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I believe that since the book typically uses partial pressures for homogeneous reactions with gases, the p in Kp is implied when they give K. It would be safe to assume that if they give you a value of K and give measures of partial pressure for homogeneous reactions with gases that K=Kp.
Yes, you can assume Kp as essentially the only way to find the K of homogenous gas is through the partial pressure. Similarly, you can assume Kc when dealing with the K of homogenous aqueous reactions because you use molarity to find it.
K in general is an expression to describe the composition of the reaction at equilibrium. Kp is for concentration of partial pressure of the reactants/products while Kc is used in concentration of the reactants/products. Solid and liquids aren't included in the equilibrium constant expression, only gases or aqueous solutions. Kc is used in molar concentration while Kp is for partial pressure.
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