Ideal gas pressure:
from moles, temperature and volume to pascals

Why pressure scales with moles and temperature

Gas pressure is the combined push of countless molecules striking the walls of their container. More molecules mean more strikes per second, so pressure rises with the amount of gas. Hotter molecules move faster and hit harder, so pressure rises with temperature as well. Give the same gas more room and the strikes thin out, which is why pressure falls as volume grows.

The ideal gas law packs all three relationships into one line. It treats molecules as points that do not attract or repel each other, an assumption that works well for common gases at moderate pressure and temperatures well above their boiling points, and degrades as gases are squeezed or cooled toward liquefaction.

The equation rearranged for pressure

The law is usually remembered as PV = nRT. Dividing both sides by V gives pressure directly. What matters in practice is that R carries its units with it: 8.314 J/(mol·K) is the same as 8.314 Pa·m³/(mol·K). That means the volume must be in cubic metres and the answer arrives in pascals.

Two unit habits cause most wrong answers. First, temperature must be on the kelvin scale, so add 273.15 to degrees Celsius. Second, a volume given in litres has to be divided by 1000 to reach cubic metres; 10 L is 0.010 m³.

Worked example: 0.5 mol of gas in a 10 L vessel

A sealed 10 L vessel holds 0.5 mol of nitrogen at 298 K, about room temperature. Convert the volume to 0.010 m³, then substitute.

Divide by 101,325 Pa per standard atmosphere and the same pressure is about 1.22 atm, or 1.24 bar. A pressure gauge on such a vessel would read the excess over the surrounding air, so about 0.22 atm if the room is at one atmosphere, because most gauges are calibrated against ambient pressure.

What happens when you change one thing at a time

Because every symbol sits in a simple product or ratio, it is easy to predict direction before touching numbers. Doubling the moles doubles the pressure when volume and temperature are held fixed. Doubling the kelvin temperature does the same. Doubling the volume halves it.

• Heat the 0.5 mol sample from 298 K to 596 K at constant volume and pressure climbs from about 124 kPa to about 248 kPa.
• Squeeze it into 5 L and pressure also doubles to about 248 kPa, assuming the temperature is held.
• Add another 0.5 mol at the original conditions and the vessel reaches about 248 kPa.

Mind the word kelvin in the temperature line: 25 °C to 50 °C is only an increase from 298 K to 323 K, about 8%, not a doubling. Pressure rises by that same 8%.

Rearranging for moles, volume or temperature

The same relationship is often needed the other way round. If pressure, volume and temperature are measured and the amount of gas is unknown, divide pressure times volume by R times T. If you need the volume a known amount will fill at a chosen pressure, multiply n, R and T and divide by pressure.

• Moles: n = pV ÷ (RT)
• Volume: V = nRT ÷ p
• Temperature: T = pV ÷ (nR)

A useful anchor is the molar volume. One mole of an ideal gas at 101,325 Pa and 298 K fills 8.314 × 298 ÷ 101,325 = 0.02445 m³, which is 24.45 L. So a 1 L flask of any ideal gas under those conditions contains about 1 ÷ 24.45 = 0.0409 mol, or 40.9 mmol. That quick figure is a good sanity check on any answer you obtain; if a litre of room-temperature gas comes out as several moles, a unit has slipped.

Pressure units you will meet

When a problem hands you pressure in a different unit, convert it to pascals before using the 8.314 constant, rather than trying to find a new gas constant for the unit. The gas in a mixture follows the same rule: add up the moles of every gas in the vessel, because the total pressure depends on the overall molecule count, not on the type of molecule.

A car tyre inflated to 32 psi gauge is at about 220 kPa above the atmosphere, roughly 320 kPa absolute. That distinction between gauge and absolute is exactly the one that decides whether a gas-law calculation matches the number on the dial.

Where the ideal gas law stops being reliable

Real molecules occupy space and attract one another. At pressures of tens of atmospheres, or near the temperature where a gas condenses, measured pressure departs from nRT/V by a visible margin. Steam near its boiling point, carbon dioxide in a high-pressure cylinder and refrigerants in a working compressor are typical cases where an equation of state with correction terms, such as the van der Waals equation, is the better tool.

For air, nitrogen, oxygen and helium at ordinary room conditions the ideal law is usually good to a fraction of a percent, which is accurate enough for most schoolwork, lab planning and rough engineering checks. When safety is involved, for example sizing a pressure vessel, use the code-approved method and the manufacturer's data rather than this shortcut.

A separate chemistry magnitude tool can keep related quantities tidy while you work through a set of problems, though the P = nRT/V step itself is simple enough to verify by hand with the unit conversions above.