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 K

_{c} is equal to the value of Q at equilibrium, Q

_{eq}.

This is strictly true only at high dilution, when activity equals concentration. Let us call this equilibrium constant K

_{c∞}.

Now we could define K

_{c} ≡ K

_{c∞}, and this is a constant. Then we would say that at higher concentrations, when activity differs from concentration, Q

_{eq} may be different from K

_{c}. K

_{c} is constant, but Q

_{eq} is variable.

Alternatively (and I don't know which practice is currently fashionable) we could define K

_{c} ≡ Q

_{eq} under all conditions, and then say that K

_{c}, which is variable, differs from K

_{c∞}, which is constant.

Now let's consider it in terms of activities, α. We define

Qα = α

_{C}α

_{D}/α

_{A}α

_{B}and Kα ≡ Qα

_{eq} = Q

_{eq}γ

_{C}γ

_{D}/γ

_{A}γ

_{B}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 = K

_{c∞}If you take the second alternative above, where K

_{c} = Q

_{eq} and is variable, then

Kα = K

_{c}γ

_{C}γ

_{D}/γ

_{A}γ

_{B}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.