Autoionization and pH

Circling back to our definitions of acids and bases, we know that there are two ions that are at the heart of acid-base chemistry: hydroxide (OH^-) which is produced by bases in solution, and hydronium/hydrogen ions (H₃O⁺/H⁺) which are produced by acids and can be thought of interchangeably for purposes of this section.

When working with acids and bases, it is often useful to know the concentrations of these ions present in solution. Sometimes this is simple: if you have a 6.0 M solution of NaOH, you know that solution contains 6.0 M hydroxide ion. However, sometimes concentrations of hydronium or hydroxide are inconveniently small to write out. So chemists sometimes use the pH scale (and its less-known cousin, the pOH scale) to indicate concentrations of hydronium and hydroxide. We'll learn about this system now.

Autoionization

Even in pure water, with no other substances added, there is a process that continually generates very small concentrations of hydronium and hydroxide ions. Very occasionally, when two water molecules collide, an acid-base reaction takes place. One water molecule acts as an acid, donating H⁺ to the other molecule, which acts as a base. This proton transfer creates one hydronium ion and one hydroxide ion. Two different representations of this process are shown below.

H₂O (l) + H₂O (l) ⇌ H₃O⁺ (aq) + OH^- (aq)

2 H₂O (l) ⇌ H₃O⁺ (aq) + OH^- (aq)

There are a few things you should know about this reaction. First, as the double arrows indicate, it is an equilibrium process. We learned about dynamic equilibrium in Lesson 4; as a refresher, an equilibrium process is a reaction that can proceed in both directions, which eventually arrives at a situation in which any given reactant or product is being both broken down and formed at the same rate, so concentrations eventually level off.

Second, the concentrations of hydronium and hydroxide present at equilibrium due to this reaction are very low. Just like the dissociation equilibria we discussed in Lesson 4 had K_sp values that determined ion concentrations in saturated solutions, this equilibrium reaction has a value we call K_w, which is equal to the products of the hydronium and hydroxide ion concentrations. At normal temperatures, the value of this constant is 1.0*10^-14.

K_w = [H₃O⁺][OH^-] K_w = 1.0*10^-14

In pure water, each hydronium ion is produced along with one hydroxide ion, so the two concentrations must be equal. In order for this to be true, each ion will have an equilibrium concentration of 1.0*10^-7 M. We will come back to this equation and discuss it in greater depth momentarily, but first let's learn about the pH scale.

The pH and pOH Scales

pH

As mentioned above, in pure water the concentrations of hydronium and hydroxide ions are each 1.0*10^-7 M. This is a very small number: one ten-millionth of a mole per liter. A scientist who regularly works with concentrations of these ions like this would have to spend a lot of time typing out numbers in scientific notation. As an alternative, chemists have developed the pH and pOH scales, which provide a more convenient way of expressing very small concentrations of hydronium and hydroxide.

The pH of a solution is defined as the negative logarithm of its hydronium concentration, as shown below. Technically, we should specify that it is the base-10 logarithm, but the convention in math is that when no base is specified it is assumed to be 10. If you don't remember much about logarithms, here is a good refresher from Khan Academy: https://youtu.be/Z5myJ8dg_rM.

pH = -log[H₃O⁺]

The pH of a solution is technically unitless, but the concentration used in calculations should always be in molar. Note that the symbol has a lowercase "p" and an uppercase "H" - this is the correct format for writing "pH."

A few things are worth noting here. First, the pH depends solely on the concentration of H₃O⁺. It is just a different way of expressing that number; it doesn't add any new information. Second, let's look at what happens when we calculate pH for solutions with a few different H₃O⁺ concentrations.

[H₃O⁺] (M): 1.0*10^-7 5.0*10^-7 1.0*10^-4 1.0*10^-11

pH: 7.0 6.3 4.0 11.0

As you can see, the base-ten logarithm means the pH is always close to the exponent in the hydronium concentration. This is what logarithms do with values in scientific notation. So this means a wide range of concentrations have relatively "friendly" pH values.

If pure water always contains the same concentration of hydronium (1.0*10^-7 M), then how do we get pH values other than 7.0? Well, when acids are added to solution they dissociate to form H₃O⁺. So our solutions with pH values of 6.3 and 4.0 must have some kind of acid in them.

There are a couple places you should be careful when dealing with pH values. First, the pH scale tends to make large changes look small. The second concentration above is five times greater than the first, but the pH value only changes by about 10%. Each step of one on the pH scale corresponds to multiplying or dividing by 10. The third concentration above is 1000 times greater than the first, or 10³ times. Thus they differ by three on the pH scale.

Also, note that the scale is "backwards." The fourth value above is the smallest concentration, which gives the largest number on the pH scale. So remember that small pH values mean high hydronium concentrations, and vice versa (this is due to the negative sign in the calculation of pH).

pOH

Most people have at least heard of the pH scale, however few know about its counterpart, the pOH scale. As the name suggests, it does the same thing pH does, just with the hydroxide ion. So the pOH of a solution is the negative logarithm of the hydroxide concentration.

pOH = -log[OH^-]

We will now discuss the various calculations you can do involving pH and pOH values and the corresponding hydronium and hydroxide concentrations.

Calculations with pH and pOH

We have already learned how to calculate pH and pOH from concentrations of H₃O⁺ and OH^-, respectively. You take the negative logarithm of the concentration. Two short examples for your practice are shown below. Given the relevant concentration, you should be able to find a pH or pOH value.

[H₃O⁺] = 6.90*10^-4 M [OH^-] = 1.45*10^-11 M

pH = -log(6.90*10^-4) pOH = -log(1.45*10^-11)

pH = 3.16 pOH = 10.84

We will also want to be able to do the reverse of these calculations, meaning to calculate a concentration from a pH or pOH. This process requires two steps. First, take the negative of your pH (or pOH) to reverse the negative sign. Second, we need to reverse the base-ten logarithm; this is done by raising ten to the power of the value you have. So we use the formulas below to calculate concentration from pH/pOH.

[H₃O⁺] = 10^-pH [OH^-] = 10^-pOH

Using these equations you should be able to calculate a concentration given a pH or pOH. You can practice your calculator skills on the examples below, as well as several in your lab workbook. Verify that you get the correct answers using your calculator. If you do not know how to raise ten to a power with your calculator, ask the lab instructor.

pH = 12.8 pOH = 1.2

[H₃O⁺] = 10^-12.8 [OH^-] = 10^-1.2

[H₃O⁺] = 1.58*10^-13 [OH^-] = 6.31*10^-2

Now let us return to the equation for water's autoionization.

2 H₂O (l) ⇌ H₃O⁺ (aq) + OH^- (aq)

K_w = [H₃O⁺][OH^-] K_w = 1.0*10^-14

Previously we used this equation to show that in pure water the concentrations of both H₃O⁺ and OH^- are equal to 1.0*10^-7 M (pH = pOH = 7.0). We call solutions like this neutral because they have equal amounts of these two ions (more on that later). But even when hydronium (from an acid) or hydroxide (from a base) are added to water, this equation still holds.

Let's say we add some acid to water until has an H₃O⁺ concentration of 4.00*10^-4 M. If we plug this value into the equation above, we can calculate what the hydroxide concentration must be.

K_w = [H₃O⁺][OH^-] K_w = 1.0*10^-14

1.0*10^-14 = 4.00*10^-4 * [OH^-]

[OH^-]= 1.0*10^-14/4.00*10^-4

[OH^-]= 2.5*10^-11

Because the product of the H₃O⁺ and OH^- concentrations is a fixed value, if one rises the other must always fall. This can be visualized in the graphic below in terms of the pH and pOH of solutions. For any given solution, the higher its pH the lower its pOH and vice versa.

The equation for K_w can be used as above to calculate hydronium concentration from hydroxide concentration or vice versa. In addition, if you look at the image above you may notice a pattern in pH and pOH values that we can also use in calculations. When the pH and pOH of a solution are added together, the sum is always equal to 14. This fact is related to the value of K_w and its derivation is shown below for mathematical interest, which uses the fact that the logarithm of a product is equal to the sum of the individual logarithms. Note: you will not be tested on this, and it is shown just for general interest.

K_w = [H₃O⁺][OH^-] K_w = 1.0*10^-14

1.0*10^-14 = [H₃O⁺][OH^-]

-log(1.0*10^-14) = -log([H₃O⁺][OH^-])

14 = -log [H₃O⁺] + (-log[OH^-])

14 = pH + pOH

As an example, let's find the pH of a solution with a hydroxide concentration of 0.0003724 M. We have to read the question carefully; the hydroxide concentration allows us to calculate the pOH, not the pH directly. Once we have the pOH we can use it and our new relationship to find the pH.

pOH = -log(0.0003724)

pOH = 3.43

pH = 14.0 - 3.43

pH = 10.57

Practice Problems

To practice calculations with hydronium and hydroxide concentrations, pH, and pOH, fill in the table below.

pH pOH [H₃O⁺] [OH^-]

3.2

8.2

6.4

0.00500

0.00750

Answers to Practice Problems

pH pOH [H₃O⁺] [OH^-]

3.2 10.8 6.31*10^-4 1 .58*10^-11

8.2 5.8 6.31*10^-9 1 .58*10^-6

7.6 6.4 2.51*10^-8 3 .98*10^-7

2.3 11.7 0.00500 2.00*10^-12

11.88 2.12 1.3*10^-12 0.00750