I'm not entirely familiar with the terminology here but let's have a try.
First, let's introduce the concept of the reaction quotient
Q. For a reaction
A + B
C + D
Q = [C][D]/[A][B ] (This may be at any point in the reaction, not necessarily equilibrium)
The equilibrium constant Kc
is equal to the value of Q at equilibrium, Qeq
This is strictly true only at high dilution, when activity equals concentration. Let us call this equilibrium constant Kc∞
Now we could define Kc
, and this is a constant. Then we would say that at higher concentrations, when activity differs from concentration, Qeq
may be different from Kc
is constant, but Qeq
Alternatively (and I don't know which practice is currently fashionable) we could define Kc
under all conditions, and then say that Kc
, which is variable, differs from Kc∞
, which is constant.
Now let's consider it in terms of activities, α. We define
Qα = αC
and Kα ≡ Qαeq
This is always true, from the definition of activity - it is that quantity which behaves as concentration ideally should. Kα is constant.
At high dilution γ = 1, so Kα = Kc∞
If you take the second alternative above, where Kc
and is variable, then
Kα = Kc
and Kα corresponds to your K(T).
I hope this helps a little. The key point is that the equilibrium constant in terms of activities is a true constant - that's what activity means.