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We omit the units because in actuality we are supposed to be calculating equilibrium constants and reaction quotients using the activities of each reactant or product of a reaction. Activities are unitless, only scalar values, so it makes sense that calculating a value K with unitless values produces another unitless value. However, for the purpose of this class, activities and concentration/partial pressure values are close enough to the activities' values that we use those to calculate K. We are concerned about the values when we calculate K or Q, not necessarily the units involved.
We are technically using chemical activities to calculate K. Activities are unitless, so that makes K unitless. To be be more exact, you should be writing the equilibrium constant by using activities. K equals the activities of products to respective stoichiometric coefficients over activities of reactants to respective stoichiometric coefficients. Keep in mind that the activity of water is 1 when it is the solvent. We use concentrations to calculate K because it is a good enough approximation to activities.
You aren't really "omitting" the units. It is just that since all of the units on the values being used to calculate K[sub]c[sub] are the same, they cancel each other out. It is sort of like when doing conversion factors you put the same units in the denominator or numerator in order to cancel them out, but this is completely unintentional.
But in this course often a question will ask for a specific K, like Kc or Kp. In this case, make sure your units match with what the question is asking for. But K does not necessarily have to have this specificity
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