Ideal Gas Law Equation of State (PV = nRT)
Calculate pressure, volume, temperature, and molar amount for ideal gas systems across SI and US customary units.
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Primary Mathematical Expression
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| P | Absolute pressure | Pa | psi | The force per unit area exerted by the gas. |
| V | Volume | m³ | ft³ | The space occupied by the gas. |
| n | Amount of gas | mol | mol | Moles of gas present in the system. |
| R | Gas constant | J/(mol·K) | J/(mol·K) | The Universal Ideal Gas Constant (8.314 J/(mol·K)). |
| T | Absolute temperature | K | R | The temperature of the system on an absolute scale. |
Step-by-Step Derivation
- 1
Boyle's Law states that at constant temperature, pressure is inversely proportional to volume: V \propto 1/P.
- 2
Charles's Law states that at constant pressure, volume is directly proportional to absolute temperature: V \propto T.
- 3
Avogadro's Law states that at constant temperature and pressure, volume is proportional to the number of moles: V \propto n.
- 4
Combining these proportional relationships gives: V \propto (n * T) / P.
- 5
Introducing the universal ideal gas constant R as the constant of proportionality yields: V = R * n * T / P, which rearranges to: P * V = n * R * T.
Worked Example Calculation
A container of volume 0.05 m³ contains 2.0 moles of helium gas at a temperature of 300 K. Calculate the absolute pressure of the gas. (R = 8.314 J/(mol·K)).
- •Identify input parameters: V = 0.05 m³, n = 2.0 mol, T = 300 K, R = 8.314 J/(mol·K).
- •Apply the Ideal Gas Law: P = n * R * T / V.
- •Compute: P = (2.0 * 8.314 * 300) / 0.05 = 4988.4 / 0.05 = 99,768 Pa = 99.8 kPa.
Engineering Assumptions
- •Gas molecules are point masses with negligible volumes.
- •Intermolecular forces between gas particles are non-existent.
- •Collisions between molecules and container walls are perfectly elastic.
Design Limitations
- •Becomes inaccurate at high pressures or low temperatures where real gas intermolecular forces and volumes cannot be ignored (requires Van der Waals or Redlich-Kwong equations).
Academic References & Standards
Cengel & Boles, Thermodynamics: An Engineering Approach, 9th Edition
Textbook covering ideal gas behavior and state equations.