Ideal Gas Laws

Ideal gases follow simple relations between $P$, $V$, and $T$. Can be applied to ideal gases.

Properties of Fluids

Fluid properties describe measurable characteristics such as pressure, temperature, and volume.

• Density: $\rho = \frac{m}{V}$.
• Specific volume: $v = \frac{V}{m} = \frac{1}{\rho}$.
• Specific gravity: ratio of fluid density to reference density (water for liquids, air for gases).

Boyle’s Law

At constant temperature, $PV = \text{constant}$.

Charles’ Law

At Constant Pressure

At constant pressure, $V \propto T$.

At Constant Volume

At constant volume, $P \propto T$.

Combined Gas Law

Derived using Boyle’s and Charles’ laws.

$$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$

Ideal Gas Law

Relates all three variables:

$$PV = mRT = \frac{RT}{v}$$

Here:

• $P$ - pressure
• $V$ - pressure
• $m$ - mass
• $R$ - specific gas constant
• $T$ - absolute temperature
• $v$ - specific volume

Universal Gas Constant

$R_u = C_p - C_v = 8.314 \;\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}$.

Gas Constant

Denoted by $R$. Defined for each gas.

$$R = \frac{R_u}{M}$$

Here $M$ is the molar mass of the gas.

Example: $R_\text{air} = 0.287\;\text{kJ}\,\text{kg}^{-1}\,\text{K}^{-1}$.

Compressibility Factor

Denoted by $Z$. Real gases deviate from ideal behavior, especially near saturation and the critical point.

$$Z = \frac{Pv}{RT} = \frac{v_\text{real}}{v_\text{ideal}}$$

$Z = 1$ for an ideal gas. $Z$ close to 1 means the gas behaves similar to ideal gas.

$Z \gt 1$ means the gas is hard to be compressed. $Z \lt 1$ means the gas is easy to be compressed.

When deviations matter

• High pressure
• Low temperature
• Near critical region

Generalized Compressibility Chart

Different gases behave similarly when compared using normalized variables.

Reduced Pressure

$$P_R = \frac{P}{P_c}$$

When $P_R \ll 1$, the gas shows ideal gas behavior.

Reduced Temperature

$$T_R = \frac{T}{T_c}$$

When $T_R \gt 2$, the gas shows ideal gas behavior.

Principle of Corresponding States

At the same $P_R$ and $T_R$, different gases have nearly the same $Z$. Deviations largest near critical point.

Charts allow estimation of $Z$ without equations.

Real-Gas Equations of State

Real-gas equations include molecular attraction and finite molecular volume.

Van der Waals Equation

$$\left(P + \frac{a}{v^2}\right)(v - b)=RT$$

Due to intermolecular forces, measured pressure would be less than ideal pressure. Thus $\frac{a}{v^2}$ is added.

Due to finite non-zero molecular size, ideal volume would be less than measured volume. Thus $b$ is subtracted.

$$a = \frac{27R^2T_\text{cr}^2}{64 P_\text{cr}} \quad \text{and} \quad b = \frac{RT_\text{cr}}{8 P_\text{cr}}$$

Accuracy limited but conceptually useful.

Beattie–Bridgeman Equation

Based on five empirical constants. Accurate for densities up to about $0.8\rho_{cr}$.

$$P = \frac{R_u T}{\bar{v}^2} \left( 1 - \frac{c}{\bar{v} T^3} \right)(\bar{v} + B) - \frac{A}{\bar{v}^2}$$

Here:

$$A = A_0 \left(1 - \frac{a}{\bar{v}}\right) \quad \text{and} \quad B = B_0 \left(1 - \frac{b}{\bar{v}}\right)$$

Benedict–Webb–Rubin Equation

Aka. BWR equation. Eight constants. Accurate up to $2.5\rho_{cr}$.

$P = \frac{R_u T}{\bar{v}} + \left( B_0 R_u T - A_0 - \frac{C_0}{T^2} \right)\frac{1}{\bar{v}^2} + \frac{b R_u T - a}{\bar{v}^3} + \frac{a \alpha}{\bar{v}^6} + \frac{c}{\bar{v}^3 T^2}\left(1 + \frac{\gamma}{\bar{v}^2}\right)e^{-\gamma/\bar{v}^2}$

Virial Equation

$$Pv = RT[1 + B(T)/v + C(T)/v^2 + \ldots]$$

$B(T)$, $C(T)$ are virial coefficients. Derived from statistical mechanics.