The ideal gas law is one of the most fundamental equations in chemistry and physics. Expressed as PV = nRT, it relates the pressure, volume, temperature, and amount of a gas in a single elegant formula. Whether you are a chemistry student solving homework problems, an engineer sizing gas vessels, or simply curious about how gases behave, understanding this law unlocks a huge range of practical calculations. This guide explains the formula, walks through solved examples, and shows how to use the Liczbnik Ideal Gas Law Calculator.
The Ideal Gas Law: PV = nRT
Each variable in the ideal gas law represents a physical quantity:
- P — Pressure of the gas (SI unit: Pascal, Pa; also commonly expressed in atm, bar, or kPa)
- V — Volume of the gas (SI unit: cubic metres, m³; also commonly expressed in litres, L)
- n — Amount of substance in moles (mol)
- R — Universal gas constant = 8.314 J·mol⁻¹·K⁻¹ (or equivalently 0.08206 L·atm·mol⁻¹·K⁻¹)
- T — Absolute temperature in Kelvin (K). To convert: K = °C + 273.15
The law assumes the gas consists of point particles with no intermolecular forces — the so-called "ideal gas" model. Real gases deviate from this model at high pressures and low temperatures, but for most everyday conditions, the ideal gas law gives excellent results.
Derivation: Where Does PV = nRT Come From?
The ideal gas law is a synthesis of three earlier empirical laws:
- Boyle's Law (1662): At constant temperature and amount, pressure and volume are inversely proportional: P ∝ 1/V, or PV = constant.
- Charles's Law (1787): At constant pressure and amount, volume is proportional to absolute temperature: V ∝ T.
- Avogadro's Law (1811): At constant temperature and pressure, volume is proportional to the number of moles: V ∝ n.
Combining these three relationships gives V ∝ nT/P, or equivalently PV ∝ nT. Introducing the proportionality constant R gives PV = nRT. The constant R is universal — it has the same value for all ideal gases — which is why the equation is so powerful.
Standard Conditions: STP and SATP
In chemistry, you will frequently encounter two standard conditions:
- STP (Standard Temperature and Pressure): T = 273.15 K (0°C), P = 100,000 Pa (1 bar). At STP, one mole of an ideal gas occupies 22.414 L.
- SATP (Standard Ambient Temperature and Pressure): T = 298.15 K (25°C), P = 100,000 Pa. At SATP, one mole occupies 24.789 L.
Note: The older definition of STP used 1 atm (101,325 Pa) rather than 1 bar. Always check which definition a textbook or problem is using.
Solved Examples
Example 1: Finding Pressure
A container holds 2.0 mol of nitrogen gas at 300 K with a volume of 50 L. What is the pressure?
Using PV = nRT, rearranged as P = nRT/V:
P = (2.0 mol × 0.08206 L·atm·mol⁻¹·K⁻¹ × 300 K) / 50 L = 0.984 atm
Example 2: Finding Volume
How much volume does 5.0 g of helium (molar mass 4.003 g/mol) occupy at 25°C and 1.00 atm?
First, find moles: n = 5.0 / 4.003 = 1.249 mol. Convert temperature: T = 25 + 273.15 = 298.15 K.
V = nRT/P = (1.249 × 0.08206 × 298.15) / 1.00 = 30.56 L
Example 3: Finding Temperature
A fixed container of volume 10 L holds 0.5 mol of gas at 2.0 atm. What is the temperature?
T = PV/(nR) = (2.0 × 10) / (0.5 × 0.08206) = 487.5 K (214.4°C)
Example 4: Finding Moles
A balloon at 1.01 atm and 20°C has a volume of 4.5 L. How many moles of gas does it contain?
n = PV/(RT) = (1.01 × 4.5) / (0.08206 × 293.15) = 0.189 mol
Combined Gas Law
When the amount of gas is constant (n is fixed) and conditions change from state 1 to state 2, the combined gas law applies:
P₁V₁/T₁ = P₂V₂/T₂
This is useful for problems where gas is compressed, heated, or expanded. Special cases:
- Isothermal (constant T): P₁V₁ = P₂V₂ (Boyle's Law)
- Isobaric (constant P): V₁/T₁ = V₂/T₂ (Charles's Law)
- Isochoric (constant V): P₁/T₁ = P₂/T₂ (Gay-Lussac's Law)
Molar Mass from Ideal Gas Law
If you know the mass of a gas sample (m) and can measure P, V, and T, you can calculate the molar mass (M):
Since n = m/M, substituting into PV = nRT gives: M = mRT/(PV)
This technique is used in laboratory experiments to identify unknown gases.
Dalton's Law of Partial Pressures
For a mixture of ideal gases, the total pressure equals the sum of the partial pressures of each component:
P_total = P₁ + P₂ + P₃ + ...
Each component behaves independently as if it alone occupied the full volume. This law is essential in respiratory physiology (partial pressures of O₂ and CO₂ in lung air), industrial gas mixing, and meteorology.
Deviations from Ideal Behaviour: The Van der Waals Equation
Real gases deviate from ideal behaviour because:
- Gas molecules occupy real volume (not zero)
- Intermolecular attractive and repulsive forces exist
The Van der Waals equation corrects for both effects:
(P + a·n²/V²)(V − nb) = nRT
Where a is the attraction parameter and b is the volume exclusion parameter, both specific to each gas. At low pressures and high temperatures, the corrections become negligible and the ideal gas law is an excellent approximation.
Using the Liczbnik Ideal Gas Law Calculator
Our calculator supports all four variants of PV = nRT: solving for P, V, n, or T. You can input values in any common unit system (atm, bar, Pa, kPa for pressure; L, mL, m³ for volume; K or °C for temperature) and the calculator handles all unit conversions automatically. It also includes the combined gas law solver for two-state problems, Dalton's law partial pressure calculator, and a molar mass finder.
Summary
The ideal gas law PV = nRT is a cornerstone of physical chemistry. It applies reliably to most gases at typical temperatures and pressures, enabling calculations of pressure, volume, temperature, and moles with just three known quantities. Understanding its derivation from Boyle's, Charles's, and Avogadro's laws builds intuition for when and why it works — and when to apply corrections like the Van der Waals equation for non-ideal conditions.
Frequently Asked Questions
What does PV = nRT mean?
PV = nRT is the ideal gas law, where P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant (8.314 J/mol·K), and T is absolute temperature in Kelvin. The equation describes the relationship between these four quantities for an ideal gas.
What is the value of the gas constant R?
The universal gas constant R = 8.314 J·mol⁻¹·K⁻¹. In other common units: R = 0.08206 L·atm·mol⁻¹·K⁻¹, or R = 8.314 Pa·m³·mol⁻¹·K⁻¹. Choose the form of R that matches your unit system to avoid conversion errors.
Why must temperature be in Kelvin?
The ideal gas law requires absolute temperature because gas properties (pressure, volume) go to zero as temperature approaches absolute zero (0 K = −273.15°C). Using Celsius would give nonsensical results (e.g., at 0°C you would divide by zero in some formulas). Always convert °C to K by adding 273.15.
When does the ideal gas law fail?
The ideal gas law breaks down at high pressures (above ~10 atm for most gases) and low temperatures (near the boiling point of the gas), where intermolecular forces and molecular volume become significant. In these conditions, use the Van der Waals equation or other real gas equations of state.
What is standard molar volume?
At STP (0°C, 1 bar), one mole of an ideal gas occupies 22.414 litres. At SATP (25°C, 1 bar), the standard molar volume is 24.789 litres per mole. These values are useful shortcuts for many calculations.
How do I convert between units for the ideal gas law?
Ensure all units are consistent with your choice of R. If using R = 0.08206 L·atm·mol⁻¹·K⁻¹, express P in atm, V in litres, n in moles, and T in Kelvin. If using R = 8.314 J·mol⁻¹·K⁻¹, express P in Pa, V in m³, n in moles, T in Kelvin. The Liczbnik calculator handles all conversions automatically.
What is Dalton's Law and how does it relate to the ideal gas law?
Dalton's Law states that the total pressure of a gas mixture equals the sum of the partial pressures of its components. Each component behaves as an ideal gas independently. This follows directly from the ideal gas law: each gas contributes a partial pressure P_i = n_i RT/V, and the total is the sum of all partial pressures.
Can I use the ideal gas law for steam (water vapour)?
Yes, water vapour behaves approximately as an ideal gas at temperatures well above its boiling point and at low pressures. Near the condensation point (100°C at 1 atm), or at high humidity levels close to saturation, deviations from ideal behaviour become significant and more precise equations of state should be used.
What is the combined gas law?
The combined gas law P₁V₁/T₁ = P₂V₂/T₂ applies when the amount of gas is constant and conditions change. It combines Boyle's, Charles's, and Gay-Lussac's laws. Setting one variable constant recovers each individual law: constant T gives Boyle's Law, constant P gives Charles's Law, and constant V gives Gay-Lussac's Law.
How is the ideal gas law used in real life?
The ideal gas law is used in countless applications: calculating tyre pressure changes with temperature, sizing compressed gas cylinders, understanding weather balloon behaviour, designing combustion engines, modelling atmospheric layers, and in respiratory medicine to describe gas exchange in lungs. It is one of the most practically useful equations in all of science.