Van't Hoff Equation (Equilibrium Constant vs. Temperature) Equation, Derivation & Rearrangements

Q: What are common mistakes with the Van't Hoff Equation (Equilibrium Constant vs. Temperature) formula?

Using Celsius instead of Kelvin for temperature. Inconsistent units for R, ΔH°, and ΔS° (e.g., kJ for ΔH° and J for R/ΔS°). Confusing the Van't Hoff equation with the Arrhenius equation (which relates rate constant to temperature).

Q: What is a real-world example of the Van't Hoff Equation (Equilibrium Constant vs. Temperature) formula?

Predicting the optimal temperature for ammonia synthesis in the Haber-Bosch process to maximize yield.

Q: What are some study tips for the Van't Hoff Equation (Equilibrium Constant vs. Temperature) formula?

Ensure temperature 'T' is always in Kelvin (K). The gas constant 'R' must be chosen with units consistent with ΔH° and ΔS° (e.g., 8.314 J mol⁻¹ K⁻¹). ΔH° and ΔS° should be in consistent units (e.g., J mol⁻¹ and J mol⁻¹ K⁻¹). A plot of ln K vs. 1/T yields a straight line with a slope of -ΔH°/R and a y-intercept of ΔS°/R.

Core idea

Overview

The Van't Hoff equation is a fundamental thermodynamic relationship that describes how the equilibrium constant (K) of a chemical reaction changes with temperature (T). It connects the macroscopic observable K with the microscopic thermodynamic properties of the reaction: the standard enthalpy change (ΔH°) and standard entropy change (ΔS°). This equation is crucial for predicting the shift in equilibrium position under varying thermal conditions and for determining thermodynamic parameters from experimental equilibrium data.

When to use: Apply this equation when you need to predict the equilibrium constant at a different temperature, or when you want to determine the standard enthalpy or entropy change of a reaction from experimental equilibrium constant data at various temperatures. It's particularly useful for understanding the temperature dependence of industrial processes and biological systems.

Why it matters: The Van't Hoff equation is vital for optimizing chemical processes, designing catalysts, and understanding natural phenomena. It allows chemists to predict how changes in temperature will affect product yield, which is critical in industrial synthesis. In biochemistry, it helps explain how temperature influences enzyme activity and biological equilibria.

Symbols

Variables

K = Equilibrium Constant, $ln$ K = Natural Log of Equilibrium Constant, $Δ$ H^ $\circ$ = Standard Enthalpy Change, R = Gas Constant, T = Temperature

K

Equilibrium Constant

dimensionless

ln K

Natural Log of Equilibrium Constant

dimensionless

Δ H^{\circ}

Standard Enthalpy Change

J/mol

R

Gas Constant

J m o l^{- 1} K^{- 1}

T

Temperature

K

Δ S^{\circ}

Standard Entropy Change

J m o l^{- 1} K^{- 1}

Walkthrough

Derivation

Formula: Van't Hoff Equation

The Van't Hoff equation describes the temperature dependence of the equilibrium constant for a chemical reaction.

ΔH° and ΔS° are constant over the temperature range considered.
The reaction is at equilibrium.
Ideal gas behavior for gaseous components or ideal solution behavior for dissolved components.

1

Relate Gibbs Free Energy to Equilibrium Constant:

The standard Gibbs free energy change (ΔG°) is related to the equilibrium constant (K) at a given temperature (T) by this fundamental thermodynamic equation, where R is the gas constant.

Δ G^{\circ} = - R T ln K

2

Relate Gibbs Free Energy to Enthalpy and Entropy:

The standard Gibbs free energy change is also defined in terms of the standard enthalpy change (ΔH°) and standard entropy change (ΔS°) of the reaction.

Δ G^{\circ} = Δ H^{\circ} - T Δ S^{\circ}

3

Equate the Expressions for ΔG°:

By equating the two expressions for ΔG°, we link the equilibrium constant to the enthalpy and entropy changes.

- R T ln K = Δ H^{\circ} - T Δ S^{\circ}

4

Rearrange to Isolate ln K:

Divide the entire equation by -RT to make ln K the subject.

ln K = - \frac{Δ H ^{\circ}}{R T} + \frac{T Δ S ^{\circ}}{R T}

5

Simplify the Expression:

Simplify the second term by canceling T from the numerator and denominator, yielding the Van't Hoff equation.

ln K = - \frac{Δ H ^{\circ}}{R T} + \frac{Δ S ^{\circ}}{R}

Result

ln K = - \frac{Δ H ^{\circ}}{R T} + \frac{Δ S ^{\circ}}{R}

Source: Physical Chemistry by P.W. Atkins and J. de Paula, 11th Edition — Chapter 6: Chemical Equilibrium

Free formulas

Rearrangements

Solve for $K$

Make K the subject

K = e^{\frac{- Δ H ^{\circ} + T Δ S ^{\circ}}{R T}}

Exact symbolic rearrangement generated deterministically for K.

Difficulty: 3/5

Solve for $Δ H^{\circ}$

Make deltaH the subject

Δ H^{\circ} = T (- R ln (K) + Δ S^{\circ})

Exact symbolic rearrangement generated deterministically for deltaH.

Difficulty: 3/5

Solve for $R$

Make R the subject

R = \frac{- \Delta H^\circ + T \Delta S^\circ}{T \ln\left(K \right)}}

Exact symbolic rearrangement generated deterministically for R.

Difficulty: 3/5

Solve for $T$

Make T the subject

T = - \frac{\Delta H^\circ}{R \ln\left(K \right)} - \Delta S^\circ}

Exact symbolic rearrangement generated deterministically for T.

Difficulty: 3/5

Solve for $Δ S^{\circ}$

Make deltaS the subject

\Delta S^\circ = \frac{\Delta H^\circ}{T} + R \ln\left(K \right)}

Exact symbolic rearrangement generated deterministically for deltaS.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a hyperbolic curve because the temperature variable is located in the denominator, resulting in a horizontal asymptote at the value of the standard entropy change divided by the gas constant. For a chemistry student, this shape illustrates that as temperature increases toward large values, the natural log of the equilibrium constant approaches a fixed limit determined by entropy, whereas at low temperatures, the enthalpy term exerts a much more dominant influence on the equilibrium. The most important feature of this curve is that it is restricted to positive temperature values and never crosses its horizontal asymptote, meaning the ratio of entropy to the gas constant represents a theoretical ceiling or floor for the reaction's equilibrium state.

Graph type: hyperbolic

Why it behaves this way

Intuition

Plotting the natural logarithm of the equilibrium constant (ln K) against the reciprocal of the absolute temperature (1/T) yields a straight line, where the slope is -ΔH°/R and the y-intercept is ΔS°/R.

K

Equilibrium constant of the reaction

A measure of the extent to which a reaction proceeds towards products at equilibrium. A large K means products are favored.

Δ H^{\circ}

Standard molar enthalpy change of the reaction

Represents the heat absorbed (positive) or released (negative) by the reaction under standard conditions. It dictates how sensitive the equilibrium is to temperature changes.

Δ S^{\circ}

Standard molar entropy change of the reaction

Represents the change in the system's disorder or the dispersal of energy and matter during the reaction. It contributes to the intrinsic drive of the reaction.

R

Ideal gas constant

A fundamental constant that scales energy changes (like enthalpy and entropy contributions) into units compatible with temperature and the dimensionless equilibrium constant.

T

Absolute temperature in Kelvin

Provides the thermal energy context for the reaction. Higher temperatures increase the kinetic energy of molecules, affecting the balance between enthalpy and entropy contributions.

Signs and relationships

-\frac{Δ H^°}{RT}: The negative sign in front of the enthalpy term indicates an inverse relationship between temperature and the equilibrium constant for exothermic reactions (ΔH° < 0), and a direct relationship for endothermic reactions

Free study cues

Insight

Canonical usage

This equation requires consistent SI units for thermodynamic quantities (J for energy, K for temperature, mol for amount) to ensure the terms on the right-hand side are dimensionless, matching the dimensionless nature of

Common confusion

A common mistake is using temperature in Celsius instead of Kelvin, or using inconsistent energy units for ΔH° (e.g., kJ mol^-1) and R (J mol^-1 K^-1) without proper conversion.

Unit systems

$K$ dimensionless · The equilibrium constant K is fundamentally defined as a ratio of activities, making it dimensionless. While often expressed using concentrations or partial pressures, its true nature is unitless.

$Δ H °$ J mol^-1 · Standard enthalpy change. Commonly reported in kJ mol^-1, but must be converted to J mol^-1 for consistency with the ideal gas constant R in J mol^-1 K^-1.

$Δ S °$ J mol^-1 K^-1 · Standard entropy change.

$R$ J mol^-1 K^-1 · The ideal gas constant (or molar gas constant).

$T$ K · Absolute temperature, always expressed in Kelvin.

One free problem

Practice Problem

For a reaction, the standard enthalpy change (ΔH°) is -50,000 J/mol and the standard entropy change (ΔS°) is -100 J/mol K. Calculate the natural logarithm of the equilibrium constant (ln K) at 298 K. Use R = 8.314 J/mol K.

Standard Enthalpy Change-50000 J/mol

Standard Entropy Change-100 J mol^{-1} K^{-1}

Temperature298 K

Gas Constant8.314 J mol^{-1} K^{-1}

Solve for: lnK

Hint: Ensure all units are consistent (Joules for energy terms).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Predicting the optimal temperature for ammonia synthesis in the Haber-Bosch process to maximize yield.

Study smarter

Tips

Ensure temperature 'T' is always in Kelvin (K).
The gas constant 'R' must be chosen with units consistent with ΔH° and ΔS° (e.g., 8.314 J mol⁻¹ K⁻¹).
ΔH° and ΔS° should be in consistent units (e.g., J mol⁻¹ and J mol⁻¹ K⁻¹).
A plot of ln K vs. 1/T yields a straight line with a slope of -ΔH°/R and a y-intercept of ΔS°/R.

Avoid these traps

Common Mistakes

Using Celsius instead of Kelvin for temperature.
Inconsistent units for R, ΔH°, and ΔS° (e.g., kJ for ΔH° and J for R/ΔS°).
Confusing the Van't Hoff equation with the Arrhenius equation (which relates rate constant to temperature).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Van't Hoff equation describes the temperature dependence of the equilibrium constant for a chemical reaction.

Apply this equation when you need to predict the equilibrium constant at a different temperature, or when you want to determine the standard enthalpy or entropy change of a reaction from experimental equilibrium constant data at various temperatures. It's particularly useful for understanding the temperature dependence of industrial processes and biological systems.

The Van't Hoff equation is vital for optimizing chemical processes, designing catalysts, and understanding natural phenomena. It allows chemists to predict how changes in temperature will affect product yield, which is critical in industrial synthesis. In biochemistry, it helps explain how temperature influences enzyme activity and biological equilibria.

Using Celsius instead of Kelvin for temperature. Inconsistent units for R, ΔH°, and ΔS° (e.g., kJ for ΔH° and J for R/ΔS°). Confusing the Van't Hoff equation with the Arrhenius equation (which relates rate constant to temperature).

Predicting the optimal temperature for ammonia synthesis in the Haber-Bosch process to maximize yield.

Ensure temperature 'T' is always in Kelvin (K). The gas constant 'R' must be chosen with units consistent with ΔH° and ΔS° (e.g., 8.314 J mol⁻¹ K⁻¹). ΔH° and ΔS° should be in consistent units (e.g., J mol⁻¹ and J mol⁻¹ K⁻¹). A plot of ln K vs. 1/T yields a straight line with a slope of -ΔH°/R and a y-intercept of ΔS°/R.

References

Sources

Atkins' Physical Chemistry
Wikipedia: Van 't Hoff equation
Callen, Herbert B. Thermodynamics and an Introduction to Thermostatistics
NIST CODATA
IUPAC Gold Book: 'equilibrium constant'
IUPAC Gold Book: 'standard enthalpy of reaction'
IUPAC Gold Book: 'standard entropy of reaction'
Atkins' Physical Chemistry, 11th ed.

Van't Hoff Equation (Equilibrium Constant vs. Temperature)

Overview

Variables

Derivation

Relate Gibbs Free Energy to Equilibrium Constant:

Relate Gibbs Free Energy to Enthalpy and Entropy:

Equate the Expressions for ΔG°:

Rearrange to Isolate ln K:

Simplify the Expression:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Gibbs Free Energy (Standard Conditions)

Gibbs Free Energy (Non-Standard Conditions)

Arrhenius Equation

Frequently Asked Questions

Sources