# Derivation of the Van 't Hoff equation

I was reading the paper Relaxation Kinetics of Ferric Thiocyanate (Goodall et. al, 1972) and I came across the passage

Reaction (1) is the simplest representation of the equilibrium between ferric and thiocyanate ions and their complex. The ratio $$\frac{k_{12}}{k_{21}}$$ is the equilibrium constant $$K_c$$, which has the value $$\pu{1391 mol-1}$$ at $$\pu{298 ^\circ K}$$ and ionic strength $$\pu{0.5 mol kg-1}$$. $$\ce{[Fe(H2O)6]^3+ + SCN^-1 <=>[k_{21}][k_{12}] [Fe(H2O)5SCN]^2+ + H2O} \tag{1}$$ The enthalpy change in the reaction is $$\pu{-5 kJ mol-1}$$, since $$\frac{\mathrm{d}\ln K_c}{\mathrm{d}T} = \frac{\Delta H}{RT^2}.$$

I am quite confused as to where the equation $$\frac{\mathrm{d}\ln K_c}{\mathrm{d}T} = \frac{\Delta H}{RT^2}$$ came from, as after integrating it, I seem to get the equation $$\ln K_c=-\frac{\Delta H}{RT}$$, which then suggests that $$\Delta H=-RT\ln K_c$$. However, $$\Delta G^\circ=-RT\ln K$$. How can these two be the same?

• You have to calculate the finite integral between 2 T. See also van't Hoff equation on Wikipedia. For the infinite integral, you mustn't omit the integration constant. Delta H =-RT(ln K_c + C) Commented Sep 2, 2023 at 5:09
• If the equation is written as it is in ($1$), the equilibrium constant $\pu{K_c}$ should not have any dimension. As this constant is given in $\pu{mol^{-1}}$, it means that there is no $\ce{H2O}$ at all in the equation, and that ($1$) should probably be replaced by : $\ce{Fe^{3+} + SCN^- -> [Fe(SCN)]^{2+}}$. Commented Sep 2, 2023 at 9:17
• Unfortunately, the title of the question has no connexion with the question. The question is not related to "Relaxation Kinetics". Commented Sep 2, 2023 at 9:25
• I have quoted the paper exactly exactly as it was written, and yes, I do apologise for the poor title choice, which I have changed. Commented Sep 2, 2023 at 9:32
• Chemistry SE (in contrary to some other SE sites) strongly recommends plain text question titles for index/search reasons and due possible displaying issues in question title lists. // Pay more attention to prior search before asking, see en.wikipedia.org/wiki/Van_%27t_Hoff_equation Commented Sep 2, 2023 at 9:56