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?
