So we must have rules telling us how to change the reaction constant if we go from from one system of measuring concentrations to another one. The problem clearly comes from the changing reference state if the mol count changes in a reaction. Well, lets go back to the final stage in the derivation of the mass action law and see what can be done. The a i are the activities, which we defined when discussing the chemical potential analogous to the fugacities for gases. Fugacities , in turn, were introduced to take care of non-ideal behavior of gases.
Lets now switch to an absolute quantity.
We take the number of mols of gas i. If this looks a bit like magic, you are encouraged to go through the motions in fiddling around the products and the sums yourself. Lets try it.