What Is the Ideal Gas Law?

You have a gas, you need to know its volume, and you have the pressure, temperature, and moles. The ideal gas law, PV = nRT, gives you that answer in one equation. It is the single most used formula in introductory chemistry and physics for predicting how a gas behaves when conditions change.

The ideal gas formula PV = nRT links four variables and one constant. Each term has specific units that must match the value of R you choose. Using the wrong R is the most common failure in gas-law calculations. The math will produce a number, but that number will be physically meaningless.

P: Pressure

Pressure is the force gas particles exert on the container walls. Common units include atmospheres (atm), pascals (Pa), kilopascals (kPa), bar, millimeters of mercury (mmHg or torr), and pounds per square inch (psi). Your choice of unit dictates which R value you must use. If you enter pressure in atm, you need R in L·atm/(mol·K), not in J/(mol·K).

V: Volume

Volume is the space the gas occupies. Typical units are liters (L), milliliters (mL), cubic meters (m³), cubic feet (ft³), and gallons (US). Volume must be in the same unit system as the gas constant. A common mismatch: using liters for volume with R = 8.314 J/(mol·K), which expects cubic meters.

n: Amount of Substance

n stands for number of moles, not mass. One mole contains Avogadro's number of particles (6.022×10²³). To convert mass to moles, divide by the molar mass. Forgetting this conversion, treating grams as moles, is a frequent error.

T: Absolute Temperature

Temperature must always be in Kelvin (K), the absolute scale. The Kelvin scale starts at absolute zero (−273.15 °C), where molecular motion stops. Convert Celsius by adding 273.15: K = °C + 273.15. Using Celsius or Fahrenheit directly breaks the equation because the law depends on the proportionality between temperature and kinetic energy, which is linear only on an absolute scale.

R: The Molar Gas Constant

R is not a single number; its value depends entirely on the units used for P, V, n, and T. The fundamental value, exact since the 2019 SI redefinition, is 8.314462618 J/(mol·K). For student work, the rounded value 8.314 J/(mol·K) is standard. The most common R in general chemistry textbooks is 0.082057 L·atm/(mol·K), derived from the conversion 1 L·atm = 101.325 J. Other useful values include 62.3637 L·Torr/(mol·K) for manometer readings in mmHg, and 10.731 psi·ft³/(lb-mol·°R) for US customary engineering. Note that the last one is per pound-mole, not per gram-mole, a factor-454 error that is a classic failure mode.

The ideal gas law formula did not appear from nowhere. It is a single equation that merges four simpler gas laws, each discovered over two centuries. OpenStax Chemistry 2e covers the individual laws in sections 9.2-9.3.

Boyle's law (1662) states that at constant n and T, pressure and volume are inversely proportional: P ∝ 1/V. Charles's law, published by Gay-Lussac in 1802 based on Charles's unpublished work, says that at constant n and P, volume is directly proportional to temperature: V ∝ T. Gay-Lussac's law (also called Amontons's law) states that at constant n and V, pressure is directly proportional to temperature: P ∝ T. Avogadro's law (1811) says that equal volumes of gases at the same temperature and pressure contain the same number of molecules: V ∝ n.

Students often misuse the combined gas law when the amount of gas changes, for example, when adding gas to a container, producing a wrong answer.

The ideal gas law rests on five assumptions from kinetic-molecular theory. These assumptions define what an ideal gas is: a hypothetical gas that obeys PV = nRT exactly at all conditions.

• Gas particles are point masses with negligible volume. The volume of the particles themselves is zero compared to the container volume.
• No intermolecular forces act between particles. There is no attraction or repulsion.
• Particles are in constant, random, straight-line motion.
• Collisions between particles and with container walls are perfectly elastic, no kinetic energy is lost.
• The average kinetic energy of the particles is proportional to the absolute temperature (T in Kelvin).

Real gases violate these assumptions. At high pressures, molecular volume becomes significant because the particles themselves occupy space. At low temperatures, intermolecular attraction pulls particles together, reducing the pressure below what the ideal gas law predicts. The van der Waals equation, (P + a(n/V)²)(V - nb) = nRT, adds correction terms: a accounts for intermolecular attraction, and b accounts for molecular volume.

Émile Clapeyron published the first combined statement of the ideal gas law in 1834 in his paper "Mémoire sur la puissance motrice de la chaleur" (Memoir on the Motive Power of Heat). Clapeyron took the earlier works of Boyle, Charles, Gay-Lussac, and Avogadro and expressed them as a single equation, using a constant R to tie the variables together. He did not determine the numerical value of R himself, that was done later experimentally by other scientists, but his formulation became the standard.

Before Clapeyron, each gas relationship was treated separately. His contribution was recognizing that all four laws were special cases of one underlying relationship, and that R was a universal proportionality constant. The 1834 paper is the origin of the form PV = nRT that appears in every modern chemistry textbook.

The ideal gas law is an approximation. It works well at high temperatures and low pressures, roughly above 1.5 times the boiling point of the gas and below about 5 atm for most diatomic gases, where the error stays under 1%. At room temperature (298 K) and 1 atm, real gases deviate by 0.1-0.5% (helium, hydrogen) to 1-3% (carbon dioxide, propane).

At higher pressures, above 10 atm, the error for nitrogen can exceed 5%. At temperatures near the condensation point, the ideal gas law fails completely because the gas is about to become a liquid. The compressibility factor Z = PV/nRT quantifies this deviation: Z = 1 for an ideal gas, and Z deviates from 1 for real gases. Engineering thermodynamics relies on generalized compressibility charts to correct for this.

For gases near their condensation point, steam at high pressure, propane in a tank, carbon dioxide near 0 °C, the van der Waals equation or a more sophisticated real-gas equation (Peng-Robinson, virial) is required. The ideal gas law will give a number, but that number will be physically wrong.