The ideal gas law
The pressure of a gas is nothing more than millions of molecules hitting the walls. Heat it, compress it or add gas and see what happens.
Initial values: 1 mol at 0 °C in 22.4 L, that is, 1 atm. These are standard conditions.
PV = nRT
The ideal gas law links the four quantities that describe the state of a gas: pressure P, volume V, amount of substance n (in moles) and absolute temperature T (in kelvin):
P · V = n · R · T with R = 8.314 J/(mol·K) = 0.08206 atm·L/(mol·K)
It combines three historical laws that you can check in the simulator by moving a single slider:
- Boyle’s law (constant T): when the volume shrinks, the pressure rises in the same proportion. The molecules have less room and hit the walls more often. On the P–V diagram you move along an isotherm, a hyperbola.
- Charles’s law (constant P): volume is proportional to temperature. A balloon shrinks in the freezer.
- Gay-Lussac’s law (constant V): pressure is proportional to temperature. When heated, the molecules move faster and hit harder and more often. That is why you should never heat an aerosol can.
What does the speed of the molecules depend on?
Temperature measures the average kinetic energy of the molecules. Their root-mean-square speed is vrms = √(3RT/M), where M is the molar mass. At the same temperature, helium (M = 4 g/mol) moves almost 3 times faster than nitrogen (28 g/mol), yet the pressure is the same: its molecules are lighter and hit more often, but with less mass. Change the gas in the simulator and see that the pressure does not change.
The speeds are not all the same: they follow the Maxwell-Boltzmann distribution, which is why the animation shows slow and fast molecules at the same time.
The ideal gas model assumes molecules with no volume of their own and no forces between them. It works very well at low pressure and high temperature; at high pressure or near condensation you need equations such as Van der Waals’.
Frequently asked questions
What is the value of the gas constant R?
R = 8.314 J/(mol·K) in SI units, with pressure in pascals and volume in cubic metres. If you work with atmospheres and litres, R = 0.08206 atm·L/(mol·K). Picking the wrong value is the most common mistake when using PV = nRT.
What volume does one mole of gas occupy?
At 0 °C and 1 atm, one mole of any ideal gas takes up 22.4 litres. At 0 °C and 1 bar, the current IUPAC standard conditions, it takes up 22.7 L, and at 25 °C and 1 atm about 24.5 L. It does not depend on which gas it is: a mole of helium and a mole of carbon dioxide occupy the same volume.
Why must temperature be in kelvin?
Because the pressure and volume of a gas are proportional to absolute temperature, measured from absolute zero (−273.15 °C). Going from 10 °C to 20 °C does not double the pressure: in kelvin it is 283 K to 293 K, an increase of only 3.5 %. To convert, add 273.15 to the Celsius value.
When does the ideal gas law stop working?
At high pressure and low temperature, close to the point where the gas condenses. Then the volume of the molecules themselves and the attraction between them are no longer negligible. In those cases real-gas models such as the Van der Waals equation are used. At atmospheric pressure and room temperature, air behaves almost like an ideal gas.