Deriving K

So far we've treated Keq empirically.  We've shown that there is such an expression by working out the values from a simulation, and have learned how to write K expressions.  However, there should be a theoretical explanation for why K exists.

Remember that science consists of two processes that work together, and unless both are done, you do not have science:
  • empirical detection of properties of nature (observation)
  • theoretical explanation of the observed properties (scientist's best attempt at creating an explanation for the observations)

The explanation for K must come from the theory of reaction rates.  After all, an equilibrium is explained by assuming that it occurs when the rates of the forward and reverse reactions become equal.

To show how we can derive an equilibrium constant, we will use an imaginary reaction

aA (g) + bB (g) yY (g) + zZ (g)

At equilibrium every step in this reaction's mechanism must also be in equilibrium.   Since the rate of a reaction depends on a rate constant, multiplied by the concentrations of the reacting species (click here if you need to review reaction rates) the rate of the forward reaction (Rf) is:

Rf = kf [A]a[B]b

(1)

and the rate of the reverse reaction (Rr) is:

Rr = kr [Y]y[Z]z

(2)

However, since this is an equilibrium and the forward (Rf) and reverse (Rr) rates are equal, then equations (1) and (2) must be equal to each other.  Therefore:

kf [A]a[B]b = kr [Y]y[Z]z

(3)

which can be rearranged to obtain the concentrations of products over the reactants, by dividing both sides by kr and [A]a[B]b:

leaving

(4)

Now, both kf and kr are just numbers, therefore if you divided them you would just get another number.  In other words, let , so that:

derivek_4.gif (316 bytes)

(5)

which is the equilibrium constant for this reaction.

So, we can derive the equilibrium constant expression from our theory of equilibrium – equal reaction rates in the forward and reverse directions – and from empirical observation.   The theory (which was created by our scientific imagination) and the observations about Keq agree.  This gives us more reason to believe that the theory of equal reaction rates at equilibrium is correct.