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Since the number that you end up with, when calculating the value of the law of mass action for a given reaction often also has a unit, I was wondering if there is any physical meaning to this given unit or if it is basically "useless"?

Example:

$$\ce{CH3COOH <=> CH3COO- + H+}$$

$$K_c = \frac{c(\ce{CH3COO-})\cdot c(\ce{H+})}{c(\ce{CH3COOH})}\qquad [K_c] = \pu{mol L-1}$$

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Well, you can calculate difference in number of moles of gases and active liquid species on reactant and product side species using units of $K_c$ and $K_p$, respectively, as these are mostly used for homogeneous reactions.

For example, the unit of $K_c$ is $\text{(concentration)}^n$ if $n > 0$, more liquid or aqueous (ionic) species on product side.

Then, you can predict whether the product is formed or reactant when you increase or decrease external pressure or volume of a container containing a mixture in equilibrium. Temperature's effect would be only heat of reaction dependent.

Hint: write $K_c$ in terms of moles and volume to check effect of volume, write $K_p$ in terms of mole fractions and total pressure to check effect of external pressure.

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