CHM 1046

CHM 1046
General Chemistry II
Dr. Michael Blaber

Acid-Base Equilibria

Relationship Between K_a and K_b

Strong acids have weak conjugate bases

Strong bases have weak conjugate acids

Is there some quantitative relationship between these observations?

The ammonia (NH₃)/ammonium ion (NH₄⁺) conjugate acid/base pair

Ammonia can act as a base and accept a proton in water

NH₃(aq) + H₂O(l) ó NH₄⁺(aq) + OH^-(aq)

Ammonium ion can act as an acid and donate a proton in water

NH₄⁺(aq) + H₂O(l) ó NH₃(aq) + H₃O⁺(aq)

The two balanced equations for the reaction of NH₃(aq) or NH₄⁺(aq) with water can be combined:

NH₃(aq) + H₂O(l) ó NH₄⁺(aq) + OH^-(aq)

NH₄⁺(aq) + H₂O(l) ó NH₃(aq) + H₃O⁺(aq)

NH₃(aq) + NH₄⁺(aq) + 2H₂O(l) ó NH₃(aq) + H₃O⁺(aq) + NH₄⁺(aq) + OH^-(aq)

Canceling similar terms on the left and right sides yields:

~~NH₃(aq)~~ + ~~NH₄⁺(aq)~~ + 2H₂O(l) ó ~~NH₃(aq)~~ + H₃O⁺(aq) + ~~NH₄⁺(aq)~~ + OH^-(aq)

2H₂O(l) ó H₃O⁺(aq) + OH^-(aq)

H₂O(l) ó H⁺(aq) + OH^-(aq)

Thus, when considering the base/conjugate acid reactions of NH₃/NH₄⁺ the combination results in an overall reaction of the ionization of H₂O(l)

Reaction 1 + Reaction 2 = Reaction 3

Base reaction of NH₃ + Conjugate acid reaction of NH₄⁺ = Ionization of H₂O

We know how to combine reactions to give an overall new reaction - it is the algebraic sum of the reactants and products

But, what is the resulting equilibrium constant for a combined reaction?

When two reactions are added to give a third reaction, the equilibrium constant for the overall reaction is equal to the product of the equilibrium constants for the two added reactions

K₃ = K₁ * K₂

For the base reaction of NH₃ and the conjugate acid reaction of NH₄⁺, the overall equilibrium constant would be:

In other words, the overall reaction of ammonia with its conjugate acid is the ionization of H₂O(l), and the overall equilibrium constant is K_w, the ion-product constant for water

This result gives provides a clue as to the relationship between K_a and K_b for acid/conjugate base pairs (or base/conjugate acid pairs):

The product of the acid-dissociation constant for an acid (i.e. K_a) and the base-dissociation constant for its conjugate base (i.e. K_b) is the ion-product constant for water

K_a * K_b = K_w

What are the implications of this relationship?

The product of K_a * K_b must always equal 1 x 10^-14
If K_a is large (i.e. if the acid is a strong acid), then K_b must be small (i.e. the conjugate base is a weak base)
If K_b is large (i.e. if the base is a strong base), then K_a must be small (i.e. the conjugate acid is a weak acid)

In tables of dissociation constants for acids or bases, often only the value for the acid-dissociation constants is given.

The base-dissociation constants can be determined from the relationship:

K_b = K_w/K_a

Also, acid-dissociation constants are sometimes given as pK_a values:

pK_a = -log(K_a)

K_a = 10^-pKa

and

pK_a + pK_b = pK_w = 14.0