The gaseous state: ideal and real gases and pV = nRT

Gas pressure

· Gas pressure is caused by collisions between gas molecules and the walls of the container.
· Each collision transfers momentum to the container wall, producing a force.
· Pressure = force per unit area, so more frequent or more forceful collisions increase pressure.
· Increasing temperature increases the average kinetic energy of gas molecules, so collisions with the walls are more frequent and forceful.
· Increasing the number of gas molecules in a fixed volume increases collision frequency, so pressure increases.
· Decreasing volume gives molecules less space, so they collide with the walls more often, increasing pressure.

Gas pressure is caused by gas particles colliding with container walls. Increasing temperature or decreasing volume increases the frequency and/or force of collisions, raising pressure. Source

Ideal gases

· An ideal gas is a theoretical gas that obeys pV = nRT exactly.
· Ideal gas particles are assumed to have zero particle volume.
· Ideal gas particles are assumed to have no intermolecular forces of attraction.
· Collisions between particles and container walls are treated as perfectly elastic.
· Ideal behaviour is most closely approached at high temperature and low pressure, when particles are far apart and attractions are minimal.
· In CIE exam answers, define ideal gas assumptions clearly: negligible molecular volume + no intermolecular forces.

The diagram illustrates gas particles moving randomly and colliding with the container wall. These collisions explain the origin of gas pressure in terms of particle motion. Source

Real gases

· Real gases do not behave perfectly ideally because particles have finite volume and can experience intermolecular forces.
· Real gases deviate most from ideal behaviour at high pressure because particle volume becomes significant.
· Real gases deviate most from ideal behaviour at low temperature because particles move more slowly and attractions have a greater effect.
· Real gases behave more like ideal gases when particles are far apart and moving quickly: low pressure, high temperature.
· Exam phrasing: “A real gas differs from an ideal gas because its molecules have non-zero volume and intermolecular attractions.”

Ideal gas equation: pV = nRT

· pV = nRT links the measurable properties of a gas.
· p = pressure, usually in Pa if using SI units.
· V = volume in m³ if using SI units.
· n = amount of gas in mol.
· R = gas constant, usually 8.31 J mol⁻¹ K⁻¹.
· T = temperature in K, never °C.
· Always convert temperature using K = °C + 273.
· If pressure is given in kPa, convert to Pa by multiplying by 1000 unless using a matching value of R.
· If volume is given in cm³, convert to m³ by dividing by 1 000 000.
· If volume is given in dm³, convert to m³ by dividing by 1000.

This diagram links the ideal gas law to the simpler gas relationships involving pressure, volume, temperature and amount. It is useful for seeing why pV = nRT combines the main gas variables into one equation. Source

Rearranging pV = nRT

· To find pressure: p = nRT ÷ V.
· To find volume: V = nRT ÷ p.
· To find moles: n = pV ÷ RT.
· To find temperature: T = pV ÷ nR.
· Check that p, V, T and R use compatible units before substituting.
· A common exam error is using °C instead of K.
· Another common error is using cm³ or dm³ directly with R = 8.31.

Determining Mr using pV = nRT

· The syllabus specifically requires use of pV = nRT in calculations, including determination of Mr.
· First calculate moles using n = pV ÷ RT.
· Then use Mr = mass ÷ moles.
· If mass is given in grams, Mr has units g mol⁻¹ numerically, but relative molecular mass itself is usually written without units.
· For volatile liquids, a known mass may be vaporised; use gas volume, pressure and temperature to find n, then calculate Mr.
· Always ensure the gas sample is treated as an ideal gas approximation unless the question states otherwise.

Exam calculation method

· Write the equation: pV = nRT.
· Convert units: p in Pa, V in m³, T in K, R = 8.31 J mol⁻¹ K⁻¹.
· Rearrange before substituting to avoid calculator errors.
· Substitute values with units shown clearly.
· Give the answer to an appropriate number of significant figures.
· Include units where required, especially for pressure, volume, temperature and amount of substance.

The isotherms show that, at constant temperature, pressure and volume are inversely related. This supports the relationship contained within pV = nRT when n and T are constant. Source