation:

$$ \left(P + \dfrac{a n^2}{V^2}\right)(V – nb) = nRT $$

where \( a \) corrects for intermolecular attraction and \( b \) corrects for molecular volume. The constants \( a \) and \( b \) are specific to each gas.

Related study notes: Avogadro’s Number, Mole Concept, Stoichiometry, Periodic Table.

Frequently Asked Questions