The Ideal Gas Law: PV = nRT
The ideal gas law is the equation PV = nRT, which relates the pressure (P), volume (V), amount in moles (n) and absolute temperature (T) of an ideal gas through the gas constant R. In brief: one mole of gas at 273.15 K and 1 atm has a volume of V = nRT/P = 1 × 0.08206 × 273.15 / 1 = 22.41 L.
The ideal gas law combines Boyle’s, Charles’s, Gay-Lussac’s and Avogadro’s laws into one equation, so it is the most useful relationship in gas chemistry. It lets you find any one of the four variables when the other three are known. This article explains the symbols and units, gives the values of R, solves several examples and notes where the law fails.
What is the ideal gas law?
The law states that PV = nRT, where:
- P is the absolute pressure,
- V is the volume of the gas,
- n is the amount of gas in moles,
- R is the universal gas constant,
- T is the absolute temperature in kelvins.
An ideal gas is a hypothetical gas whose particles have no volume and exert no forces on one another. Real gases behave almost ideally at low pressure and high temperature. The law reduces to the earlier gas laws when one or more variables are held constant, for example Boyle’s law (n and T constant) and Charles’s law (n and P constant).
What is the gas constant R?
R is a proportionality constant that has the same value for all gases. Its numerical value depends on the units of pressure and volume. Since the 2019 redefinition of the SI units, R = NA × kB is an exact value, 8.314462618… J/(mol·K), which is usually rounded to 8.314. The units of R must match the units of P and V; this is the most important practical decision in any ideal gas problem.
Values of R in different units
| Value of R | Units | Use when pressure is in |
|---|---|---|
| 0.08206 | L·atm/(mol·K) | atm, volume in L |
| 62.36 | L·mmHg/(mol·K) | mmHg or torr, volume in L |
| 8.314 | L·kPa/(mol·K) | kPa, volume in L |
| 8.314 | J/(mol·K) or Pa·m3/(mol·K) | Pa, volume in m3 |
| 0.08314 | L·bar/(mol·K) | bar, volume in L |
How do you solve ideal gas law problems?
- List the known quantities and identify the unknown.
- Convert the temperature to kelvins.
- Choose a value of R whose units match the pressure and volume units.
- Rearrange the equation: P = nRT/V, V = nRT/P, n = PV/(RT) or T = PV/(nR).
- Substitute and calculate. Check that the answer is physically reasonable.
Worked examples
Example 1: finding the pressure
A 10.0 L tank contains 2.50 mol of nitrogen at 27 °C. What is the pressure?
T = 300.15 K. P = nRT / V = 2.50 × 0.08206 × 300.15 / 10.0 = 6.16 atm.
Example 2: finding the moles
How many moles of gas are in a 500 mL flask at 98.0 kPa and 20 °C?
V = 0.500 L and T = 293.15 K. n = PV / RT = 98.0 × 0.500 / (8.314 × 293.15) = 49.0 / 2437.2 = 0.0201 mol.
Example 3: finding the volume
What volume is occupied by 4.00 g of hydrogen gas at 1.50 atm and 25 °C?
- n = 4.00 g / 2.016 g/mol = 1.984 mol.
- T = 298.15 K.
- V = nRT / P = 1.984 × 0.08206 × 298.15 / 1.50 = 48.54 / 1.50 = 32.4 L.
Example 4: finding the temperature
A 2.00 L container holds 0.100 mol of gas at 1.50 atm. What is the temperature?
T = PV / nR = 1.50 × 2.00 / (0.100 × 0.08206) = 3.00 / 0.008206 = 365.6 K, which is about 92 °C.
How was the ideal gas law developed?
The individual laws were found over two centuries: Boyle in 1662, Charles around 1787 (published by Gay-Lussac in 1802) and Avogadro’s hypothesis in 1811. Émile Clapeyron combined the early laws into a single equation in 1834. The form PV = nRT, with the molar gas constant, became standard after the work of Dmitri Mendeleev, who also derived a version of the equation, in 1874. For this reason the law is sometimes called the Clapeyron–Mendeleev equation, especially in Russian-language texts.
When is a gas not ideal?
Real gas molecules have a finite volume and attract each other. These effects matter at high pressures, where the molecules are close together, and at low temperatures, where they move slowly and attract each other more effectively. For example, CO2 at 1 atm and 25 °C obeys the law within about 0.5 %, but at 50 atm the error exceeds 10 %. Gases with strong intermolecular forces, such as ammonia and water vapor, deviate more than helium or hydrogen.
Which mistakes are common?
- Using Celsius for T: always add 273.15.
- Mismatched units of R: 0.08206 only works with atmospheres and liters.
- Forgetting to convert mass to moles: n must be in moles, so divide the mass by the molar mass.
- Using gauge pressure: the law needs the absolute pressure.
- Using volume in mL with R in liters: convert to liters first.