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Ideal Gas Law
Relates pressure, volume, temperature, and moles of an ideal gas.
Interpretation
The ideal gas law relates the pressure, volume, temperature, and amount of an ideal gas. It is expressed as PV = nRT, where R is the universal gas constant. This equation is a combination of Boyle's, Charles's, and Avogadro's laws and is fundamental in thermodynamics.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| P | Pressure | Pa |
| V | Volume | m³ |
| n | Moles | mol |
| R | Gas constant | J/(mol·K) |
| T | Temperature | K |
What it means
The ideal gas law is an equation of state that describes the behavior of an ideal gas under various conditions. It states that the product of pressure (P) and volume (V) is proportional to the number of moles (n) and absolute temperature (T): PV = nRT, with R = 8.314 J/(mol·K). This law assumes that gas particles have negligible volume and no intermolecular forces. It works well for real gases at low pressures and high temperatures. The ideal gas law combines Boyle's law (P ∝ 1/V at constant T), Charles's law (V ∝ T at constant P), and Avogadro's law (V ∝ n at constant P and T). It is used in many fields, including chemical engineering, meteorology, and aerodynamics. It allows calculation of unknown properties when three are known. The law is also the starting point for more complex equations of state (e.g., van der Waals). Applications include sizing gas cylinders, calculating buoyancy, and designing internal combustion engines.
Worked example
| # | P (Pa) | V (m³) | n (mol) | R (J/mol·K) | T (K) |
|---|---|---|---|---|---|
| 1 | 101325 | 0.0245 | 1.00 | 8.314 | 298.15 |
| 2 | 202650 | 0.0245 | 2.00 | 8.314 | 298.15 |
| 3 | 101325 | 0.0490 | 2.00 | 8.314 | 298.15 |
| 4 | 101325 | 0.0245 | 1.00 | 8.314 | 596.30 |
| 5 | 150000 | 0.0300 | 1.50 | 8.314 | 360.87 |
| 6 | 200000 | 0.0200 | 2.00 | 8.314 | 240.52 |
| 7 | 80000 | 0.0400 | 1.50 | 8.314 | 256.64 |
| 8 | 120000 | 0.0250 | 1.20 | 8.314 | 300.65 |
| 9 | 250000 | 0.0100 | 2.00 | 8.314 | 150.33 |
| 10 | 101325 | 0.0735 | 3.00 | 8.314 | 298.15 |
| 11 | 90000 | 0.0450 | 2.00 | 8.314 | 243.63 |
| 12 | 180000 | 0.0280 | 2.50 | 8.314 | 242.76 |
| 13 | 110000 | 0.0330 | 1.80 | 8.314 | 242.88 |
| 14 | 130000 | 0.0220 | 1.40 | 8.314 | 244.40 |
| 15 | 160000 | 0.0260 | 2.20 | 8.314 | 228.63 |
Common mistakes
- Wrong R value: Use R = 8.314 J/(mol·K) for SI; with other units (e.g., L·atm) use R = 0.0821.
- Temperature in Kelvin: Always use absolute temperature – never °C or °F.
- Volume units: Use m³ with Pa, or litres with the appropriate R.
- Ideal gas assumption: Valid only at low pressure and high temperature; not near condensation.
- Number of moles: n = mass / molar mass; do not use total mass directly.
Applications
The ideal gas law PV = nRT is a fundamental equation that relates pressure, volume, temperature, and amount of gas for ideal gases. It is indispensable in chemistry, physics, and engineering for predicting the behaviour of gases under various conditions. In industrial processes, it is used to size gas storage tanks, design pneumatic systems, and control chemical reactors. In meteorology, it helps model atmospheric pressure and weather patterns. The law also underpins the operation of internal combustion engines, gas turbines, and refrigeration cycles, where gas properties change during compression and expansion. It is also applied in the calibration of gas flow meters and in the study of thermodynamic cycles. Even though real gases deviate, the ideal gas law provides a close approximation for many engineering calculations.
- Design of pressure vessels and gas pipelines
- Pneumatic actuators and control systems
- Gas turbine and jet engine performance modelling
- Air conditioning and refrigeration cycle analysis
- Meteorological forecasting and atmospheric studies
Frequently Asked Questions
The Ideal Gas Law combines Boyle’s, Charles’s, and Avogadro’s laws into a single equation: PV = nRT, where P is absolute pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is absolute temperature. It describes the behaviour of an ideal gas – a theoretical gas with negligible molecular volume and no intermolecular forces.
- SI: R = 8.314 J/(mol·K) = 8.314 kPa·L/(mol·K) (when using kPa and L).
- Metric: R = 0.08206 L·atm/(mol·K).
- Imperial: R = 10.73 ft³·psi/(lb·mol·°R).
Assumptions:
- Gas molecules have zero volume (point particles).
- No intermolecular forces (no attraction or repulsion).
- Collisions are perfectly elastic.
- Using the wrong R value – using R in J/(mol·K) when pressure is in atm and volume in L.
- Not using absolute temperature – T must be in Kelvin (K) or Rankine (°R); using Celsius or Fahrenheit gives nonsense.
- Ignoring the mole count – n must be the number of moles; for mixtures, use the total moles or partial pressures.
- Applying it to liquids or solids – the ideal gas law is only for gases in the ideal region.
From kinetic theory, the pressure exerted by a gas is P = (1/3) ρ ⟨v²⟩, where ρ is density and ⟨v²⟩ is the mean square speed. Since ρ = m·n/V and ⟨v²⟩ is proportional to T, combining gives PV = nRT. The derivation establishes the link between macroscopic pressure and temperature and the microscopic molecular motion.
The van der Waals equation accounts for two effects:
- Molecular volume – subtracts a constant b from the volume: (V – nb).
- Intermolecular attraction – adds a correction to pressure: (P + a n²/V²).
Density ρ = m/V. From PV = nRT, we have n = m/M (M = molar mass). Then ρ = m/V = (P M) / (R T). This is very useful for estimating gas densities under various conditions, e.g., air at STP has ρ ≈ 1.225 kg/m³.
STP is defined as 0°C (273.15 K) and 1 atm (101.325 kPa). At STP, one mole of an ideal gas occupies 22.414 L (0.022414 m³). This is a useful reference for gas calculations.
For a mixture of ideal gases, each gas behaves independently. The partial pressure of each gas is P_i = n_i RT / V. The total pressure is the sum of partial pressures (Dalton’s law). The total mixture obeys PV = n_total RT. This is essential in combustion, atmospheric science, and chemical engineering.
- HVAC system design – calculating air flow rates and densities.
- Compressed air systems – sizing storage tanks and estimating consumption.
- Internal combustion engines – modelling the intake and compression strokes.
- Scuba diving – computing gas consumption at depth.
- Chemical reactors – determining gas volumes for stoichiometric calculations.