How To Calculate The Ideal Gas Law

The ideal gas law is the equation of state written PV = nRT. It states that for a fixed amount of gas, the product of pressure and volume is proportional to the absolute temperature, the constant of proportionality being nR. Because four quantities are linked, fixing any three fixes the fourth, which is what makes the equation so widely used. Boyle's law, Charles's law and Avogadro's law are all special cases of it, recovered by holding two of the variables constant. That economy is the reason it is taught as a single relation rather than as three separate ones.

The letter R is called the gas constant, and it is the part of the equation that confuses people most. R is a genuine physical constant, equal to the Avogadro constant multiplied by the Boltzmann constant, but its numerical value depends entirely on the units chosen for pressure and volume. Write pressure in kilopascals and volume in litres and R is 8.314462618; write pressure in atmospheres and volume in litres and R is 0.08206. The constant has not changed; only the way it is expressed has. Understanding this before substituting anything is the single most useful habit you can build on this topic.

Four values of R cover nearly every school and laboratory problem. With pressure in kilopascals and volume in litres, R is 8.314462618. With pressure in atmospheres and volume in litres, R is 0.08206. With pressure in pascals and volume in cubic metres, R is again 8.314, because the pascal multiplied by the cubic metre is the joule, the same case in disguise. With pressure in millimetres of mercury and volume in litres, R is 62.36. Pairing any of these with the wrong pressure or volume unit is the most frequent error on the topic, and it is the one this page exists to prevent.

Temperature in the ideal gas law is always absolute, which for practical purposes means kelvin. The equation is built on a scale that begins at absolute zero, so a Celsius reading has to be shifted by 273.15 before it can be used. Substituting 25 for a room temperature of 25 C, when the correct value is 298.15 K, understates the temperature by a factor of roughly twelve. The resulting error is not a rounding difference but an order-of-magnitude one, and it usually shows up as an absurd number of moles. Convert first, then substitute.

At standard temperature and pressure, one mole of an ideal gas occupies a volume that textbooks round to 22.4 litres. The exact value at 0 C and 101.325 kPa is 22.4139695 litres, so 22.4 is a convenience rather than a definition. This is why the worked calculation on this page returns 0.9993767483 moles for 22.4 litres instead of exactly one mole. The shortfall of about 0.06 per cent is the rounding in the textbook figure showing through the arithmetic. Treating 22.4 as exact quietly caps the accuracy of everything built on it.

The two standard states that appear in questions are not interchangeable. STP, standard temperature and pressure, is defined at 0 C, while SATP, standard ambient temperature and pressure, is defined at 25 C. The reference pressure has also shifted over time, from one atmosphere at 101.325 kPa to exactly 100 kPa under the current IUPAC convention. Because SATP is the warmer of the two, the molar volume there is larger than at STP, and the difference is not negligible in careful work. Always establish which standard a question intends before quoting a molar volume, because the two answers differ by a few per cent.

The word ideal in the name is a warning as much as a description. An ideal gas is an imaginary substance whose particles have no volume of their own and exert no forces on one another. Real gases approach that picture at low pressure and high temperature, where the molecules are far apart and moving quickly. At high pressure or near the point of condensation the approximations break down, because molecular volume and intermolecular attraction both start to matter. The van der Waals equation adds two correction terms to patch these failures, which is why it appears whenever real-gas accuracy is needed.

Rearranging the law is straightforward once the units are settled. Solving for amount gives n = PV / RT; solving for volume gives V = nRT / P; solving for pressure gives P = nRT / V. A useful variant replaces n with mass divided by molar mass, producing PM = dRT, which connects the law to gas density and lets a measured density yield a molar mass. Dividing through by n gives the molar volume Vm = V / n = RT / P, the quantity that turns 22.4139695 litres per mole into a property of the conditions rather than of the gas. Every one of these forms is the same equation seen from a different angle.

Precision should follow the inputs, as it does everywhere else. The gas constant is known to many digits, but the pressure and temperature you measure rarely are, so carry full precision through the arithmetic and round only the final answer. A temperature read to the nearest degree does not justify six decimal places in the result. The one place where extra digits genuinely matter is the gap between 22.4 and 22.4139695, which is small but real and occasionally the difference between a correct and an incorrect exam answer. Report an answer that the measurement can defend.