Gas-Law Formulae

Introduction

Gas formulae link measurable pressure, volume and temperature to amount of substance. The ideal-gas equation is versatile because it also describes each component of an ideal mixture. Yet it is a model, not a universal identity. The page's central habit is to define the gas state, convert temperature to kelvin and inspect whether nonideal effects could change the answer.

Core explanation

The ideal-gas equation is PV = nRT, with pressure P, volume V, gas amount n and absolute temperature T. In SI, P is Pa, V is m³ and R ≈ 8.314 J mol⁻¹ K⁻¹. Since Pa m³ = J, the units balance. Another consistent choice is L, atm and R ≈ 0.082057 L atm mol⁻¹ K⁻¹. Mixing these sets without conversion causes systematic errors.

At fixed n, the model predicts PV/T = constant between states. If T stays constant, P is inversely proportional to V; if P stays constant, V is proportional to T. These relationships follow directly from the equation and require temperature in kelvin. A rise from 20 °C to 40 °C is not a doubling of thermodynamic temperature.

For an ideal mixture, each component i satisfies P i V = n i RT in the same total volume at common T. Summing gives P total = ΣP i. Since x i = n i/n total, Dalton's relation is P i = x iP total. The partial pressure refers to the gas phase, not the mole fraction of a component in an adjacent liquid.

Gas collected over water contains water vapour in the headspace. If the gas mixture is ideal and liquid water is present at equilibrium, dry-gas pressure is P dry = P total − P H₂O at the collection temperature. Ignoring vapour pressure overestimates dry gas amount. The water vapour pressure is temperature dependent and should be measured or obtained for the actual conditions.

Molar volume is not a universal fixed 22.4 L. From V m = RT/P, it depends on stated T and P. At 273.15 K and 100 kPa, ideal molar volume is about 22.71 L mol⁻¹; at 1 atm the result differs. Always specify the standard-state convention before using a memorized molar volume.

Real gases depart from ideality when attractions and finite molecular volume matter, particularly at high pressure or near condensation. The compressibility factor Z = PV/(nRT) equals one for an ideal gas. If Z is measured or modeled, n = PV/(ZRT). A value Z ≠ 1 shows that an ideal estimate needs correction, but a single Z may vary with P, T and mixture composition.

Gas reactions also use stoichiometry. Mole fractions and partial pressures can change as reaction occurs even at constant total pressure. In a rigid sealed vessel, total pressure may change with total gas moles; in a piston at fixed pressure, volume may change. Identify which state variables are held fixed before applying a combined-gas relation.

Step-by-step reasoning

List gas components and whether the reported pressure includes vapour. Convert T to K and choose a consistent P–V–R unit set. Use PV = nRT for total or component amounts. For mixtures, compute mole fractions and partial pressures. Consider Z or other corrections if pressure is high, temperature low or accuracy requirements are tight.

Visual explanation

Sketch one container with red and blue gas particles. Label its common V and T, then write P red = n redRT/V and P blue = n blueRT/V. Their sum gives total P. Add a small droplet at the bottom to remind the reader that wet collected gas includes water vapour.

Real-world analogy

Several groups sharing a room each contribute to crowding in proportion to their numbers if interactions are negligible. The total crowding is the sum of group contributions. This resembles ideal partial pressures, but real molecules may attract and exclude volume in ways people do not model precisely.

Real-world example

A gas collection experiment reports 100.0 kPa total pressure and 3.2 kPa water vapour pressure. The dry reaction gas contributes 96.8 kPa. Using total pressure in PV = nRT would overcount dry gas by about 3.3% under the same volume and temperature.

Why?

Gas formulae support reaction yields, atmospheric mixtures, combustion and laboratory gas collection. Their transparent assumptions prevent accidental misuse of a convenient but imperfect model. This matters especially when pressure or temperature makes ideality doubtful.

Common misconception

“Standard molar volume is always 22.4 L” ignores the chosen pressure standard and temperature. Another error assigns partial pressure from mass fraction rather than gas-phase mole fraction.

Worked example

A rigid 10.0 L container at 298 K contains 0.100 mol N₂ and 0.0500 mol O₂. With R = 0.082057 L atm mol⁻¹ K⁻¹, total pressure is (0.150 × 0.082057 × 298)/10.0 ≈ 0.367 atm. O₂ mole fraction is 1/3, so its partial pressure is about 0.122 atm; N₂ contributes about 0.245 atm. Their sum checks the total.

Quick check

1. Why must 25 °C be converted before using PV = nRT? Answer: The ideal-gas equation requires absolute temperature, 298.15 K.

Exam focus

State the gas model and unit set. Use P i = x iP total only for the gas mixture under ideal behavior. Correct gas collected over water and identify when a real-gas Z factor is warranted.

Advanced insight

For mixtures at high pressure, component fugacities replace simple partial pressures in rigorous equilibrium calculations. This extends the same principle seen elsewhere: a classroom concentration or pressure expression is an approximation to a thermodynamic activity.

Summary

PV = nRT connects ideal-gas state variables, and Dalton's law partitions ideal-mixture pressure by mole fraction. Temperature and units must be consistent, wet gases need vapour correction, and high-pressure or near-condensation conditions may require nonideal treatment.

Practice questions

1. What is x A for 2 mol A and 3 mol B? Answer: x A = 2/5 = 0.40. 2. If total ideal-mixture pressure is 200 kPa, what is P A at x A = 0.40? Answer: P A = 80 kPa. 3. What does Z = 1 mean in the compressibility definition? Answer: The measured state follows the ideal-gas PV = nRT relation. 4. Why subtract water vapour pressure from a wet collected-gas pressure? Answer: The reported total includes vapour as well as the dry target gas.

Sources

- BIPM SI Brochure. - IUPAC Gold Book.