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When you have a reaction that consists of only gasses, you use their partial pressures in the equation that we learned to find the Kp value. When the reaction consists of compounds in a non gas state (for instance aqueous) you would plug in the concentrations of the products and reactants into the same equation to now find the Kc value (omitting solids and liquids). The p and c just denote what is being used to find the equilibrium constant (partial pressure or concentration).
I agree that the p or c denotes what unit is being used and that liquids and solids must be omitted, but I believe you can use partial pressure or concentration for gases. Using the ideal gas law, we can convert partial pressure to concentration by calculating concentration=P over R times T since this is an equivalent value it can be used to calculate Kc and Qc even if you were given partial pressures. This is to the best of my knowledge according to what has been taught in class.
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