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Solution chemistry, equilibria and kinetics
Gas dissociation fraction at fixed total pressure
Fraction of A₂ dissociated at equilibrium in A₂(g) ⇌ 2A(g) when starting from pure A₂ at a fixed total pressure P.
- Ref
- SCH-10
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- α
Starting with pure A₂, let a fraction α dissociate in A₂(g) ⇌ 2A(g) at total pressure P. The mole fractions are (1 − α)/(1 + α) and 2α/(1 + α), and Kp = 4α²/(1 − α²) × P/p°, which gives α = √[Kp/(Kp + 4P/p°)]. Raising the pressure lowers α, since the reaction makes more gas moles. It assumes ideal gases and no inert gas.
Fraction of A₂ dissociated at equilibrium in A₂(g) ⇌ 2A(g) when starting from pure A₂ at a fixed total pressure P. Ideal gases and no inert gas. The constant Kp is dimensionless and p° is the standard pressure in the same units as P.
Calculator
- Partial pressure of A₂, (1 − α)P/(1 + α)0.25
- Partial pressure of A, 2αP/(1 + α)0.5
- Moles of gas per mole of A₂ started, 1 + α1.5
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 0.75 bar, 50 % of the A₂ has dissociated, so the gas holds 1.5 mol per mole of A₂ started.
- Quadrupling the pressure roughly halves α for small α, since α varies as P^(−1/2) in that limit.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Kp Dimensionless equilibrium constant | 1e-4 to 10,000 | 0.005773 | 0.9999 | 0.9941 | +10% | +3.59 % |
| P Equilibrium total pressure | 0.001 to 100 bar | 0.04994 | 0.998 | 0.9481 | +10% | −3.55 % |
| p° Standard pressure (same unit) | 0.1 to 10 bar | 0.1796 | 0.8771 | 0.6975 | +10% | +3.59 % |
Sweep of Equilibrium total pressure
| Equilibrium total pressure (bar) | Fraction dissociated |
|---|---|
| 0.001 | 0.998 |
| 0.004217 | 0.9917 |
| 0.01778 | 0.9662 |
| 0.07499 | 0.8771 |
| 0.3162 | 0.6645 |
| 1.334 | 0.3973 |
| 5.623 | 0.2063 |
| 23.71 | 0.1021 |
| 100 | 0.04994 |
Sweep one variable
Variables and how each one contributes
Fraction dissociated
What the equation gives you. Everything else feeds it.
Dimensionless equilibrium constant
The dimensionless equilibrium constant of A₂ ⇌ 2A. A larger Kp raises the dissociation fraction. Raising it by 10 % (to 1.1) moves the result by +3.59 %, from 0.5 to 0.518.
Equilibrium total pressure
The equilibrium total pressure. A higher pressure suppresses the dissociation. Raising it by 10 % (to 0.825) moves the result by −3.55 %, from 0.5 to 0.4822.
Standard pressure (same unit)
The standard pressure, in the same unit as P. A larger standard pressure lowers P/p°, so more dissociation occurs. Raising it by 10 % (to 1.1) moves the result by +3.59 %, from 0.5 to 0.518.
Worked example with the values above
- Pressure term: 4P/p° = 4 × 0.75 / 1 = 3.
- Ratio: Kp/(Kp + 3) = 0.25.
- α = √0.25 = 0.5.
Assumptions and limits
- Constants are dimensionless thermodynamic values for the same temperature and standard states, or consistent concentration-based values.
- Ideal solutions or ideal gases, with activities replaced by concentrations or pressures.
Common mistakes
- Using this formula when an inert gas is present, which changes the partial pressures.
- Mixing the pressure unit of P with the standard pressure.
- Using Kp for a reaction written in a different form.
Statistics and uncertainty
Equilibrium constants are known only to within a factor that depends on the source, often 5 % to 30 %, and the constant-ΔH assumption in the van’t Hoff equation limits it to modest temperature ranges.
Where it is used
- Combining step equilibria, and extrapolating a constant to another temperature.
- Judging which way a mixture will react and how far it is from equilibrium.
Maths you need
- Equilibrium constant: K, the ratio of products to reactants at equilibrium
- Reaction quotient: Q, the same ratio at any composition
- Gibbs energy: ΔG° = −RT ln K