The Van't Hoff equation is meant to show the Temperature dependance of K, and while it is derived from an expression that relates G to K, in its ultimate for the equation excludes G.
2) Divide both sides by -RT to isolate ln(K):
3) Use the above equation, plug in K1 at T1 to make the equation for initial K and plug in K2 at T2 to make the equation for Final K. Subtract the final K equation by initial K equation:
4) Simplify the equation by using ln(a)-ln(b) = ln(a/b) and cancelling out the entropy terms:
5) Now you factor out -delta H/R to get the final form that we use:
Note how the process of deriving this equation eventually allows us to compute a final K @ Tfinal from an initial K @ Tini knowing only the enthalpy of reaction.