# 16.3: Equilibrium Constants for Acids and Bases

## Introduction

In general chemistry 1 we calculated the pH of strong acids and bases by considering them to completely dissociate, that is, undergo 100% ionization. We will now look at weak acids and bases, which do not completely dissociate, and use equilibrium constants to calculate equilibrium concentrations. We could solve all these problems using the techniques from the last chapter on equilbria, but instead we are going to develop short cut techniques, and identify when they are valid. But first, we need to define what are equilibrium constants for acid base reactions.

## Weak Acids

$HA(aq)+H_2O(l)⇌H_3O^+(aq)+A^-(aq)$

At first glance this gives an equilibrium constant of

$K=\frac{[H_{3}O^{+}][A^{-}]}{[HA][H_{2}O]}$

But we can consider the water concentration constant because it is much greater than of acid that has ionized. There are two factors at work here, first that the water is the solvent and so [H2O] is larger than [HA], and second, that [HA] is a weak acid, and so at equilibrium the amount ionized is smaller than [HA]. It should be noted that this is a homogenous equlibria, and although we are ignoring the water and treating it as a liquid, it is for a different reason than was used in the last chapter for heterogeneous equilibria.

This results in Acid Dissociation Constant (Ka) for aqueous systems:

$K_{a}=\frac{[H_{3}O^{+}][A^{-}]}{[HA]}$

where, $$K_a=K[H_2O]$$

Ka is only used for weak acids. Strong acids have a large Ka and completely dissociate and so you just state the reaction goes to completion.

## Weak Bases

There are two types of weak bases, those as modeled by ammonia and amines, which grab a proton from water, and the conjugate bases of weak acids, which are ions, and grab the proton to form the weak acid.

### Type 1:

$B(aq) + H_2O(l) ⇌ HB^+(aq) + OH^-(aq)$

Note, in this reaction the base removes a proton from the water and following the same logic for weak acids, we consider the water concentration to stay constant because only a small fraction of it reacts with the weak base, so:

$K_b=\frac{[HB^+][OH^-]}{[B]}$

An example of the first type would be that of methyl amine, CH3NH2.

$CH_3NH_2(aq) + H_2O(l) ⇌CH_3NH_3^+(aq)+OH^- (aq) \\ \\ K=\frac{[CH_3NH_3^+][OH^-]}{[CH_3NH_2]} = 5.0x10^{-4}$

### Type 2:

$A^-(aq) + H_2O(l) ⇌ HA(aq) + OH^-(aq)$

$K'_b=\frac{[HA][OH^-]}{[A^-]} \\ \text{ where} \; K_b \; \text{is the basic equilibrium constant of the conjugate base} \; A^- \; \text{of the weak acid HA}$

## Acid Base Conjugate Pairs

We will use K(a or b) to represent the acid or base equilibrium constant and K'(b or a) to represent the equilibrium constant of the conjugate pair.

For an Acid Base Conjugate Pair

$\large K_aK_{b'}=K_w$

Consider the generic acid HA which has the reaction and equilibrium constant of

$HA(aq)+H_2O(l)⇌H_3O^+(aq)+A^-(aq), \; K_{a}=\frac{[H_{3}O^{+}][A^{-}]}{[HA]}$

its conjugate base A- has the reaction and equilibrium constant of:

$A^-(aq) + H_2O(l) ⇌ HA(aq) + OH^-(aq), K'_b=\frac{[HA][OH^-]}{[A^-]}$

$K_aK'_{b}=\left ( \frac{[H_{3}O^{+}] \textcolor{red}{\cancel{[A^{-}]}}}{ \textcolor{blue}{\cancel{[HA]}}}\right )\left (\frac{ \textcolor{blue}{\cancel{[HA]}}[OH^-]}{ \textcolor{red}{\cancel{[A^-]}}} \right )=[H_{3}O^{+}][OH^-]=K_w=10^{-14}$

So there is an inverse relationship across the conjugate pair

• The Stronger an Acid the Weaker it's Conjugate Base
• The Weaker an Acid the Stronger it's Conjugate Base
• The Stronger the Base the Weaker it's Conjugate Acid
• The Weaker the Base the Stronger it's Conjugate Acid

Figure$$\PageIndex{1}$$: Relationship between acid or base strength and that of their conjugate base or acid.

## Ka and Kb Values for PolyProtic Acids

PolyProtic Acids

It is always harder to remove a second proton from an acid because you are removing it from a negative charged species, and even harder to remove the third, as you are removing it from a dianion.

Consider a triprotic acid

$H_3A + H_2O ⇌H_2A^- +H_3O^+ \; \; K_{a1}$
$H_2A^- + H_2O ⇌HA^{-2} +H_3O^+ \; \; K_{a2}$
$HA^{2}- + H_2O ⇌A^{-3} +H_3O^+ \; \; K_{a3}$

$K_{a1}>K_{a2}>K_{a3}$

## pKa and pKb

Because pKa and pKb values are so small they are often recorded a pX values, where pX= -logX. So

pKa = -logKa and Ka =10-pka
pKb = -logKb​​​​​​​ and Kb​​​​​​​ =10-pkb​​​​​​​

## Table of Ka

Table $$\PageIndex{1}$$: Table of Acid Ionization Constants

Name Formula K a1 pKa1 K a2 pKa2 K a3 pKa3 K a4 pKa4
Acetic acid CH3CO2H 1.75 × 10−5 4.756
Arsenic acid H3AsO4 5.5 × 10−3 2.26 1.7 × 10−7 6.76 5.1 × 10−12 11.29
Benzoic acid C6H5CO2H 6.25 × 10−5 4.204
Boric acid H3BO3 5.4 × 10−10* 9.27* >1 × 10−14* >14*
Bromoacetic acid CH2BrCO2H 1.3 × 10−3 2.90
Carbonic acid H2CO3 4.5 × 10−7 6.35 4.7 × 10−11 10.33
Chloroacetic acid CH2ClCO2H 1.3 × 10−3 2.87
Chlorous acid HClO2 1.1 × 10−2 1.94
Chromic acid H2CrO4 1.8 × 10−1 0.74 3.2 × 10−7 6.49
Citric acid C6H8O7 7.4 × 10−4 3.13 1.7 × 10−5 4.76 4.0 × 10−7 6.40
Cyanic acid HCNO 3.5 × 10−4 3.46
Dichloroacetic acid CHCl2CO2H 4.5 × 10−2 1.35
Fluoroacetic acid CH2FCO2H 2.6 × 10−3 2.59
Formic acid CH2O2 1.8 × 10−4 3.75
Hydrazoic acid HN3 2.5 × 10−5 4.6
Hydrocyanic acid HCN 6.2 × 10−10 9.21
Hydrofluoric acid HF 6.3 × 10−4 3.20
Hydrogen selenide H2Se 1.3 × 10−4 3.89 1.0× 10−11 11.0
Hydrogen sulfide H2S 8.9 × 10−8 7.05 1 × 10−19 19
Hydrogen telluride H2Te 2.5 × 10−3‡ 2.6 1 × 10−11 11
Hypobromous acid HBrO 2.8 × 10−9 8.55
Hypochlorous acid HClO 4.0 × 10−8 7.40
Hypoiodous acid HIO 3.2 × 10−11 10.5
Iodic acid HIO3 1.7 × 10−1 0.78
Iodoacetic acid CH2ICO2H 6.6 × 10−4 3.18
Nitrous acid HNO2 5.6 × 10−4 3.25
Oxalic acid C2H2O4 5.6 × 10−2 1.25 1.5 × 10−4 3.81
Periodic acid HIO4 2.3 × 10−2 1.64
Phenol C6H5OH 1.0 × 10−10 9.99
Phosphoric acid H3PO4 6.9 × 10−3 2.16 6.2 × 10−8 7.21 4.8 × 10−13 12.32
Phosphorous acid H3PO3 5.0 × 10−2* 1.3* 2.0 × 10−7* 6.70*
Pyrophosphoric acid H4P2O7 1.2 × 10−1 0.91 7.9 × 10−3 2.10 2.0 × 10−7 6.70 4.8 × 10−10 9.32
Resorcinol C6H4(OH)2 4.8 × 10−10 9.32 7.9 × 10−12 11.1
Selenic acid H2SeO4 Strong Strong 2.0 × 10−2 1.7
Selenious acid H2SeO3 2.4 × 10−3 2.62 4.8 × 10−9 8.32
Sulfuric acid H2SO4 Strong Strong 1.0 × 10−2 1.99
Sulfurous acid H2SO3 1.4 × 10−2 1.85 6.3 × 10−8 7.2
meso-Tartaric acid C4H6O6 6.8 × 10−4 3.17 1.2 × 10−5 4.91
Telluric acid H2TeO4 2.1 × 10−8‡ 7.68 1.0 × 10−11‡ 11.0
Tellurous acid H2TeO3 5.4 × 10−7 6.27 3.7 × 10−9 8.43
Trichloroacetic acid CCl3CO2H 2.2 × 10−1 0.66
Trifluoroacetic acid CF3CO2H 3.0 × 10−1 0.52
* Measured at 20°C, not 25°C.
‡ Measured at 18°C, not 25°C.

## Table of Kb

Table$$\PageIndex{2}$$: Base Ionization Constants

Name Formula $$K_b$$ $$pK_b$$
Ammonia NH3 1.8 × 10−5 4.75
Aniline C6H5NH2 7.4 × 10−10 9.13
n-Butylamine C4H9NH2 4.0 × 10−4 3.40
sec-Butylamine (CH3)2CHCH2NH2 3.6 × 10−4 3.44
tert-Butylamine (CH3)3CNH2 4.8 × 10−4 3.32
Dimethylamine (CH3)2NH 5.4 × 10−4 3.27
Ethylamine C2H5NH2 4.5 × 10−4 3.35
Hydrazine N2H4 1.3 × 10−6 5.9
Hydroxylamine NH2OH 8.7 × 10−9 8.06
Methylamine CH3NH2 4.6 × 10−4 3.34
Propylamine C3H7NH2 3.5 × 10−4 3.46
Pyridine C5H5N 1.7 × 10−9 8.77
Trimethylamine (CH3)3N 6.3 × 10−5 4.20

Robert E. Belford (University of Arkansas Little Rock; Department of Chemistry). The breadth, depth and veracity of this work is the responsibility of Robert E. Belford, rebelford@ualr.edu. You should contact him if you have any concerns. This material has both original contributions, and content built upon prior contributions of the LibreTexts Community and other resources, including but not limited to: