The equation that agrees with the ideal gas law is PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant, and T is temperature in kelvins. This single equation ties together all the classic gas laws you may have learned separately, including Boyle’s law, Charles’s law, and Avogadro’s law. Each of those older laws is simply a special case of PV = nRT with one or two variables held constant.
What PV = nRT Actually Means
The ideal gas law says that the pressure a gas exerts, multiplied by the volume it occupies, equals the number of moles times a constant times the absolute temperature. In practical terms, if you know any three of the four variables (P, V, n, T), you can solve for the fourth.
The constant R, called the universal gas constant, has different numerical values depending on the units you use. The two most common are 0.08206 L·atm/(mol·K) when working with liters and atmospheres, and 8.3145 J/(mol·K) when working in SI energy units. Picking the right value of R is just a matter of matching your pressure and volume units.
The Three Laws Built Into PV = nRT
Boyle’s law, Charles’s law, and Avogadro’s law all “agree with” the ideal gas law because each one drops out of PV = nRT when you hold certain variables fixed.
- Boyle’s law: Hold temperature and moles constant, and nRT becomes a single constant. The equation reduces to PV = constant, meaning pressure and volume are inversely proportional. Double the pressure on a sealed container at constant temperature, and the volume halves.
- Charles’s law: Hold pressure and moles constant. Now V is directly proportional to T. Heat a balloon and it expands; cool it and it shrinks.
- Avogadro’s law: Hold pressure and temperature constant. Volume becomes directly proportional to the number of moles. Add more gas to a flexible container at the same pressure and temperature, and the volume increases in lockstep.
You can also combine any two of these into the combined gas law: P₁V₁/T₁ = P₂V₂/T₂. This version is useful when a gas sample changes conditions but the amount of gas stays the same. Because n and R don’t change, they cancel out of both sides of the equation.
The Molecular-Scale Version
There is a second form of the ideal gas law that counts individual molecules instead of moles: PV = NkT. Here N is the total number of molecules and k is the Boltzmann constant (1.381 × 10⁻²³ J/K). The two forms are equivalent because the Boltzmann constant is simply R divided by Avogadro’s number (6.022 × 10²³). Physicists tend to prefer this version; chemists usually stick with PV = nRT.
Density and Molar Mass Form
The ideal gas law can be rearranged to connect a gas’s density to its molar mass. Since the number of moles equals mass divided by molar mass (n = m/M), substituting into PV = nRT and rearranging gives d = MP/(RT), where d is density and M is molar mass. This is handy when you need to identify an unknown gas: measure the density at a known temperature and pressure, and you can solve for M directly.
Dalton’s Law and Gas Mixtures
The ideal gas law also underpins Dalton’s law of partial pressures. If you hold volume and temperature constant, pressure is directly proportional to the number of moles: P = n(RT/V). For a mixture of gases, each component contributes its own partial pressure as though the other gases weren’t there. The total pressure is simply the sum of those partial pressures: P_total = P₁ + P₂ + P₃ and so on. Each partial pressure individually follows PV = nRT.
When the Ideal Gas Law Breaks Down
PV = nRT assumes gas molecules have no size and no attraction to one another. Those assumptions work well for dilute gases at moderate temperatures, but they fail at very high pressures (where molecules are squeezed close together) and very low temperatures (where intermolecular attractions become significant). The van der Waals equation adds two correction terms to account for molecular volume and intermolecular forces. As those correction factors approach zero, the van der Waals equation collapses back to PV = nRT, confirming that the ideal gas law is the simplified baseline all other gas equations build on.