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K can mean many things. However, in equations where we take a logarithm of K, it must be unitless. Otherwise it would be meaningless, as a logarithm is just an inverse of an exponentiation. So K is actually expressed in the unitless activities of species. Still, activity is not that useful a concept in the lab, where we work in concentrations. So the thing is, if you want to use something like the van't Hoff equation, you need to be in the unitless K, which really is the only K that makes sense given the form of the equation. But obviously the same basic concepts of K are the same for Kc, which you would expect as activity is closely related to concentration.
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