Ideal Gas Law Calculator — PV = nRT Simulator & Guide

What Is the Ideal Gas Law?

An ideal gas is a theoretical model in which gas molecules have negligible volume and interact only through perfectly elastic collisions — no intermolecular attractions or repulsions. For such a gas, three experimental observations — Boyle's Law, Charles' Law, and Avogadro's Law — can be combined into a single equation:

where:

• P — pressure (kPa or Pa)
• V — volume (L or m³)
• n — amount of substance (moles)
• R — universal gas constant = 8.314 J/(mol·K)
• T — absolute temperature (Kelvin; K = °C + 273.15)

Real gases (e.g. CO₂ at high pressure, steam near its boiling point) deviate from this model, but for most engineering conditions at moderate pressures and temperatures PV = nRT gives answers within 1–2% of measured values.

The Four Gas Variables — Solve for Any One

The beauty of PV = nRT is that it can be rearranged to solve for whichever variable is unknown, given the other three:

In the simulator, the four Lock tabs let you select which variable to solve for. The locked variable's slider becomes read-only and updates automatically as you move the other three.

Worked Example: Pressure of Air in a Container

The simulator's default "Solve for P" state uses n = 0.5 mol, T = 300 K (27°C), V = 2.0 L:

That is about 6.15 atmospheres — roughly the pressure inside a compressed-air cylinder. The simulator confirms this reading, and the live P-V diagram plots the current state dot on the isotherm for T = 300 K.

Vary the temperature slider from 200 K to 800 K and watch the state dot move along isotherms on the P-V diagram. This is Boyle's Law in action within a single temperature change.

Boyle's, Charles' and Gay-Lussac's Laws

Each of the three classical gas laws is a special case of PV = nRT with two variables held constant:

Boyle's Law — constant T and n

Pressure and volume are inversely proportional. Halving the volume doubles the pressure. In the simulator: lock to "Solve for P", hold T and n fixed, and drag the V slider — the state dot traces the current isotherm on the P-V diagram.

Charles' Law — constant P and n

Volume is directly proportional to absolute temperature at fixed pressure. Gas at 300 K occupying 3 L expands to 6 L when heated to 600 K. This explains why a balloon shrinks in the cold — it obeys Charles' Law.

Gay-Lussac's Law — constant V and n

Pressure is directly proportional to temperature at fixed volume. A sealed aerosol can at 20°C (293 K) heated to 60°C (333 K) increases in pressure by a factor of 333/293 = 1.14 — a 14% rise that could rupture a container near its rated burst pressure.

The Combined Gas Law

When a gas changes state — all three of P, V, T change while n stays fixed — the combined gas law is the most convenient form:

Example: A gas at P₁ = 100 kPa, V₁ = 3 L, T₁ = 300 K is compressed to V₂ = 1 L and heated to T₂ = 400 K:

This scenario appears in the simulator's "Combined" section of the Explore mode, with a worked example and graph.

Molar Volume at STP

At Standard Temperature and Pressure (0°C = 273.15 K, 101.325 kPa), one mole of any ideal gas occupies exactly 22.414 litres. Verification with PV = nRT:

This is a crucial benchmark: if you calculate a molar volume significantly different from 22.4 L/mol at STP, check your unit conversions — most errors arise from mixing kPa and Pa, or Celsius and Kelvin.

Real-World Applications

• Tyre pressure — A car tyre at 20°C (293 K) pumped to 220 kPa will read approximately 220 × 313/293 = 235 kPa after driving to 40°C (313 K), because V and n are constant (Gay-Lussac).
• Scuba tank — A 12-litre tank at 200 bar (20 000 kPa) and 20°C contains n = PV/RT = 20 000 × 12 / (8.314 × 293) ≈ 98 mol of air — enough for about 50 minutes of diving.
• Weather balloons — As a balloon rises, external pressure falls, so the gas expands (Boyle's Law). At 30 km altitude, where P ≈ 1.2 kPa, a balloon that was 1 m³ at sea level expands to about 84 m³.
• Engine combustion — The rapid pressure rise during the power stroke of a four-stroke engine follows Gay-Lussac's Law: combustion raises T, pressure rises dramatically in the fixed-volume cylinder, driving the piston.

Using the Ideal Gas Law Simulator

Open the Ideal Gas Law Simulator and explore these four modes:

1. Simulate — Choose "Solve for" P, V, T, or n using the Lock tabs. Adjust the three independent sliders and watch the locked variable update instantly. The P-V diagram plots the state dot on the correct isotherm, and the animated gas container shows particles speeding up as T increases.
2. Explore — Read concept cards grouped by Laws, Equation, Combined processes, and Applications. Each card has a formula, description, and a worked numerical example.
3. Practice — Solve 12 randomised problems: find P from n, T, V; find V using Boyle's Law; apply Gay-Lussac to a tyre; calculate moles in a scuba tank. Full step-by-step solutions are shown on request.
4. Quiz — Test yourself with 5 random questions from a pool of 15, covering all gas laws and their applications.

Switch the gas type between Air, Helium, CO₂, and Nitrogen to see how the colour of the particle animation changes, representing the different thermal behaviour at the same temperature.

Key Takeaways

• The ideal gas law \(PV = nRT\) relates all four gas variables; rearrange to solve for any one.
• R = 8.314 J/(mol·K) — always use absolute temperature in Kelvin, never Celsius.
• Boyle's Law: \(P_1V_1 = P_2V_2\) at constant T and n — pressure and volume are inversely proportional.
• Charles' Law: \(V/T = \text{const}\) at constant P — volume proportional to absolute temperature.
• Gay-Lussac's Law: \(P/T = \text{const}\) at constant V — sealed containers pressure-up when heated.
• At STP (0°C, 101.325 kPa), 1 mol of ideal gas occupies exactly 22.4 L.
• Real gases deviate from PV = nRT at high pressure or near their boiling point.