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Solution chemistry, equilibria and kinetics

Equilibrium constant at a second temperature

Equilibrium constant at T₂ from its value at T₁ and the standard reaction enthalpy, using the integrated van’t Hoff equation with constant ΔH°.

Ref: SCH-07
Rev: v1
Field: Solution chemistry, equilibria and kinetics
Result: K₂

K_{2} = K_{1} \exp [\frac{Δ H^{} c i r c}{R} (\frac{1}{T_{1}} - \frac{1}{T_{2}})]

The van’t Hoff equation gives the temperature dependence of K. For a constant standard reaction enthalpy, K₂ = K₁ exp[(ΔH°/R)(1/T₁ − 1/T₂)]. Endothermic reactions (ΔH° > 0) have a larger K at higher temperature, and exothermic ones a smaller K. Use absolute temperature and ΔH° in J/mol. Over wide temperature ranges the enthalpy itself changes.

Equilibrium constant at T₂ from its value at T₁ and the standard reaction enthalpy, using the integrated van’t Hoff equation with constant ΔH°. Temperatures are absolute, and ΔH° is in J/mol.

Calculator

Solve for

K₂Equilibrium constant at T₂Computed

K₁Equilibrium constant at T₁

1e-61,000,000

ΔH°Standard reaction enthalpy

J/mol

−300,000300,000

T₁First temperature

200600

T₂Second temperature

200600

Ratio K₂/K₁2.073
Change in log₁₀ K0.3166
Van’t Hoff slope, ΔH°/R2,405 K

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.

Equilibrium constant at T₂ () current value 4.146

K₁Equilibrium constant at T₁Key driver

2.07e-62,070,000

ΔH°Standard reaction enthalpy

3.57e-5112,000

T₁First temperature

0.0752228

T₂Second temperature

0.0363110

Going from 300 K to 330 K changes K from 2 to 4.146, a factor of 2.073.
A 20 kJ/mol reaction roughly doubles its K over a 30 K rise near room temperature; the effect grows with |ΔH°|.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
K₁ Equilibrium constant at T₁	1e-6 to 1,000,000	2.073e-6	2,073,000	2,073,000	+10%	+10 %
ΔH° Standard reaction enthalpy	−300,000 to 300,000 J/mol	3.569e-5	112,100	112,100	+12000 J/mol	+54.9 %
T₁ First temperature	200 to 600 K	0.07524	228.4	228.3	+10%	−51.8 %
T₂ Second temperature	200 to 600 K	0.0363	110.2	110.2	+10%	+94 %

Sweep of Second temperature

Second temperature (K)	Equilibrium constant at T₂
200	0.0363
250	0.4023
300	2
350	6.288
400	14.85
450	28.96
500	49.42
550	76.54
600	110.2

Sweep one variable

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Equilibrium constant at T₂ ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

K₂

Equilibrium constant at T₂

Result ·

What the equation gives you. Everything else feeds it.

K₁

Equilibrium constant at T₁

· typical 1e-6 to 1,000,000

The equilibrium constant at T₁. Raises K₂ in direct proportion. Raising it by 10 % (to 2.2) moves the result by +10 %, from 4.146 to 4.56.

ΔH°

Standard reaction enthalpy

J/mol · typical −300,000 to 300,000

The standard reaction enthalpy, in J/mol (signed). A more endothermic reaction (when T₂ > T₁) gives a larger K₂, exponentially. Raising it by 10 % (to 22,000) moves the result by +7.56 %, from 4.146 to 4.459.

T₁

First temperature

K · typical 200 to 600

The starting temperature, in K. A higher T₁ shrinks the temperature step, so K₂ moves less from K₁. Raising it by 10 % (to 330) moves the result by −51.8 %, from 4.146 to 2.

T₂

Second temperature

K · typical 200 to 600

The final temperature, in K. A higher T₂ raises K₂ for an endothermic reaction. Raising it by 10 % (to 363) moves the result by +94 %, from 4.146 to 8.042.

Worked example with the values above

Temperature term: 1/T₁ − 1/T₂ = 1/300 − 1/330 = 3.0303×10⁻⁴ 1/K.
Exponent: ΔH°/R × that = 2,405.4 × 3.0303×10⁻⁴ = 0.72892.
K₂ = 2 × e^(0.72892) = 4.1457.

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 constants at different temperatures in one combination.
Using °C in place of K in the van’t Hoff or Gibbs-energy equation.
Reversing a reaction without inverting K, or scaling it without raising K to the same power.

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

Related formulas

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