How to Do a Step-by-Step Derivation of the Ideal Gas Law

The ideal gas law, (PV = nRT), connects the pressure, volume, temperature, and amount of a gas in a single equation. You can derive it by combining three simpler gas laws discovered in the 17th and 18th centuries: Boyle’s law, Charles’s law, and Avogadro’s law. Each one describes how two of these properties change while the others are held constant, and putting them together gives you the complete relationship.

The Three Building Blocks

Boyle’s Law (Pressure-Volume Relationship)

Robert Boyle found in 1662 that if you keep the temperature and amount of gas constant, the pressure and volume are inversely proportional. Mathematically:

(P \propto \frac{1}{V}) (at constant (n) and (T))

If you double the pressure on a gas, its volume drops to half. This relationship holds well for most gases at moderate pressures and temperatures Royal Society of Chemistry.

Charles’s Law (Volume-Temperature Relationship)

Jacques Charles and Joseph Louis Gay-Lussac later showed that at constant pressure and amount, the volume of a gas is directly proportional to its absolute temperature (measured in kelvin):

(V \propto T) (at constant (n) and (P))

A gas heated to twice its original absolute temperature will occupy twice the volume, provided the pressure does not change.

Avogadro’s Law (Amount-Volume Relationship)

Amedeo Avogadro proposed in 1811 that equal volumes of gases at the same temperature and pressure contain the same number of particles. This gives:

(V \propto n) (at constant (P) and (T))

If you double the number of gas molecules, the volume doubles.

Combining the Proportionalities

To get the ideal gas law, you combine these three proportionalities into a single statement. Since volume depends on all three other variables, you can write:

(V \propto \frac{nT}{P})

To turn a proportionality into an equation, you introduce a constant of proportionality called the ideal gas constant, (R):

(V = R \cdot \frac{nT}{P})

Rearranging gives the standard form:

(PV = nRT)

The value of (R) depends on the units you use. A common value is 8.314 J/(mol·K), which works when pressure is in pascals and volume in cubic meters. For liters and atmospheres, you use 0.08206 L·atm/(mol·K). Both values are well-established by the National Institute of Standards and Technology National Institute of Standards and Technology.

Where the Constant Comes From

The constant (R) comes from experimental measurements of gases at low pressure, where real gases behave almost ideally. At standard temperature and pressure (273.15 K and 1 atm), one mole of an ideal gas occupies 22.414 L. Plugging these numbers into (PV = nRT) and solving for (R) gives:

(R = \frac{PV}{nT} = \frac{(1\ \text{atm})(22.414\ \text{L})}{(1\ \text{mol})(273.15\ \text{K})} = 0.08206\ \text{L·atm/(mol·K)})

This value is a measured result, not a theoretical one. IUPAC defines standard conditions for gases to make these comparisons consistent across laboratories IUPAC.

A Worked Example

Consider a sample of gas with 2.00 moles at a temperature of 300 K and a pressure of 1.50 atm. You can calculate its volume directly from the ideal gas law:

(V = \frac{nRT}{P} = \frac{(2.00\ \text{mol})(0.08206\ \text{L·atm/(mol·K)})(300\ \text{K})}{1.50\ \text{atm}})

(V = \frac{49.236}{1.50} = 32.8\ \text{L})

If you only had Boyle’s law, you would not be able to find this volume because you would not know how temperature and amount affect it. The derivation of the ideal gas law is what makes such a calculation possible.

Why the Derivation Matters for Real Gases

The ideal gas law is a model. It assumes gas molecules have no volume and do not interact with each other. Real gases deviate from this behavior at high pressures and low temperatures, where molecular size and attractions become significant. But the derivation gives you a baseline. You can then apply corrections, like those in the van der Waals equation, to account for non-ideal behavior.

Scientists use the ideal gas law in fields from meteorology to engineering. For example, it helps calculate how air density changes with altitude, or how much gas is produced in a chemical reaction.

Key Takeaways

• The ideal gas law (PV = nRT) is derived by combining Boyle’s law, Charles’s law, and Avogadro’s law.
• Each component law describes how two gas properties change while the others are held constant.
• The constant (R) is determined experimentally and has different values depending on the units used.
• The ideal gas law is an approximation that works best at low pressures and high temperatures, but it provides a foundation for more accurate models.