At equilibrium there is a relationship between the concentrations (partial pressures) of the products and the reactants. We will see that these are the concentration that minimize the total free energy. But for now we will simply invoke this idea. For different types of chemical species, the free energy at equilibrium is related to different standard states. As such, we end up with different "types" of equilibrium constants. This idea can be unified by using a new term which has different definitions in different contexts. This is the "activity." The activity of a compound tells us how the free energy (or more correctly the chemical potential) of that compound varies with the reaction. The equilibrium constant is then a ratio of the activities of the products and the reactants. The activity of each chemical species is raised to the power of its stoichiometric coefficient in the balanced equations. Therefore, to be interpreted, an equilibrium constant must always be accompanied by a corresponding balanced equation. For example, look at the generic reaction of
\[ a{\rm A} + b{\rm B} \rightleftharpoons c{\rm C} + d{\rm D} \]
for this reaction the equilibrium constant will be
\[ K = {{{a_{\rm C}^c \; a_{\rm D}^d}\over a_{\rm A}^a \; a_{\rm B}^b} }\]
where \(a_{\rm i}\) is the activity of compound i. The expression on the right is the mass action expression (or law of mass action). Technically the "law of mass action" states that at equilibrium this expression is a constant. We typically referred to this constant as the equilibrium constant. This ratio of the activities at equilibrium gives us the value of the equilibrium constant. At any other conditions, these activities are simply the activities "now" and we give the mass action expression a different name "the reaction quotient", generally denoted by the a capital Q.
To give a concrete example, let's look at the reaction
\[\rm{2H_2(g)+O_2(g) \rightleftharpoons 2H_2O(g)}\]
the equilibrium constant for this reaction will be
\[K = {{a_{\rm H_2O}^2} \over {a_{\rm H_2}^2 \; a_{\rm O_2}}}\]
The activity of a chemical species varies depending on its phase (since the free energy will change differently in different phases during a reaction). For gases the activity is defined as
\[a_i = {P_i \over P^{\circ}}\]
where \(P^{\circ}\) is the standard pressure of 1 atm (or 1 bar). This means that the activity is simply the partial pressure in atm, but since we divide by one atm the activity is unit less.
For solutions, the activity is nearly identical except that it is the concentration
\[a_i = {[i] \over C^{\circ}}\]
where the standard concentration, \(C^{\circ}\) is just 1 M. So the activity is simply the concentration in Molarity (but since it is divided by 1 M it is also unit less).
For pure solids and pure liquids the free energy doesn't change from the standard value during a reaction and thus the activity is just defined as 1.
\[a_i = 1\]
So anywhere we have the activity of a solid or a liquid in the equilibrium expression it "disappears" since it is replaced by the number 1.
For example the equilibrium constant for the reaction
\[\rm{CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)}\]
is
\[ K = {{a_{\rm CaO(s)} \; a_{\rm CO_2(g)}} \over {a_{\rm CaCO_3(s)}}} = P_{\rm CO_2}\]
Any example of an aqueous equilibrium is
\[\rm{HF(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + F^-(aq)}\]
for which the equilibrium constant would be
\[ K = {{a_{\rm F^-(aq)} \; a_{\rm H_3O^+(aq)}} \over {a_{\rm HF(aq)} \;{a_{\rm H_2O(l)} }}} ={{{\rm [F^-]} \; {\rm [H_3O^+]}} \over {{\rm [HF]}}} \]
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