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Q is the the reaction quotient and K is the equilibrium constant. Both Q and K are calculated the same way (by using pressures or concentrations) but they are applied differently depending on the equation that you are using. Because they are calculated the same, it shouldn't make too much of a difference if you are calculating one or the other but the problem and equation should specify that for you.
In much of what we do, Q is used when we want to calculate some sort of potential at non-standard conditions (Gibbs Free Energy, Nernst Equation). K is used in our calculations when we want to calculate the potential at standard conditions, usually.
K can be used calculate the final concentrations at equilibrium for a reaction using an ICE table and the natural progression of the reaction, from left to right or from right to left. Q expresses the relative ratio of products to reactants at a given instant.
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