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K and Q can have different values because the concentrations used the calculate them are different even though the method to do so is the same. Because Q can be calculated at any point during the reaction, the concentrations won't be constant like they would if you were calculating K.
Q, the reaction quotient, is calculated at any time during the reaction when the concentrations are different than at equilibrium and is usually compared to K, the equilibrium constant, to determine which direction a reaction will proceed. In this case, K will already be known so if the calculated value does not match the K value, it will be considered Q. If Q is less than K at some point during the reaction, then there are more reactants than products and the forward reaction is favored. If Q is greater than K at some point during the reaction, then there are more products than reactants and the reverse reaction is favored.
The reaction quotient is used to calculate the ratio of the reactant and product concentrations at any point as the reaction takes place. When the reaction quotient equals the equilibrium constant k, the reaction reaches equilibrium. Thus, the reaction quotient is used to calculate whether the forward or reverse reaction proceeds by looking at the concentrations of the products and reactants and comparing it to the k value. If the reaction quotient is smaller than k, the forward reaction is favored because there are more reactants than products, and if the reaction quotient is greater than k, the reverse reaction is favored since there are more products than reactants.
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