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Kc is used to calculate the equilibrium constant using the concentration of the products and reactants at equilibrium. Kc is essentially a more specific way of saying K, because it specifies that you are calculating it using concentration, whereas Kp is calculated using partial pressure. Q, the reaction quotient, is what you want to find when determining which way a reaction will proceed. It is found the same way that K is, except that it can be calculated at any point during the reaction, not just at equilibrium like when finding K. When Q is less than K, it means that there are more reactants (denominator) than products (numerator). In order to reach equilibrium, Q will have to get larger, so the amount of product will have to increase (larger numerator = larger answer) so that the overall calculation of Q will increase. To get more product, the forward reaction would be favored.
Its not that Kc is greater or less than K, but it is greater or less than Q. K is used in terms of equilibrium. On the other hand Q is used at any time during the reaction. When the mols of reactants are greater than those of products, the reaction proceed forward in order to create more products for the reaction to reach equilibrium.
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