Ideal Gas Law

Overview

The ideal gas law relates pressure, volume, temperature, and amount of gas. It works well for most gases at moderate pressures and temperatures.

Variables

Formula

$PV = nRT$

Calculator

Notes

• Results are approximate and should be verified for critical applications
• Input values should be within reasonable engineering ranges

Key Formulas

The ideal gas law can be expressed in several equivalent forms:

Universal Gas Constant Form: $pV = nR_uT$ where $R_u = 8.3145 \, \text{J/(mol·K)}$

Individual Gas Constant Form: $pV = mRT$ $p = \rho RT$ where $R$ is the specific gas constant for the particular gas.

Combined Gas Law (for fixed amount of gas): $\frac{p_1V_1}{T_1} = \frac{p_2V_2}{T_2}$

Non-Ideal Gas Law (with Compressibility Factor): $pV = ZnR_uT$ where $Z$ is the dimensionless compressibility factor.

Molecular Form: $pV = NkT$ where $N$ is the number of molecules and $k = 1.38066 \times 10^{-23} \, \text{J/K}$ is the Boltzmann constant.

Important Definitions

• Ideal Gas: A theoretical gas composed of randomly moving, non-interacting point particles. The ideal gas law is most accurate at low pressures and high temperatures.
• Universal Gas Constant ($R_u$): A fundamental constant that appears in the ideal gas law, independent of the type of gas. Its value is approximately $8.3145 \, \text{J/(mol·K)}$.
• Individual Gas Constant ($R$): Specific to a particular gas, related to the universal gas constant by $R = R_u / M$, where $M$ is the molar mass of the gas.
• Compressibility Factor ($Z$): A correction factor that accounts for the deviation of real gases from ideal gas behavior. $Z = 1$ for an ideal gas. Its value depends on the gas's pressure and temperature.

Example Calculation

For air in a 1 ft³ tank at 70°F and 50 psi gauge pressure:

1. Absolute pressure: $p = 50 \, \text{psi} + 14.7 \, \text{psi} = 64.7 \, \text{psi}$.
2. Convert to consistent units: $p = 64.7 \times 144 = 9316.8 \, \text{lb/ft}^2$.
3. Convert temperature to Rankine: $T = 70 + 460 = 530 \, \text{°R}$.
4. The individual gas constant for air is $R = 1716 \, \text{ft·lb/(slug·°R)}$.
5. Density calculation: $\rho = \frac{p}{RT} = \frac{9316.8}{1716 \times 530} \approx 0.0102 \, \text{slugs/ft}^3$.
6. Weight of the air: $w = \rho g V = 0.0102 \times 32.2 \times 1 \approx 0.328 \, \text{lb}$.

Compressibility Factor (Z) for Air

This table provides the compressibility factor ($Z$) for air, showing how its behavior deviates from an ideal gas across various temperatures and pressures.

References

• Original Source