Derivation of the Van't Hoff Isochore Equation from Free Energy
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 Kp (1)
By differentiating equation (1) with respect to temperature (T) at constant pressure (p):
[∂(ΔG°) / ∂T]p = -R ln Kp - RT [d(ln Kp) / dT]p (2)
Multiply equation (2) by T:
T [∂(ΔG°) / ∂T]p = -RT ln Kp - RT2 [d(ln Kp) / dT]p
Since ΔG° = -RT ln Kp, we can substitute this into the equation:
T [∂(ΔG°) / ∂T]p = ΔG° - RT2 [d ln Kp / dT] (3)
Therefore: ΔG° - T [∂(ΔG°) / ∂T]p = RT2 [d ln Kp / dT]
Integration with the Gibbs-Helmholtz Equation
Now, the Gibbs-Helmholtz equation at standard state is:
ΔG° = ΔH° + T [∂(ΔG°) / ∂T]p
ΔH°...