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Kc is the equilibrium constant, referring to when the reactants and products are in terms of molarity. Kp is the equilibrium constant when the products and reactants are given in terms of atm (usually when they're gases) and so it's know as the equilibrium constant of partial pressures. Qc and Qp basically refers to molarity and partial pressure respectively, but instead of equilibrium, they refer to at some point in time of the reaction. It could be before, after, or even at the equilibrium depending on whether or not Q is equal to K.
Some of the difference can be seen in the names. We call Q the reaction quotient and K the equilibrium constant. "Constant" implies that this value does not change – it is a set value that the reaction wants to attain for the given set of conditions – while Q can have different values throughout the course of a reaction. The values K and Q are calculated in the exact same way, so they are easy to compare to see which direction the reaction will proceed.
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