13.2 Properties of ideal gas mixtures

Dalton's Law of Partial Pressures

Fundamentals of Dalton's Law

Dalton's law states that the total pressure of an ideal gas mixture equals the sum of the partial pressures of each individual gas. A partial pressure is the pressure a gas would exert if it alone occupied the entire volume of the mixture at the same temperature.

This law rests on two assumptions: the gases don't react chemically with each other, and the volume of the gas molecules themselves is negligible compared to the container volume. Both assumptions hold well for ideal gas behavior.

Mole Fractions and Partial Pressures

The mole fraction of a component is the ratio of its moles to the total moles in the mixture:

$y_i = \frac{n_i}{n_{\text{total}}}$

For example, a mixture of 2 moles of nitrogen and 3 moles of oxygen has $y_{N_2} = 2/5 = 0.4$ and $y_{O_2} = 3/5 = 0.6$. Note that all mole fractions in a mixture must sum to 1.

The partial pressure of any component equals its mole fraction times the total pressure:

$P_i = y_i \cdot P_{\text{total}}$

So if the total pressure is 1 atm and $y_{N_2} = 0.4$, then $P_{N_2} = 0.4 \text{ atm}$.

Ideal Gas Mixture Calculations

Partial Pressure, Total Pressure, and Volume Calculations

Total pressure is simply the sum of all partial pressures:

$P_{\text{total}} = P_1 + P_2 + \cdots + P_n$

If a three-component mixture has partial pressures of 0.2 atm, 0.5 atm, and 0.8 atm, the total pressure is 1.5 atm.

Each component in the mixture individually obeys the ideal gas law. You can find the volume a component would occupy at its own partial pressure using:

$V_i = \frac{n_i R T}{P_i}$

However, since every component in the mixture shares the same total volume $V$ and temperature $T$, a more useful form is to apply the ideal gas law to the entire mixture at once:

$P_{\text{total}} V = n_{\text{total}} R T$

This is typically the equation you'll reach for first in mixture problems.

Molar Mass and Density Calculations

The apparent molar mass of a mixture is the mole-fraction-weighted average of the component molar masses:

$M_{\text{mix}} = \sum y_i M_i$

For a mixture of 60% $N_2$ ($M = 28 \text{ g/mol}$) and 40% $O_2$ ($M = 32 \text{ g/mol}$):

$M_{\text{mix}} = 0.6 \times 28 + 0.4 \times 32 = 29.6 \text{ g/mol}$

Once you have $M_{\text{mix}}$, you can find the density of the mixture directly from the ideal gas law rearranged:

$\rho = \frac{P \cdot M_{\text{mix}}}{R T}$

where $P$ is total pressure, $R$ is the universal gas constant, and $T$ is absolute temperature. This is the same form you'd use for a pure ideal gas, just with $M_{\text{mix}}$ in place of a single molar mass.

Properties of Ideal Gas Mixtures

Specific Heat Capacities

For ideal gas mixtures, extensive properties are additive. The molar specific heat capacities are mole-fraction-weighted averages of the component values:

$\bar{c}_{p,\text{mix}} = \sum y_i \, \bar{c}_{p,i} \qquad \bar{c}_{v,\text{mix}} = \sum y_i \, \bar{c}_{v,i}$

If you're working on a mass basis instead, use mass fractions ($mf_i$) with the mass-based specific heats:

$c_{p,\text{mix}} = \sum mf_i \, c_{p,i} \qquad c_{v,\text{mix}} = \sum mf_i \, c_{v,i}$

Don't mix these up. Mole fractions pair with molar specific heats; mass fractions pair with mass-based specific heats. This is a common source of errors on exams.

The relationship $\bar{c}_p - \bar{c}_v = R$ still holds for the mixture on a molar basis, since each ideal gas component satisfies it individually.

Enthalpy and Internal Energy

For ideal gases, enthalpy and internal energy depend only on temperature, not on pressure. The mixture values are again mole-fraction-weighted sums:

$\bar{h}_{\text{mix}} = \sum y_i \, \bar{h}_i \qquad \bar{u}_{\text{mix}} = \sum y_i \, \bar{u}_i$

Changes in these properties follow the same pattern. For a temperature change from $T_1$ to $T_2$:

$\Delta \bar{h}_{\text{mix}} = \sum y_i \, \bar{c}_{p,i} \, (T_2 - T_1)$

$\Delta \bar{u}_{\text{mix}} = \sum y_i \, \bar{c}_{v,i} \, (T_2 - T_1)$

These expressions assume constant specific heats. For large temperature ranges, you'd use tabulated enthalpy values for each component instead.

Ideal Gas Mixture Behavior

Ideal Gas Equation of State

The ideal gas equation $PV = n_{\text{total}} R T$ applies directly to the mixture as a whole. You can also define a mixture-specific gas constant on a mass basis:

$R_{\text{mix}} = \frac{R}{M_{\text{mix}}}$

where $R$ is the universal gas constant and $M_{\text{mix}}$ is the apparent molar mass. This lets you write $Pv = R_{\text{mix}} T$ using specific volume $v$.

The compressibility factor $Z$ for any ideal gas mixture equals 1 by definition. Real gas mixtures deviate from this, but for this course the ideal assumption is standard unless stated otherwise.

Amagat's Law and Volume Fractions

Amagat's law (the law of additive volumes) states that when ideal gases are mixed at constant temperature and pressure, the total volume equals the sum of the component volumes:

$V_{\text{total}} = V_1 + V_2 + \cdots + V_n$

A component volume (sometimes called a partial volume) is the volume that component $i$ would occupy alone at the mixture's temperature and total pressure. For ideal gases, the volume fraction of each component equals its mole fraction:

$\frac{V_i}{V_{\text{total}}} = y_i$

So if you mix 2 L of nitrogen and 3 L of oxygen at the same $T$ and $P$, the mixture occupies 5 L, and the volume fractions are 0.4 and 0.6 respectively.