Deriving the Van't Hoff Isochore Equation

Chemist Concept of Free Energy

Derivation of the Van't Hoff Isochore Equation

To discuss the Van't Hoff Isochore Equation, we begin with the equation for the Van't Hoff isotherm:

ΔG° = -RT ln K_p (1)

By differentiating equation (1) with respect to temperature (T) at constant pressure (p):

[∂(ΔG°) / ∂T]_p = -R ln K_p - RT [d(ln K_p) / dT]_p (2)

Multiply equation (2) by T:

T [∂(ΔG°) / ∂T]_p = -RT ln K_p - RT² [d(ln K_p) / dT]_p

Since ΔG° = -RT ln K_p, we can substitute this into the equation:

T [∂(ΔG°) / ∂T]_p = ΔG° - RT² [d ln K_p / dT] (3)

Therefore: ΔG° - T [∂(ΔG°) / ∂T]_p = RT² [d ln K_p / dT]

Integration with the Gibbs-Helmholtz Equation

Now, the Gibbs-Helmholtz equation at standard state is:

ΔG° = ΔH° + T [∂(ΔG°) / ∂T]_p

ΔH° = ΔG° - T [∂(ΔG°) / ∂T]_p (4)

From equation (4), we can equate the terms:

ΔH° = RT² [d ln K_p / dT]

(ΔH°) / (RT²) = d ln K_p / dT

Where ΔH° is the enthalpy change of the substance at standard state and pressure. Generally, it is observed that with a change in external pressure, the substance is not affected significantly. Thus, without expressing external pressure, the equation remains:

(ΔH) / (RT²) = d ln K_p / dT (8)

Equation (8) is known as the Van't Hoff Equation.

Published By: BHARNT

Free Energy and Chemical Equilibrium

From the above equation, if we plot a graph of ln K_p vs. 1/T, the slope of the graph will be -ΔH / R. Equation (8) shows that for a reaction of gases, the change in the equilibrium constant with temperature is expressed in terms of enthalpy change.

The term isochore used earlier for this equation is not strictly proper because the ΔH of reaction is the heat of reaction at constant pressure, not at constant volume. If the value of ΔH° is taken as constant for a definite period of temperature, then the integration of the Van't Hoff equation will be:

∫ d ln K_p = ∫ (ΔH°) / (RT²) dT

ln K_p = (-ΔH°) / (RT) + C

Therefore: log K_p = -ΔH° / (2.303 RT) + C

Graphical Representation and Slope Analysis

Compare the above equation with y = mx + c (a straight line equation). A plot of log K_p vs. 1/T will be a straight line having a slope value of -ΔH° / 2.303R. Integrating the Van't Hoff equation between two limits of temperature (T₁ and T₂):

∫_Kp1^K_p2 d ln K_p = ∫_T1^T₂ (ΔH° / RT²) dT

ln (K_p2 / K_p1) = - (ΔH° / R) [1/T]_T1^T₂

ln (K_p2 / K_p1) = - (ΔH° / R) [1/T₂ - 1/T₁]

log (K_p2 / K_p1) = (ΔH° / 2.303R) [ (T₂ - T₁) / (T₁T₂) ] (10)

Applications of the Van't Hoff Equation

Where K_p2 and K_p1 are equilibrium constants at temperatures T₂ and T₁ in Kelvin, respectively.

• If ΔH = +ve (Endothermic), then with an increase in T, the value of K also increases.
• If ΔH = -ve (Exothermic), then with an increase in T, the value of K decreases.

Published By: BHAF