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Relationship between Ka and Kb

Relationship between Ka of a weak acid and Kb for its conjugate base. Equations for converting between Ka and Kb, and converting between pKa and pKb. 

Key points

For conjugate-acid base pairs, the acid dissociation constant K, start subscript, start text, a, end text, end subscript and base ionization constant K, start subscript, start text, b, end text, end subscript are related by the following equations:
  • K, start subscript, start text, a, end text, end subscript, dot, K, start subscript, start text, b, end text, end subscript, equals, K, start subscript, start text, w, end text, end subscript
    where K, start subscript, start text, w, end text, end subscript is the autoionization constant
  • start text, p, end text, K, start subscript, start text, a, end text, end subscript, plus, start text, p, end text, K, start subscript, start text, b, end text, end subscript, equals, 14, space, space, start text, a, t, space, end text, 25, degrees, start text, C, end text

Introduction: Weak acid and bases ionize reversibly

Weak acids, generically abbreviated as start color #1fab54, start text, H, A, end text, end color #1fab54, donate start text, H, end text, start superscript, plus, end superscript (or proton) to water to form the conjugate base start color #1fab54, start text, A, end text, start superscript, minus, end superscript, end color #1fab54 and start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript:
start text, H, A, end text, left parenthesis, a, q, right parenthesis, plus, start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, l, right parenthesis, \rightleftharpoons, start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript, left parenthesis, a, q, right parenthesis, plus, start text, A, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis
space, space, start color #1fab54, start text, a, c, i, d, end text, end color #1fab54, space, space, space, space, space, space, space, space, space, space, start text, b, a, s, e, end text, space, space, space, space, space, space, space, space, space, space, space, space, start text, a, c, i, d, end text, space, space, space, space, space, space, space, space, space, space, space, space, space, space, start color #1fab54, start text, b, a, s, e, end text, end color #1fab54
Similarly, a base (abbreviated as start color #aa87ff, start text, B, end text, end color #aa87ff) will accept a proton in water to form the conjugate acid, start color #aa87ff, start text, H, B, end text, start superscript, plus, end superscript, end color #aa87ff, and start text, O, H, end text, start superscript, minus, end superscript:
start text, B, end text, left parenthesis, a, q, right parenthesis, plus, start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, l, right parenthesis, \rightleftharpoons, start text, H, B, end text, start superscript, plus, end superscript, left parenthesis, a, q, right parenthesis, plus, start text, O, H, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis
space, space, start color #aa87ff, start text, b, a, s, e, end text, end color #aa87ff, space, space, space, space, space, space, space, space, space, start text, a, c, i, d, end text, space, space, space, space, space, space, space, space, space, space, start color #aa87ff, start text, a, c, i, d, end text, end color #aa87ff, space, space, space, space, space, space, space, space, space, space, start text, b, a, s, e, end text
For a weak acid or base, the equilibrium constant for the ionization reaction quantifies the relative amounts of each species. In this article, we will discuss the relationship between the equilibrium constants K, start subscript, start text, a, end text, end subscript and K, start subscript, start text, b, end text, end subscript for a conjugate acid-base pair.
Diagram of beaker containing aqueous solution of hydrofluoric acid. Ions of water, neutral HF molecules, and fluoride and hydrogen ions are all shown in solution.
An aqueous solution of hydrofluoric acid, a weak acid, contains undissociated start text, H, F, end text molecules which are in equilibrium with protons and fluoride ions.
Note: For this article, all solutions will be assumed to be aqueous solutions.

Finding K, start subscript, start text, a, end text, end subscript for start text, H, A, end text reacting as an acid

Let's look more closely at the dissociation reaction for a monoprotic weak acid start text, H, A, end text:
start text, H, A, end text, left parenthesis, a, q, right parenthesis, plus, start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, l, right parenthesis, \rightleftharpoons, start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript, left parenthesis, a, q, right parenthesis, plus, start text, A, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis
The products of this reversible reaction are start text, A, end text, start superscript, minus, end superscript, the conjugate base of start text, H, A, end text, and start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript. We can write the following expression for the equilibrium constant K, start subscript, start text, a, end text, end subscript:
K, start subscript, start text, a, end text, end subscript, equals, start fraction, open bracket, start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript, close bracket, open bracket, start text, A, end text, start superscript, minus, end superscript, close bracket, divided by, open bracket, start text, H, A, end text, close bracket, end fraction

Finding K, start subscript, start text, b, end text, end subscript for start text, A, end text, start superscript, minus, end superscript reacting as a base

Since start text, A, end text, start superscript, minus, end superscript is a base, we can also write the reversible reaction for start text, A, end text, start superscript, minus, end superscript acting as a base by accepting a proton from water :
start text, A, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis, plus, start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, l, right parenthesis, \rightleftharpoons, start text, H, A, end text, left parenthesis, a, q, right parenthesis, plus, start text, O, H, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis
The products of this reaction are start text, H, A, end text and start text, O, H, end text, start superscript, minus, end superscript. We can write out the equilibrium constant K, start subscript, start text, b, end text, end subscript for the reaction where start text, A, end text, start superscript, minus, end superscript acts as a base:
K, start subscript, start text, b, end text, end subscript, equals, start fraction, open bracket, start text, H, A, end text, close bracket, open bracket, start text, O, H, end text, start superscript, minus, end superscript, close bracket, divided by, open bracket, start text, A, end text, start superscript, minus, end superscript, close bracket, end fraction
Even though this almost looks like the reverse of start text, H, A, end text acting as an acid, they are actually very different reactions. When start text, H, A, end text acts as an acid, one of the products is start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript. When the conjugate base start text, A, end text, start superscript, minus, end superscript acts as a base, one of the products is start text, O, H, end text, start superscript, minus, end superscript.

Relationship between K, start subscript, start text, a, end text, end subscript and K, start subscript, start text, b, end text, end subscript for conjugate acid-base pair

If we multiply K, start subscript, start text, a, end text, end subscript for start text, H, A, end text with the K, start subscript, start text, b, end text, end subscript of its conjugate base start text, A, end text, start superscript, minus, end superscript, that gives:
KaKb=([H3O+][A][HA])([HA][OH][A])=[H3O+][OH]=Kw\begin{aligned}K_\text{a}\cdot K_\text{b}&=\Bigg(\dfrac{[\text{H}_3\text{O}^+]\cancel{[\text{A}^-]}}{\cancel{[\text{HA}]}}\Bigg)\Bigg(\dfrac{\cancel{[\text{HA}]}[\text{OH}^-]}{\cancel{[\text{A}^-]}}\Bigg)\\ \\ &=[\text{H}_3\text{O}^+][\text{OH}^-]\\ \\ &=K_\text{w}\end{aligned}
where K, start subscript, start text, w, end text, end subscript is the water dissociation constant. This relationship is very useful for relating K, start subscript, start text, a, end text, end subscript and K, start subscript, start text, b, end text, end subscript for a conjugate acid-base pair!! We can also use the value of K, start subscript, start text, w, end text, end subscript at 25, degrees, start text, C, end text to derive other handy equations:
KaKb=Kw=1.0×1014 at 25C(Eq. 1)\begin{aligned}K_\text{a}\cdot K_\text{b}&=K_\text{w}\\ \\ &=1.0\times 10^{-14}~\text{at }25\,^\circ \text C\quad\quad\quad\quad(\text{Eq. 1})\end{aligned}
If we take the negative log, start base, 10, end base of both sides of the Eq. 1, we get:
start text, p, end text, K, start subscript, start text, a, end text, end subscript, plus, start text, p, end text, K, start subscript, start text, b, end text, end subscript, equals, 14, space, space, start text, a, t, space, end text, 25, degrees, start text, C, end text, left parenthesis, start text, E, q, point, space, 2, end text, right parenthesis
We can use these equations to determine K, start subscript, start text, b, end text, end subscript (or start text, p, end text, K, start subscript, start text, b, end text, end subscript) of a weak base given K, start subscript, start text, a, end text, end subscript of the conjugate acid. We can also calculate the K, start subscript, start text, a, end text, end subscript (or start text, p, end text, K, start subscript, start text, a, end text, end subscript) of a weak acid given K, start subscript, start text, b, end text, end subscript of the conjugate base.
An important thing to remember is that these equations only work for conjugate acid-base pairs!! For a quick review on how to identify conjugate acid-base pairs, check out the video on conjugate acid-base pairs.
Concept check: Which of the following values can we calculate if we know the K, start subscript, start text, b, end text, end subscript of start text, N, H, end text, start subscript, 3, end subscript at 25, degrees, start text, C, end text?
Choose 1 answer:

Example: Finding K, start subscript, start text, b, end text, end subscript of a weak base

The start text, p, end text, K, start subscript, start text, a, end text, end subscript of hydrofluoric acid left parenthesis, start text, H, F, end text, right parenthesis is 3, point, 36 at 25, degrees, start text, C, end text.
What is K, start subscript, start text, b, end text, end subscript for fluoride, start text, F, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis?
Let's work through this problem step-by-step.

Step 1: Make sure we have a conjugate acid-base pair

We can check the conjugate acid-base pair relationship by writing out the dissociation reaction for start text, H, F, end text:
start text, H, F, end text, left parenthesis, a, q, right parenthesis, plus, start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, l, right parenthesis, \rightleftharpoons, start text, F, end text, start superscript, minus, end superscript, left parenthesis, a, q, right parenthesis, plus, start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript, left parenthesis, a, q, right parenthesis
We can see that start text, H, F, end text donates its proton to water to form start text, H, end text, start subscript, 3, end subscript, start text, O, end text, start superscript, plus, end superscript and start text, F, end text, start superscript, minus, end superscript. Therefore, start text, F, end text, start superscript, minus, end superscript is the conjugate base of start text, H, F, end text. That means we can use the start text, p, end text, K, start subscript, start text, a, end text, end subscript of start text, H, F, end text to find the start text, p, end text, K, start subscript, start text, b, end text, end subscript of start text, F, end text, start superscript, minus, end superscript. Hooray!

Step 2: Use Eq. 2 to find start text, p, end text, K, start subscript, start text, b, end text, end subscript from start text, p, end text, K, start subscript, start text, a, end text, end subscript

Rearranging Eq. 2 to solve for start text, p, end text, K, start subscript, start text, b, end text, end subscript, we have:
start text, p, end text, K, start subscript, start text, b, end text, end subscript, equals, 14, point, 00, minus, start text, p, end text, K, start subscript, start text, a, end text, end subscript
Plugging in our known start text, p, end text, K, start subscript, start text, a, end text, end subscript for start text, H, F, end text, we get:
start text, p, end text, K, start subscript, start text, b, end text, end subscript, equals, 14, point, 00, minus, left parenthesis, 3, point, 36, right parenthesis, equals, 10, point, 64
Therefore, the start text, p, end text, K, start subscript, start text, b, end text, end subscript for start text, F, end text, start superscript, minus, end superscript is 10, point, 64.

Step 3: Calculate K, start subscript, start text, b, end text, end subscript from start text, p, end text, K, start subscript, start text, b, end text, end subscript

Finally, we can convert start text, p, end text, K, start subscript, start text, b, end text, end subscript to K, start subscript, start text, b, end text, end subscript using the following equation:
start text, p, end text, K, start subscript, start text, b, end text, end subscript, equals, minus, log, left parenthesis, K, start subscript, start text, b, end text, end subscript, right parenthesis
Solving this equation for K, start subscript, start text, b, end text, end subscript, we get:
K, start subscript, start text, b, end text, end subscript, equals, 10, start superscript, minus, start text, p, end text, K, start subscript, start text, b, end text, end subscript, end superscript
Substituting our known value of start text, p, end text, K, start subscript, start text, b, end text, end subscript and solving, we get:
K, start subscript, start text, b, end text, end subscript, equals, 10, start superscript, minus, left parenthesis, 10, point, 64, right parenthesis, end superscript, equals, 2, point, 3, times, 10, start superscript, minus, 11, end superscript
Therefore, the K, start subscript, start text, b, end text, end subscript of start text, F, end text, start superscript, minus, end superscript is 2, point, 3, times, 10, start superscript, minus, 11, end superscript.

Summary

For conjugate-acid base pairs, the acid dissociation constant K, start subscript, start text, a, end text, end subscript and base ionization constant K, start subscript, start text, b, end text, end subscript are related by the following equations:
  • K, start subscript, start text, w, end text, end subscript, equals, K, start subscript, start text, a, end text, end subscript, dot, K, start subscript, start text, b, end text, end subscript
  • start text, p, end text, K, start subscript, start text, a, end text, end subscript, plus, start text, p, end text, K, start subscript, start text, b, end text, end subscript, equals, 14, space, space, start text, a, t, space, end text, 25, degrees, start text, C, end text

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