Gas Laws: From Boyle's Law to PV = nRT

Four variables that describe a gas sample

A gas sample is described by its pressure P, volume V, temperature T and amount n. Pressure appears in several units: one atmosphere is 101.325 kPa, or 760 mmHg, and for all practical purposes one torr equals one mmHg. The amount is always in moles, not grams.

Temperature must be in kelvins: add 273.15 to a Celsius reading. The gas laws are proportionalities, and a proportionality only works on a scale whose zero means zero. Going from 10 °C to 20 °C does not double a gas's volume, but doubling it from 283 K to 566 K does double the volume of a gas held at constant pressure.

Boyle, Charles and Avogadro as slices of one law

Each classic law holds two variables constant. Boyle's law: at constant n and T, P₁V₁ = P₂V₂, so squeezing a gas into half the volume doubles its pressure. Charles's law: at constant n and P, V₁/T₁ = V₂/T₂, so volume rises in proportion to Kelvin temperature. Avogadro's law: at constant P and T, V₁/n₁ = V₂/n₂, so equal gas volumes hold equal numbers of molecules.

Together they give the combined gas law for a fixed amount of gas, P₁V₁/T₁ = P₂V₂/T₂. Once you trust this form, cross out whatever stays constant and the remaining variables give the correct special case.

Worked example 1. A cylinder with a movable piston holds 1.75 L of nitrogen at 18.0 °C and 0.985 atm. The gas is cooled to −8.0 °C while the pressure on the piston is lowered to 0.712 atm. Convert temperatures first: T₁ = 291.15 K and T₂ = 265.15 K. Solve for V₂ = V₁ × (P₁/P₂) × (T₂/T₁) = 1.75 L × (0.985 atm / 0.712 atm) × (265.15 K / 291.15 K) = 2.20 L. Check each factor's direction: lower pressure expands the gas (ratio above 1), and cooling shrinks it (ratio below 1).

Solving problems with PV = nRT

Adding the amount of gas to the combined law gives the ideal gas law, PV = nRT. Each earlier law is a special case of it.

With pressure in atmospheres and volume in liters, use R = 0.08206 L·atm/(mol·K); with pressure in pascals and volume in cubic meters, use R = 8.314 J/(mol·K).

At 273.15 K and 1 atm, one mole of an ideal gas occupies 22.41 L. Since 1982, IUPAC has defined standard pressure as 1 bar, where the molar volume at 273.15 K is 22.71 L. Check which convention your course uses before relying on 22.4 L per mole as a shortcut.

Worked example 2. A 2.50 L steel cylinder contains 0.180 mol of argon at 22.0 °C. Find the pressure from P = nRT/V. T = 22.0 + 273.15 = 295.15 K. P = (0.180 mol)(0.08206 L·atm/(mol·K))(295.15 K) / 2.50 L = 1.74 atm. In kilopascals the result is 1.74 atm × 101.325 kPa/atm = 177 kPa.

Gas density and molar mass from a flask measurement

Because n = m/M, where m is mass and M is molar mass, the ideal gas law can be rewritten as PM = dRT, with d = m/V the gas density. At fixed temperature and pressure, gas density is proportional to molar mass, so weighing a known volume of gas gives its molar mass.

Worked example 3. A 0.500 L flask filled with an unknown gas at 35.0 °C and 0.968 atm contains 0.967 g of the gas. Substituting n = m/M into the ideal gas law and rearranging gives M = (m × R × T)/(P × V) = (0.967 g)(0.08206 L·atm/(mol·K))(308.15 K) / [(0.968 atm)(0.500 L)] = 50.5 g/mol. Using IUPAC standard atomic weights rounded to two decimals (H 1.01, C 12.01, Cl 35.45), chloromethane, CH₃Cl, has M = 12.01 + 3(1.01) + 35.45 = 50.49 g/mol, so the data are consistent with that gas. Consistent is not proven: another gas with a similar molar mass would give the same result.

Partial pressures in mixtures and gases collected over water

In a mixture of nonreacting gases, each gas behaves as if it were alone. Dalton's law states that the total pressure is the sum of the partial pressures: P_total = P_A + P_B + …. Each partial pressure equals the mole fraction of that gas times the total pressure, P_A = x_A × P_total, where x_A = n_A/n_total.

Worked example 4. A 4.00 L vessel at 25.0 °C holds 0.150 mol N₂ and 0.0450 mol CO₂. The total amount is 0.195 mol, so P_total = (0.195 mol)(0.08206 L·atm/(mol·K))(298.15 K) / 4.00 L = 1.19 atm. The mole fraction of N₂ is 0.150/0.195 = 0.769, giving P(N₂) = 0.917 atm, and P(CO₂) = 0.275 atm. They sum to 1.19 atm, as they must.

Gas collected over water is mixed with water vapor, so subtract the vapor pressure of water at that temperature first. For hydrogen collected at 23 °C with a total pressure of 754 torr, the water vapor pressure is 21.1 torr (OpenStax Table 9.2), so P(H₂) = 754 − 21.1 = 733 torr, or 0.964 atm. In a 0.312 L sample, n = (P × V)/(R × T) = (0.964 atm)(0.312 L) / [(0.08206 L·atm/(mol·K))(296.15 K)] = 0.0124 mol H₂.

Gas-law errors to catch before you submit

Most wrong answers in gas problems come from a few predictable slips, each with a specific check.

The most useful habit is to predict whether the answer should grow or shrink before calculating. If the arithmetic disagrees, look for an inverted ratio.

• Celsius in the equation: a temperature of 25 instead of 298.15 K gives an answer off by a factor of about 12. For ordinary lab conditions, every T you substitute should be roughly 250 K to 400 K.
• Mismatched R: using 8.314 with atm and L puts the answer off by a factor of about 100.
• Grams for moles: n in PV = nRT is an amount in moles. If the problem gives mass, divide by molar mass first, or use PM = dRT.
• Forgetting water vapor: a gas collected over water is not dry. Subtract the vapor pressure at the collection temperature.

When real gases stop behaving ideally

The ideal gas law assumes that molecules take up no volume and do not attract one another. Real gases approximate this at low pressure and high temperature. At high pressure, the molecules' own volume becomes a noticeable fraction of the container, so the gas is less compressible than predicted. At low temperature, attractions pull molecules together and the measured pressure falls below the ideal value.

The van der Waals equation adds two corrections: a term in a for attractions and a term in b for molecular size. Gases with strong intermolecular forces have larger a values and deviate more: OpenStax Table 9.3 lists a = 5.46 L²·atm/mol² for water vapor against 1.39 for nitrogen and 0.0342 for helium. Near 1 atm and room temperature the ideal gas law is usually good to a few percent, but do not trust it close to the point where a gas condenses.