MECH3260 Chap.3 Gas Mixtures
Gas Mixtures
Week 3 builds the mixtures toolkit that the psychrometry and combustion chapters both depend on. It starts with composition, converting between mass fractions and mole fractions and from there to the mixture molar mass and gas constant, then sets out the two exact ideal-gas mixture models.
The Dalton model gives every component the whole volume and splits the pressure; the Amagat model holds the pressure common and splits the volume. They agree on pressure, volume and temperature and part company on entropy and condensation, where only Dalton is valid.
The chapter closes with the two repairs available when the ideal-gas assumption fails, namely a mixture compressibility factor or Kay's pseudocritical rule, and with the Gibbs-Dalton law for combining internal energy, enthalpy and entropy across components.
What this chapter covers
- 01
Mass fraction mf_i = m_i/m and mole fraction y_i = n_i/n, with n_i = m_i/M_i
- 02
Mixture molar mass M = sum(y_i M_i) and mixture gas constant R = R_u/M
- 03
The Dalton model: each component fills the full volume, partial pressure P_i = y_i P -- the valid model for entropy and condensation calculations
- 04
The Amagat model: each component exerts the full pressure, partial volume V_i = y_i V -- exact for P-v-T but not valid for entropy/condensation
- 05
Non-ideal mixtures: compressibility factor Z = sum(y_i Z_i), or Kay's rule pseudocritical properties
- 06
The Gibbs-Dalton law: extensive properties (U, H, S) of a mixture sum as if each component alone occupied the mixture volume at the mixture temperature
Composition and partial pressures of a three-gas fuel-blend cylinder
- +2Moles: n_CH4=0.250, n_C2H6=0.200, n_N2=0.357 kmol; n=0.807 kmol.
- +2Mole fractions: y_CH4=0.310, y_C2H6=0.248, y_N2=0.442.
- +2Mixture M=24.78 kg/kmol, R=0.3355 kJ/kg.K.
- +2Total pressure P=268.4 kPa; partial pressures 83.2, 66.6, 118.6 kPa (sum checks).
Key terms
- Mole fraction
- The moles of one component divided by the total moles of the mixture. It is the quantity that partial pressures and partial volumes are proportional to, which is why almost every mixture question converts to it first.
- Mass fraction
- The mass of one component divided by the total mixture mass. It is what a question usually gives you and almost never what a formula wants, so the conversion through molar mass is the standard opening move.
- Mixture gas constant
- The universal gas constant divided by the mixture molar mass, where the mixture molar mass is the mole-fraction weighted sum of the component molar masses. It lets the whole mixture be treated with a single ideal-gas law.
- Dalton model
- A mixture model in which each component fills the entire volume at the mixture temperature and contributes its own partial pressure, equal to its mole fraction times the total pressure. This is the model to use for entropy and condensation.
- Amagat model
- A mixture model in which each component is imagined compressed to the full mixture pressure and occupies its own partial volume, equal to its mole fraction times the total volume. It is exact for pressure, volume and temperature only.
- Kay's rule
- A shortcut for non-ideal mixtures that builds a single pseudocritical pressure and temperature as mole-fraction weighted averages of the component critical properties, then reads one compressibility factor for the whole mixture.
- Gibbs-Dalton law
- Extensive mixture properties add: internal energy, enthalpy and entropy are the sums of each component's contribution evaluated at the mixture temperature. The entropy sum must use partial pressures, not the total pressure.
Gas Mixtures FAQ
When should I use Dalton instead of Amagat?
For any ideal-gas question about pressure, volume and temperature the two models give identical exact answers, so the choice is free. It stops being free the moment entropy or condensation enters, because the unit's materials flag Amagat as not valid there. Since the psychrometry chapter is entirely about water condensing out of air, everything downstream of this chapter runs on Dalton partial pressures.
Why do I keep converting from mass to moles at the start of every problem?
Because the formulas that follow are written in mole fractions, while the data a question supplies is nearly always masses or mass flow rates. The chain runs mass, then moles by dividing by molar mass, then mole fraction, then mixture molar mass, then the mixture gas constant. It is the same five steps every time, and getting it automatic frees your attention for the part of the question that is actually new.
What checks catch a mixture arithmetic slip before it reaches the answer?
Two, and both are free. Mole fractions must sum to one, and the partial pressures you computed from them must sum back to the total pressure you started from. A partial volume calculation gets the equivalent check against the total volume. Do them before moving on, because a composition error here propagates silently into every later chapter that treats moist air or combustion products as a mixture.
Do I need the compressibility route often?
Not for most of this unit, where the working mixtures sit well inside the ideal-gas range. It matters when a question puts a mixture at a high pressure, and then you have three options: component compressibility factors read at the mixture temperature and volume, the same read at the mixture temperature and pressure, or Kay's rule. They give slightly different numbers, so state which one you used.
A named method protects a small numerical difference from looking like an error.
Exam move
Drill the mass to mole to fraction chain until it runs without thought, since almost every mixtures question is a composition conversion in disguise before any model formula is needed. Then learn Dalton and Amagat as one table with two columns rather than as two separate results, and memorise the single line that separates them: entropy and condensation belong to Dalton alone.
Give the Gibbs-Dalton law its own pass, paying attention to the partial-pressure requirement in the entropy sum, because that is the one place where the internal-energy and enthalpy habits will actively mislead you. Leave the non-ideal material until the rest is solid, then learn it as three named routes you choose between and declare, not as a formula to reproduce.
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