Gas Mixtures: Mole Fractions, Dalton's Law, and Amagat's Law

Mole and Mass Fractions in Gas Mixtures

Gas mixtures are described by two main ways: mole fractions and mass fractions.
Mole fraction for component i: $Yi = \frac{ni}{n{\text{tot}}}$ where $ni$ is the moles of component i and $n{\text{tot}} = \sumi n_i$ is the total moles.
Mass fraction for component i: $wi = \frac{mi}{m{\text{tot}}}$ where $mi = ni Mi$ (Mi is the molar mass of component i) and $m{\text{tot}} = \sumi mi = \sumi ni M_i.$
Relationship between mole and mass fractions:
- The mean molar mass of the mixture: $\bar{M} = \sumi Yi Mi = \frac{m{\text{tot}}}{n_{\text{tot}}}.$
- Mass fraction in terms of mole fractions: $wi = \frac{Yi M_i}{\bar{M}}.$
Example: dry air composition (typical values)
- Nitrogen (N$2$): $Y{\text{N}_2} \approx 0.78$
- Oxygen (O$2$): $Y{\text{O}_2} \approx 0.21$
- Argon (Ar): $Y_{\text{Ar}} \approx 0.01$
Air also contains CO$_2$ and trace gases (as mentioned in the transcript).
Practical note: air is a mixture of N$2$, O$2$, Ar, CO$_2$, and traces of other gases; composition is central to engineering applications (air conditioning, rocket propulsion, etc.).

Dalton's Law of Partial Pressures

For a mixture of ideal gases:
- Total pressure: $P{\text{tot}} = \sumi P_i.$
- Partial pressure of component i: $Pi = Yi P_{\text{tot}}.$
Intuition: each gas behaves as if it is alone in the container, contributing to the total pressure in proportion to its mole fraction.
Example from transcript (numerical illustration):
- If $P{\text{tot}} = 200\ \text{kPa}$ and $Y{\text{O}_2} = 0.21$ , then
- $P{\text{O}2} = Y{\text{O}2} P_{\text{tot}} = 0.21 \times 200\ \text{kPa} = 42\ \text{kPa}.$

Amagat's Law: Additive Volumes

For a mixture at the same temperature and pressure:
- Total volume: $V{\text{tot}} = \sumi V_i.$
- Volume of component i if it occupied the container alone at the same T and P: $V_i = \text{volume that gas i would occupy alone}.$
Key distinction: Daltons's law uses pressures; Amagat's law uses volumes.
For ideal gases, Amagat's law and Dalton's law give consistent results when applied to the corresponding intensive properties (P or V).

Ideal-Gas Mixtures: Consistency of Dalton's and Amagat's Laws

In the ideal-gas limit, both Dalton's law (partial pressures) and Amagat's law (additive volumes) describe the same physics from different perspectives and are mathematically consistent with each other for ideal mixtures.
This underpins the ability to analyze mixtures using either approach, depending on which property is known or easier to measure.

Properties of Gas Mixtures

To analyze mixtures, we often define average or apparent properties (mean properties) for the mixture.
Key properties to average: molar mass, the gas constant, internal energy, enthalpy, and specific heats.
Molar mass of the mixture (same as bar{M} above): $\bar{M} = \sumi Yi M_i.$
Gas constant for the mixture (specific gas constant):
- Using the universal gas constant $Ru$ : $R{\text{mix}} = \frac{R_u}{\bar{M}}.$
- If you prefer the convention using $R$ as the universal constant in some texts, the same relation holds with the appropriate symbol choice.
Internal energy and enthalpy per mole (for mixtures of ideal gases, as averages):
- Mixture average internal energy per mole:
 $\bar{u} = \sumi Yi u_i.$
- Mixture average enthalpy per mole:
 $\bar{h} = \sumi Yi h_i.$
Corresponding total quantities for the mixture: $U = n{\text{tot}} \bar{u}, \quad H = n{\text{tot}} \bar{h}.$
Specific heats (molar basis) for mixtures (ideal-gas assumption):
- Mixture molar heat capacity at constant volume: $\bar{c}v = \sumi Yi c{v,i}.$
- Mixture molar heat capacity at constant pressure: $\bar{c}p = \sumi Yi c{p,i}.$
Summary: for ideal-gas mixtures, many properties are nonlinearly affected by composition, but simple mole-fraction weighting provides a convenient and accurate average when applying the laws of Dalton and Amagat and when computing derived properties like $R{\text{mix}}, \bar{M}, \bar{u}, \bar{h}, \bar{c}v, \bar{c}_p.$

Applications and Real-World Context

Air and humidity: moist air \approx dry air + water vapor; moisture changes partial pressures and overall properties (e.g., density, specific heats).
Combustion: fuel + oxidizer mixtures determine flame temperature and efficiency; composition critically affects performance and emissions.
Aerospace and life-support: oxygen-nitrogen mixtures for breathing in spacecraft; precise control of partial pressures is essential for safety and comfort.
Environmental engineering: tracking CO$_2$ concentrations and other gas species in the atmosphere for climate and air-quality assessments.
Key takeaway: gas mixtures can be analyzed with the same tools as pure gases by accounting for composition with mole or mass fractions and applying Dalton's and Amagat's laws as appropriate.

Practical Guidelines for Analyzing Gas Mixtures

Always start by describing the composition using either mole fractions $Yi$ or mass fractions $wi$ ;
- If you know $ni$ , compute $Yi = ni / n{\text{tot}}.$
- If you know masses, compute $wi = mi / m{\text{tot}}$ and relate to mole fractions via $wi = \frac{Yi Mi}{\bar{M}}.$
Use Dalton's law to relate partial pressures to total pressure: $Pi = Yi P{\text{tot}}, \quad P{\text{tot}} = \sumi Pi.$
Use Amagat's law to relate volumes to total volume: $Vi = \text{volume if gas i were alone at the same } T \text{ and } P; \quad V{\text{tot}} = \sumi Vi.$
For ideal gases, ensure consistency between pressure-based (Dalton) and volume-based (Amagat) descriptions; both should agree when applied to the appropriate property.
Compute mixture properties by mole-fraction weighting: $\bar{M} = \sumi Yi Mi, \quad R{\text{mix}} = \frac{Ru}{\bar{M}}, \quad \bar{u} = \sumi Yi ui, \quad \bar{h} = \sumi Yi hi, \quad \bar{c}v = \sumi Yi c{v,i}, \quad \bar{c}p = \sumi Yi c_{p,i}.$

Quick Recap of Key Equations (LaTeX)

Mole fraction: $Yi = \frac{ni}{n_{\text{tot}}}.$
Mass fraction: $wi = \frac{mi}{m{\text{tot}}}, \quad mi = ni Mi, \quad m{\text{tot}} = \sumj nj Mj.$
Mean molar mass: $\bar{M} = \sumi Yi Mi = \frac{m{\text{tot}}}{n_{\text{tot}}}.$
Relation between fractions: $wi = \frac{Yi M_i}{\bar{M}}.$
Dalton's law: $P{\text{tot}} = \sumi Pi, \quad Pi = Yi P{\text{tot}}.$
Example: $P{\text{O}2} = Y{\text{O}2} P_{\text{tot}} = 0.21 \times 200\ \text{kPa} = 42\ \text{kPa}.$
Amagat's law: $V{\text{tot}} = \sumi Vi, \quad Vi = \text{volume gas i would occupy alone at same } T, P.$
Mixture properties: $\bar{M} = \sumi Yi Mi, \quad R{\text{mix}} = \frac{Ru}{\bar{M}}, \quad \bar{u} = \sumi Yi ui, \quad \bar{h} = \sumi Yi hi, \quad \bar{c}v = \sumi Yi c{v,i}, \quad \bar{c}p = \sumi Yi c_{p,i}.$