We have seen that there is a relationship between the pKa values of the conjugate acids involved in an acid-base reaction and the value of Keq for that reaction. Since the principles involved in calculating Keq for an acid-base reaction may be extended to other reactions, we will take the time to derive the relationship between pKa values and Keq.
Consider the reaction between acetic acid and trifluoroacetate ion shown in Equation 1.
The expression for the equilibrium constant for this reaction is
but how do you determine the numerical value of Keq? A simple way involves treating each acid and its conjugate base separately. Consider the dissociation of acetic acid in water as shown in Equation 2.
As we have already seen, the equilibrium expression for this reaction is
Equation 3 describes the dissociation of trifluoroacetic acid in water.
The equilibrium expression for this reaction is
Dividing the equilibrium expression for Equation 2 by that for Equation 3 gives
Clearing terms and rearranging yields
which is the equilibrium expression for Equation 1. So, by independently measuring the degree of ionization of each acid in water, we can calculate the equilibrium constant for a reaction involving the two acid/conjugate-base pairs.
It is possible to arrive at the value of Keq for Equation 1 by another method. The pKa of acetic acid is 4.75, while that of trifluoroacetic acid is 0.18. The difference in these values is (0.18-4.75) = -4.57. The antilogarithm of -4.57 = 2.69 x 10-5. Fortunately, if you use integral values for the pKa values of each acid, you can do the math in your head. Using integral pKa values for acetic acid (5) and trifluoroacetic acid (0) gives a difference of (0-5) = -5, and the antilogarithm of -5 = 1 x 10-5. Conversely, the antilogarithm of a positive integer, say 5, is 1 x 105.
In order to calculate the equilibrium constant for any acid-base reaction follow these steps:
Here's an example using the reaction of phenol with sodium hydroxide to produce the phenolate ion and water.
The value of this type of calculation lies in its application to related systems. Consider the simple substitution reaction shown in Equation 4.
This is not an acid-base reaction in the same sense that the previous examples were. There is no proton transfer from one atom to another. Still, we can assess the equilibrium constant by considering the relative stabilities of the hydroxide and chloride ions. If chloride ion is less basic than hydroxide ion, the equilibrium will favor the products. If hydroxide ion is less basic than chloride ion, the equilibrium will favor the reactants. To assess the relative stabilities of chloride and hydroxide ion we need to write an equation for a reaction involving these two ions and their conjugate acids. Equation 5 is one possibility.
Using the method outlined above, it is easy to determine that the equilibrium constant for this reaction is 1 x 1023. In other words, chloride ion is much less basic than hydroxide ion. As a first approximation, the difference in stability of chloride and hydroxide ion should be the same in Equation 4 as it is in Equation 5. Thus it is reasonable to conclude that the equilibrium constant for reaction 4 should also be approximately 1 x 1023.
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