[ChickenKungPao]

My Alberty Silbey pchem book says d/dT ln K = DHr/(R T^2) @ const P for DGr = - R T ln K

Why can't I do this:

DGr = - R T ln K
ln K = -DGr / (R T)

From dG = -S dT + V dP and in terms of deltas and const P:
dDGr = - DSr dT = - DHr / T dT
d/dT DGr = - DHr  / T

Combine:
d/dT ln K = DGr/(R T^2) - (d/dT DGr) / R T = DGr/(R T^2) + DHr/(R T^2)
d/dT ln K = ln K + DHr/(R T^2)

Well, the last expression is close but no cigar ( ln K =  DGr/(R T^2) = 0 to jive w/ my pchem book - but clearly this can't be according to my derivation)  Where did I go wrong?

[ChickenKungPao]

DGr = DHr - T DSr

- dGR/(R T)  = lnK
ln K = -DHr/(RT) + DSr/R

Take a the derivative wrt T (d/dT DSr/R drops out)

Still looking trying to find the wrong assumption in my derivation though..

[ChickenKungPao]

It seems DGr may have been specified/constrained (const T & P)  when the Keq equation was derivied ie : DGr = DHr - T DSr

So, d/dT DGr is meaningless within the constraints of the Keq equation.  Funny, if you don't keep track of constraints, all of the variables start looking the same.  I seems I must be careful to remember that variables/functions have context.

[sdekivit]

dG(r) is the standard Gibbs free energy and is a constant for an equilibrium at standard temperature.

So when dG(r) = constant you should take the derivative over T and not over dG(r). Then the derivation is easy, since you must determine the derivative of 1/T with -dG(r)/R as a constant. Then you'll come up with the equation

d/dT ln K = dG(r) / (RT^2)

(the expression d/dT implicitly tells us the derivative must be taken over T)