Ideal Gas Law Calculator

About Ideal Gas Law Calculator

The Ideal Gas Law Formula

PV = nRT relates pressure, volume, temperature, and the number of moles of gas. It combines three earlier gas laws discovered between 1662 and 1808 by Boyle, Charles, and Avogadro.

Solve For	Formula	Example
Pressure	P = nRT / V	1 mol x 8.314 x 298 K / 0.0244 m³ = 101,543 Pa
Volume	V = nRT / P	1 mol x 8.314 x 273.15 K / 101,325 Pa = 0.02241 m³ (22.41 L)
Moles	n = PV / RT	101,325 x 0.01 / (8.314 x 300) = 0.406 mol
Temperature	T = PV / nR	200,000 x 0.005 / (0.5 x 8.314) = 240.5 K

Worked example: a balloon contains 0.05 mol of helium at 25 °C (298.15 K) and 1 atm (101,325 Pa). What is the volume? V = nRT/P = 0.05 x 8.314 x 298.15 / 101,325 = 0.001224 m³ = 1.224 litres. This matches the expected small size of a partially inflated helium balloon.

The Gas Constant R in Different Unit Systems

Value of R	Units	When to Use
8.314	J/(mol K) or Pa m³/(mol K)	SI units (Pa and m³)
8.314	kPa L/(mol K)	Pressure in kPa, volume in litres
0.08206	atm L/(mol K)	Pressure in atm, volume in litres
62.36	mmHg L/(mol K)	Pressure in mmHg, volume in litres
1.987	cal/(mol K)	Thermochemistry

The calculator handles unit matching automatically. You can enter pressure in any supported unit and volume in any supported unit, and it converts internally before applying the formula.

Standard Conditions Reference

Standard	Temperature	Pressure	Molar Volume	Used In
STP (IUPAC, current)	273.15 K (0 °C)	100 kPa (1 bar)	22.711 L/mol	Modern chemistry
STP (old definition)	273.15 K (0 °C)	101.325 kPa (1 atm)	22.414 L/mol	Older textbooks
NTP	293.15 K (20 °C)	101.325 kPa (1 atm)	24.04 L/mol	Engineering, NIST
SATP	298.15 K (25 °C)	100 kPa	24.79 L/mol	Thermodynamics tables

Be careful with STP - the IUPAC changed the definition in 1982 from 1 atm to 1 bar, which changes molar volume from 22.414 to 22.711 L/mol. Many textbooks still use the older value.

The Three Component Gas Laws

The ideal gas law combines three simpler relationships. Each holds when the other variables are constant.

Law	Relationship	Constant	Formula	Discovered
Boyle's Law	P inversely proportional to V	T, n	P₁V₁ = P₂V₂	Robert Boyle, 1662
Charles's Law	V proportional to T	P, n	V₁/T₁ = V₂/T₂	Jacques Charles, 1787
Avogadro's Law	V proportional to n	P, T	V₁/n₁ = V₂/n₂	Amedeo Avogadro, 1811

Example using Boyle's Law: a gas at 2 atm and 5 L is compressed to 2 L at constant temperature. New pressure = P₁V₁/V₂ = 2 x 5 / 2 = 5 atm. You can verify this with the ideal gas law calculator by keeping T and n the same and changing V.

When Does the Ideal Gas Law Break Down?

Condition	Why It Fails	Better Model
High pressure (above ~10 atm)	Gas molecules are pushed close enough that their volume matters	Van der Waals equation
Low temperature (near boiling point)	Intermolecular forces become significant as molecules slow down	Van der Waals or Redlich-Kwong
Polar molecules (NH₃, H₂O, HCl)	Strong dipole-dipole forces between molecules	Van der Waals with molecule-specific constants
Very light gases (H₂, He) at low T	Quantum effects at very low temperatures	Quantum statistical mechanics

For most everyday conditions - room temperature, atmospheric pressure, common gases like N₂, O₂, Ar, CO₂ - the ideal gas law is accurate to within 1-2%. It is the standard starting point for gas calculations in chemistry and engineering.

Common Applications

Application	What You Calculate	Typical Values
Stoichiometry (gas volumes)	Volume of gas produced in a reaction	At STP, 1 mol = 22.4 L
Tyre pressure with temperature	Pressure change from cold to hot driving	30 psi at 20 °C becomes ~33 psi at 60 °C
Altitude and pressure	Air pressure decreases with altitude	~84 kPa at 1,500 m vs 101.3 kPa at sea level
Scuba tank fill	Moles of air in a pressurised cylinder	12 L tank at 200 bar holds ~100 mol
Weather balloon sizing	Volume at altitude as pressure drops	1 m³ at ground expands to ~100 m³ at 30 km

For liquid and hydrostatic pressure problems, the pressure calculator handles P = F/A and P = rho x g x h. For material density values used in gas density calculations (d = PM/RT), the density calculator covers solids and liquids. All calculations run entirely in your browser with no data sent anywhere.

How Accurate Is the Ideal Gas Law in Practice?

For most everyday gases near room temperature and atmospheric pressure, PV = nRT predicts behaviour within about 1% of experimental values. The deviation is quantified by the compressibility factor Z = PV/(nRT), which equals exactly 1 for an ideal gas. Real gases deviate from Z = 1 in predictable ways as conditions change.

Gas	Z at 1 atm, 25 °C	Z at 100 atm, 25 °C	Notes
Helium	1.0005	1.05	Nearly ideal across wide range
Nitrogen	0.9998	1.01	Close to ideal under STP
Oxygen	0.9994	0.95	Slight attraction at moderate P
Carbon dioxide	0.9949	0.25	Strong deviation near critical point (31 °C, 73.8 atm)
Water vapour	0.9832	Condenses	Polar molecules deviate early

When Z drops below 1, attractive forces between molecules are pulling them closer than the ideal model predicts. When Z rises above 1, the molecules' own volume starts to dominate and gas becomes harder to compress. The Van der Waals equation adds two correction constants (a for attraction, b for molecular volume) per gas and typically brings Z within 0.5% of measured values up to 50 atm.

Kinetic Theory Behind PV = nRT

The ideal gas law emerges directly from the kinetic theory of gases, first formalised by James Clerk Maxwell and Ludwig Boltzmann in the 1860s. Starting from a cloud of point particles in random motion that collide only elastically, you can derive PV = (1/3)Nm<v²>, where N is the number of molecules, m is molecular mass, and <v²> is the mean-square velocity. Linking kinetic energy to temperature via (1/2)m<v²> = (3/2)k_B T gives PV = Nk_B T, which becomes PV = nRT when you group Avogadro's number into the gas constant (R = N_A × k_B).

This derivation means R is not an arbitrary empirical constant - it connects the macroscopic properties of a gas (pressure, volume, temperature) to the microscopic kinetic energy of its molecules. The 2019 SI redefinition made this link exact by fixing the Boltzmann constant k_B = 1.380649 × 10⁻²³ J/K by definition, which in turn fixes R = 8.314462618 J/(mol K) exactly per the CODATA 2022 adjustment published by NIST.

Temperature Must Always Be in Kelvin

Using Celsius or Fahrenheit in PV = nRT produces wildly wrong answers because these scales have arbitrary zero points. A gas at 20 °C has a pressure determined by its absolute temperature of 293.15 K, not by the number 20. Double the Celsius temperature to 40 °C and the pressure increases only by the factor 313.15/293.15 = 1.068, not by 2. The calculator converts Celsius and Fahrenheit inputs to Kelvin internally before applying the formula, but if you are working through a problem by hand, the conversion is the single most common source of errors in introductory chemistry.

Worked example - tyre pressure on a hot day: a car tyre reads 32 psi (2.207 bar gauge, 3.22 bar absolute) at 15 °C (288.15 K). After motorway driving the tyre temperature reaches 55 °C (328.15 K). Volume and moles are essentially constant, so P₂ = P₁ × T₂/T₁ = 3.22 × 328.15/288.15 = 3.667 bar absolute, which is 2.653 bar gauge or 38.5 psi. This is why tyre manufacturers specify cold pressures - the same tyre reads 6-7 psi higher after a long drive.

Common Mistakes That Throw Off Gas Law Calculations

• Using gauge pressure instead of absolute. Tyre pressures, bike pumps, and scuba gauges all read pressure above atmospheric. PV = nRT needs absolute pressure, so add 1 atm (14.7 psi, 1.013 bar) to any gauge reading before using it.
• Forgetting to convert litres to m³. Mixing units produces answers off by factors of 1,000. If you use Pa for pressure and L for volume, you must remember 1 L = 0.001 m³ or switch to kPa × L, which pairs cleanly with R = 8.314 kPa L/(mol K).
• Treating water vapour as ideal at low temperature. Below the dew point, water condenses and the gas law fails outright. For humid air at room temperature, the dry-air and water-vapour partial pressures need to be handled separately.
• Using the 1 atm STP value for modern problems. IUPAC redefined STP to 100 kPa in 1982. Molar volume dropped from 22.414 L/mol (old) to 22.711 L/mol (current). Older textbooks and engineering references still quote 22.4, so check which convention your problem uses.
• Ignoring the compressibility factor near the critical point. CO₂ at room temperature and 70 atm behaves nothing like an ideal gas - Z drops below 0.3 as the gas approaches its liquid phase. For pressurised industrial systems, the Redlich-Kwong or Peng-Robinson equations give much better predictions.

How the Gas Laws Fit Together Historically

The path from Boyle to Avogadro took almost 150 years and involved multiple dead ends. Robert Boyle published his pressure-volume relationship in 1662 using a J-shaped glass tube partially filled with mercury, compressing trapped air at constant temperature. Jacques Charles discovered the temperature-volume relationship around 1787 but never published it - the law bears his name because Joseph Louis Gay-Lussac cited his work in 1802. Amedeo Avogadro proposed in 1811 that equal volumes of gases at the same temperature and pressure contain equal numbers of particles, but the idea was ignored for nearly 50 years until Stanislao Cannizzaro championed it at the 1860 Karlsruhe Conference.

The combined ideal gas law in the form PV = nRT was formalised by French engineer Benoît Paul Émile Clapeyron in 1834. His contribution was recognising that the three earlier laws could be unified into a single equation of state, and identifying the gas constant as a universal quantity rather than one that varied between gases. For more on gas properties, see the density calculator, which handles gas density via d = PM/RT where M is molar mass in kg/mol.

Sources

• NIST - CODATA Value: Molar Gas Constant R
• Wikipedia - Ideal Gas Law (PV = nRT, derivation and history)
• IUPAC Gold Book - Standard Conditions for Gases (STP)
• Wikipedia - Van der Waals Equation and Compressibility Factor
• NIST - Fundamental Physical Constants (CODATA 2022)
• Encyclopaedia Britannica - Boyle's, Charles's, and Avogadro's Laws