Every breath you take, every balloon you inflate, every weather system on Earth is governed by one elegant relationship: the ideal gas law. PV = nRT connects the pressure, volume, temperature, and amount of a gas in a single equation, and from it you can predict how any gas behaves as conditions change. It is the foundation of thermodynamics, atmospheric science, engine design, and industrial chemistry.
PV = nRT
P = pressure (Pa or N/m²)
V = volume (m³)
n = number of moles of gas (mol)
R = 8.314 J/mol·K (the universal gas constant)
T = absolute temperature (K — must be kelvin, never Celsius)
An ideal gas is one in which molecules have negligible volume and no intermolecular forces (except during perfectly elastic collisions).
Where the Ideal Gas Law Comes From
The ideal gas law is a synthesis of three experimental laws discovered independently:
Boyle's Law (1662): at constant temperature (T) and amount (n), pressure and volume are inversely proportional: PV = constant, or P₁V₁ = P₂V₂.
Charles's Law (1787): at constant pressure (P) and amount (n), volume is proportional to absolute temperature: V/T = constant, or V₁/T₁ = V₂/T₂.
Gay-Lussac's Law (1808): at constant volume (V) and amount (n), pressure is proportional to absolute temperature: P/T = constant, or P₁/T₁ = P₂/T₂.
Combining all three gives PV/T = constant (for fixed n), or equivalently PV = nRT where R is the proportionality constant determined by experiment: R = 8.314 J/mol·K.
The ideal gas law connects directly to specific heat capacity and to the second law of thermodynamics. It can also be written using the number of molecules N and Boltzmann's constant k_B = R/N_A:
where N_A = 6.022 × 10²³ is Avogadro's number.
The Three Component Laws
| Law | Constant | Relationship | Example |
|---|---|---|---|
| Boyle's | T, n | P ∝ 1/V | Compressing a syringe |
| Charles's | P, n | V ∝ T | Hot air balloon rising |
| Gay-Lussac's | V, n | P ∝ T | Pressure cooker or car tyre heating up |
Worked Examples
Example 1: Direct application of PV = nRT
2.0 mol of ideal gas at 300 K and pressure 1.5 × 10⁵ Pa. What is its volume?
Example 2: Boyle's Law — compressed gas
A gas occupies 0.50 m³ at 2.0 × 10⁵ Pa. It is compressed isothermally to 0.20 m³. What is the new pressure?
Example 3: Charles's Law — hot air balloon
Air at 15°C (288 K) in a balloon is heated to 100°C (373 K) at constant pressure. By what factor does volume increase?
Volume increases by ~29.5%. The less dense hot air makes the balloon buoyant. Always convert to kelvin — using Celsius gives completely wrong answers.
Example 4: Combined gas law
A gas at 1.0 × 10⁵ Pa, 0.010 m³, 300 K is compressed to 0.004 m³ and heated to 400 K. Find the new pressure.
Connection to the Kinetic Theory of Gases
The ideal gas law is not just an empirical observation — it can be derived from first principles using the kinetic theory of gases. Starting from Newton's laws applied to molecules bouncing off container walls, and assuming:
• Molecules are point masses (negligible volume)
• No intermolecular forces except during elastic collisions
• Molecular speeds follow the Maxwell-Boltzmann distribution
…the derivation yields PV = Nk_B T = nRT. The kinetic theory also shows that temperature is a measure of average molecular kinetic energy: ⟨KE⟩ = (3/2)k_B T per molecule.
Real vs Ideal Gases: When the Law Breaks Down
No real gas is perfectly ideal. Real gases deviate from PV = nRT when:
• Pressure is very high — molecules are close together, so intermolecular forces and finite molecular volumes matter.
• Temperature is very low — molecules move slowly, so attractive forces become significant and the gas may liquefy.
The van der Waals equation corrects for both effects:
where a accounts for intermolecular attraction and b for finite molecular volume. At high T and low P, van der Waals → ideal gas law (a and b terms become negligible). Most common gases (nitrogen, oxygen, helium, argon) behave nearly ideally at room temperature and atmospheric pressure — within 1–2% of ideal.
Real-World Applications
Weather and atmospheric pressure: the ideal gas law governs how air pressure and density change with altitude. The barometric formula (pressure decreasing exponentially with height) is derived from PV = nRT combined with the hydrostatic equation. Weather is essentially the ideal gas law playing out at planetary scale.
Car tyres: tyre pressure increases on a hot day (Gay-Lussac's Law — same volume, higher temperature → higher pressure). Manufacturers specify tyre pressure when cold for this reason.
Hot air balloons: heating the air inside increases its temperature (at constant pressure), decreasing density (Charles's Law). The lower-density hot air inside creates buoyancy.
Medical anaesthesia: anaesthetic gases are dosed by concentration (moles per volume) at body temperature. The ideal gas law allows conversion between volumes at different temperatures and pressures during preparation and delivery.
Frequently Asked Questions
What is the ideal gas law?
The ideal gas law PV = nRT relates pressure (P), volume (V), moles of gas (n), the gas constant R = 8.314 J/mol·K, and absolute temperature (T). It describes the behaviour of an ideal gas — one with no intermolecular forces and negligible molecular volume. Real gases approximate it well at high temperature and low pressure.
What is Boyle's Law?
Boyle's Law: at constant temperature and fixed amount of gas, pressure and volume are inversely proportional — P₁V₁ = P₂V₂. Double the pressure, halve the volume. This is the PV = nRT equation with n, R, T constant, so PV = constant.
Why must temperature be in kelvin for gas law calculations?
The gas laws involve temperature proportionality (V ∝ T, P ∝ T) — which requires an absolute temperature scale starting at true zero (absolute zero, 0 K = −273.15°C). At 0°C, molecules still have kinetic energy — they are not at rest. Using Celsius gives physically nonsensical results (e.g., negative volumes). Always convert: T(K) = T(°C) + 273.15.
What is an ideal gas?
An ideal gas has: (1) molecules with negligible volume, (2) no intermolecular forces except during elastic collisions, and (3) random molecular motion following Maxwell-Boltzmann statistics. Real gases approximate this at high temperature and low pressure. Noble gases (He, Ne, Ar) are closest to ideal at room conditions.
What is the value of the gas constant R?
R = 8.314 J/(mol·K) — the universal gas constant. It appears in PV = nRT, the Boltzmann relation (R = N_A k_B), and many thermodynamic equations. Its value is determined experimentally by measuring gas behaviour and is universal — the same for all ideal gases.
When does the ideal gas law fail?
At very high pressures (molecules are close — intermolecular forces and finite volumes matter) and very low temperatures (molecules slow down — attractive forces dominate and gas may liquefy). The van der Waals equation improves accuracy in these conditions. For most engineering and chemistry at room temperature and atmospheric pressure, ideal gas law is accurate to within 1–2%.
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Written by
Dr. Sarah KimThermodynamics researcher with a PhD from MIT, specializing in statistical mechanics and energy transfer. Passionate about connecting molecular physics to everyday phenomena.
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