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In kinetics, k usually refers to rate constant k. I'm pretty sure that k should always be positive or >1 because rates of reactions are always positive values. This includes rates that occur in the reverse direction.
The equilibrium constant K is equal to k/k'. Thus if K is greater than 1, it means that k' (the rate of the reverse reaction) is very small or slow. This makes sense since when K is greater than one, the products are favored in the reaction, and vice versa.
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