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K is the general term for an equilibrium constant, which the textbook defines in terms of activities. (which Dr. Lavelle said was more accurate, but unnecessary for an undergraduate-level class) Kp is the equilibrium constant in terms of partial pressures, which applies only to homogenous gas reactions. Kc is the equilibrium constant in terms of concentration (which is what the c stands for, which is molarity, number of moles/volume) You can convert between partial pressure and molarity using the ideal gas law. (PV=nRT ---> P=MRT) Therefore if you have values in Kp and want them in Kc you can convert them. The reverse is true only if all of your substances are gases, in which case the value would likely have been given in terms of Kp in the first place.
Essentially Kc represents the equilibrium constant in terms of concentration (mol/L). You can use Kc with aqueous solutions and gases since both aqueous solutions and gases have concentration. Kp represents the equilibrium constant in terms of partial pressure and can only be used with gases since aqueous solutions do not have partial pressure. In homogenous equilibria, where all the reactants and products are gases, the textbook considers K and Kp to be the same. Hope this helps!
Kp is K specifically in terms of partial pressures. When referring to a reaction that includes exclusively gases, K is inferred to mean Kp. Kc is K in terms of molarity, and can also be used for gases, but if a question is referring to or asking for Kc, it will specify that this is the K it is asking for.
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