## 10.4 Autoionization of Water

### Learning Objectives

By the end of this module, you will be able to:

• Describe the autoionization of water.
• Calculate the concentrations of H+ and OH in solutions, knowing the other concentration.

We have already seen that H2O can act as an acid or a base:

Water acting as a acid,

$\large{\text{NH}}_{3}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\longrightarrow {\text{NH}}_{4}^{\text{+}}\left(aq\right)+{\text{OH}}^{-}\left(aq\right)$

Water acting as a base,

$\large{\text{HCl}}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\longrightarrow {\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)+{\text{Cl}}^{-}\left(aq\right)$

It may not surprise you to learn, then, that within any given sample of water, some H2O molecules are acting as acids, and other H2O molecules are acting as bases. The chemical equation is as follows:

$\large{\text{H}}_{2}\text{O}\left(l\right)+{\text{H}}_{2}\text{O}\left(l\right)\longrightarrow {\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)+{\text{OH}}^{-}\left(aq\right)$

This occurs only to a very small degree: only about 6 in 108 H2O molecules are participating in this process, which is called the autoionizaton of water.  At this level, the concentration of both H+(aq) and OH(aq) in a sample of pure H2O is about 1.0 × 10−7 M. If we use square brackets—[ ]—around a dissolved species to imply the molar concentration of that species, we have

$\large{[\text{H}}_{3}{\text{O}}^{\text{+}}]={[\text{OH}}^{-}]=1.0\times {10}^{-7}M$

for any sample of pure water because H2O can act as both an acid and a base. The product of these two concentrations is 1.0 × 10−14:

$\large{[\text{H}}_{3}{\text{O}}^{\text{+}}]\times{{[\text{OH}}^{-}]}=(2.0\times {10}^{-6})\times(5.0\times {10}^{-9})=1.0\times {10}^{-14}$

In acids, the concentration of H+(aq)—[H+]—is greater than 1.0 × 10−7 M, while for bases the concentration of OH(aq)—[OH]—is greater than 1.0 × 10−7 M. However, the product of the two concentrations—[H+][OH]—is always equal to 1.0 × 10−14, no matter whether the aqueous solution is an acid, a base, or neutral.

This value of the product of concentrations is so important for aqueous solutions that it is called the autoionization constant of water and is denoted Kw:

$\large \text{K}_w={[\text{H}}_{3}{\text{O}}^{\text{+}}]{{[\text{OH}}^{-}]}=1.0\times {10}^{-14}$

This means that if you know [H+] for a solution, you can calculate what [OH] has to be for the product to equal 1.0 × 10−14, or if you know [OH], you can calculate [H+]. This also implies that as one concentration goes up, the other must go down to compensate so that their product always equals the value of Kw.

### Example 1: Ion Concentrations in Pure Water

What are the hydronium ion concentration and the hydroxide ion concentration in pure water at 25 °C?

Example 2 demonstrates the quantitative aspects of this relation between hydronium and hydroxide ion concentrations.

### Example 2: The Inverse Proportionality of [H3O+] and [OH−]

A solution of an acid in water has a hydronium ion concentration of 2.0 $\times$ 10−6M. What is the concentration of hydroxide ion at 25 °C?

What is the hydronium ion concentration in an aqueous solution with a hydroxide ion concentration of 0.001 M at 25 °C?

### Key concepts and summary

In any aqueous solution, the product of [H+] and [OH] equals 1.0 × 10−14.

#### Key Equations

• ${K}_{\text{w}}=\left[{\text{H}}_{2}{\text{O}}^{\text{+}}\right]\left[{\text{OH}}^{-}\right]=1.0\times 1{0}^{-14}\text{ at }25^{\circ}\text{C}$

### Exercises

1. Does [H+] remain constant in all aqueous solutions? Why or why not?

2. Does [OH] remain constant in all aqueous solutions? Why or why not?

3. What is the relationship between [H+] and Kw? Write a mathematical expression that relates them.

4. What is the relationship between [OH] and Kw? Write a mathematical expression that relates them.

5. Write the chemical equation for the autoionization of water and label the conjugate acid-base pairs.

6. Write the reverse of the reaction for the autoionization of water. It is still an acid-base reaction? If so, label the acid and base.

7. For a given aqueous solution, if [H+] = 1.0 × 10−3 M, what is [OH]?

8. For a given aqueous solution, if [H+] = 1.0 × 10−9 M, what is [OH]?

9. For a given aqueous solution, if [H+] = 7.92 × 10−5 M, what is [OH]?

10. For a given aqueous solution, if [H+] = 2.07 × 10−11 M, what is [H+]?

11. For a given aqueous solution, if [OH] = 1.0 × 10−5 M, what is [H+]?

12. For a given aqueous solution, if [OH] = 1.0 × 10−12 M, what is [H+]?

13. For a given aqueous solution, if [OH] = 3.77 × 10−4 M, what is [H+]?

14. For a given aqueous solution, if [OH] = 7.11 × 10−10 M, what is [H+]?

15. What are [H+] and [OH] in a 0.344 M solution of HNO3?

16. What are [H+] and [OH] in a 2.86 M solution of HBr?

17. What are [H+] and [OH] in a 0.00338 M solution of KOH?

18. What are [H+] and [OH] in a 6.02 × 10−4 M solution of Ca(OH)2?