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K is the equilibrium constant of an reaction that is already at equilibrium, while Q (reaction quotient) is used when the reaction is not already at equilibrium. Q is also used to show in which direction the reaction shifts (for example, when K>Q, then the reaction shifts to the right).
Q is calculated the same way as K, but Q is used to calculate the compositions for reactants and products at ANY TIME during the reaction, not just at equilibrium. You can compare Q to K to see if the reaction is at equilibrium, shifted to the left, or shifted to the right.
K is appropriate to use when the reaction is already at equilibrium. There is only one K value for each reaction and its circumstances (temperature, pressure, etc.) Q is appropriate to use at any point in time of the reaction, and is used to compare to K.
You can think of Q on a "number line" with K where Q is on either side of K and K is in the middle like 0. When a reaction is not at equilibrium you want to find Q to determine which way the rxn will shift to reach K at equilibrium. If Q is less than K the rxn will move right towards K, the equilibrium, in the middle of the "number line".
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