11.5: The pH Scale

Learning Objectives

• Define the pH scale and use it to describe acids and bases.
• Calculate pH, pOH, [H₃O⁺], and [OH^-]

The Ion-Product of Water

As we have already seen, H₂O can act as an acid or a base. Within any given sample of water, some \(\ce{H2O}\) molecules are acting as acids, and other \(\ce{H2O}\) molecules are acting as bases. The chemical equation is as follows:

\[\color{red}{\underbrace{\ce{H2O}}_{\text{acid}}} + \color{blue}{\underbrace{\ce{H2O}}_{\text{base}}} \color{black} \ce{\rightleftharpoons H3O^{+} + OH^{−}} \label{Auto}\]

Figure \(\PageIndex{2}\): Autoionization of water, giving hydroxide and hydronium ions.

We often use the simplified form of the reaction:

\[\ce{H2O(l) \rightleftharpoons H^+(aq) + OH^{−}(aq)}\]

No matter what the pH, there is a relationship between the concentrations of hydronium ions and hydroxide ions. When those concentrations are multiplied together they give a constant called the ion-product of water \(K_w\).  

\[K_w = [\ce{H_{3}O^{+}}][\ce{OH^-}]\]

The variables with square brackets around chemical formulas, \([H_{3}O^+]\) and \([OH^-]\), indicate the concentration of each substance in a unit of molarity. The value of \(K_w\) is very small because the concentrations of hydronium and hydroxide ions are small. At 25^oC, the experimentally determined value of \(K_w\) in pure water is 1.0×10⁻¹⁴. The unit for \(K_w\) is customarily omitted.

\[K_w = [\ce{H_{3}O^{+}}][\ce{OH^{−}}] = 1.0 \times 10^{−14}\]

Although the concentrations are small, they can be detected and measured experimentally. In a sample of pure water, the concentrations of hydronium and hydroxide ions are equal to one another. Pure water or any other aqueous solution in which this ratio holds is said to be neutral. 

[H₃O⁺] = [OH⁻] = 1.0×10⁻⁷ at pH 7

The product of these two concentrations is 1.0×10⁻¹⁴

\[\color{red}{\ce{[H^{+}]}} \color{black}{\times} \color{blue}{\ce{[OH^{-}]}} \color{black} = (1.0 \times 10^{-7})( 1.0 \times 10^{-7}) = 1.0 \times 10^{-14}\]

• For acids, the concentration of H⁺ or [H₃O⁺]) is greater than 1.0×10⁻⁷ M
• For bases, the concentration of OH⁻ or [OH⁻] is greater than 1.0×10⁻⁷ M.

Aqueous HCl is an example of acidic solution. Hydrogen chloride (HCl) ionizes to produce H⁺ and Cl⁻ ions upon dissolving in water.

\[\ce{HCl(g) -> H^{+}(aq) + Cl^{−}(aq)} \nonumber\]

This increases the concentration of H₃O⁺ ions in the solution. According to Le Chatelier's principle, the equilibrium represented by

\[\ce{H2O(l) \leftrightharpoons H_{3}O^{+}(aq) + OH^{−}(aq)} \nonumber\]

is forced to the left, towards the reactant. As a result, the concentration of the hydroxide ion decreases.

Now, consider KOH (aq), a basic solution. Solid potassium hydroxide (KOH) dissociates in water to yield potassium ions and hydroxide ions.

KOH(s) → K⁺(aq) + OH⁻(aq)

The increase in concentration of the OH⁻ ions will cause a decrease in the concentration of the H⁺ ions because they react to form water.

No matter whether the aqueous solution is an acid, a base, or neutral:and the ion-product of [H⁺][OH⁻] remains constant.

• For acidic solutions, [H₃O⁺] is greater than [OH⁻].
• For basic solutions, [OH⁻] is greater than [H₃O⁺].
• For neutral solutions, [H₃O⁺] = [OH⁻] = 1.0×10⁻⁷M

This means that if you know \(\ce{[H3O^{+}]}\) for a solution, you can calculate what \(\ce{[OH^{−}]}\) has to be for the product to equal \(1.0 \times 10^{−14}\), or if you know \(\ce{[OH^{−}]}\)), you can calculate \(\ce{[H3O^{+}]}\). 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 \(K_w\).

\[K_w = \color{red}{\ce{[H_3O^+]}} \color{blue}{\ce{[OH^{-}]}} \color{black} = 1.0 \times 10^{-14} \label{eq10}\]

Example \(\PageIndex{2}\)

Hydrochloric acid (HCl) is a strong acid, meaning it is 100% ionized in solution. What is the [H⁺] and the [OH⁻] in a solution of 2.0×10⁻³ M HCl?

Solution

Step 1: List the known values and plan the problem.

Known

• [HCl] = 2.0×10⁻³ M
• Kw = 1.0×10⁻¹⁴

Unknown

• [H⁺]=?M
• [OH⁻]=?M

Because HCl is 100% ionized, the concentration of H+ ions in solution will be equal to the original concentration of HCl. Each HCl molecule that was originally present ionizes into one H+ ion and one Cl− ion. The concentration of OH− can then be determined from the [H+] and Kw.

Step 2: Solve.

[H⁺]=2.0×10⁻³ M

Kw = [H⁺][OH⁻] = 1.0×10⁻¹⁴

\[ [OH^−] = \dfrac{K_w}{[H^+]} = \dfrac{1.0×10^{−14}}{2.0×10^{−3}} = 5.0×10^{−12} M \]

Step 3: Think about your result.

The [H₃O⁺] is much higher than the [OH⁻] because the solution is acidic.

Exercise \(\PageIndex{2}\)

Sodium hydroxide (NaOH) is a strong base. What is the [H⁺] and the [OH⁻] in a 0.001 M NaOH solution at 25 °C?

Answer

[OH⁻] = 0.001M or 1 x 10^-3M; [H⁺]=1×10⁻¹¹M.

The pH Scale

One measure of the strength of an acid or a base solution is the pH scale, which is based on the concentration of the hydronium (or hydrogen) ion in aqueous solution.

\[pH = -\log[H^+]\]

or

\[pH = -\log[H_3O^+]\]

Figure \(\PageIndex{3}\) illustrates this relationship, along with some examples of various solutions. Because hydrogen ion concentrations are generally less than one (for example \(1.3 \times 10^{-3}\,M\)), the log of the number will be a negative number. To make pH even easier to work with, pH is defined as the negative log of \([H_3O^+]\), which will give a positive value for pH.

Figure \(\PageIndex{3}\): The relationship between [H⁺] and \(pH\) values for several common materials.

A neutral (neither acidic nor basic) solution has a pH of 7. A pH below 7 means that a solution is acidic, with lower values of pH corresponding to increasingly acidic solutions. A pH greater than 7 indicates a basic solution, with higher values of pH corresponding to increasingly basic solutions. Thus, given the pH of several solutions, you can state which ones are acidic, which ones are basic, and which are more acidic or basic than others. These are summarized in Table \(\PageIndex{4}\).

Table \(\PageIndex{4}\): Acidic, Basic and Neutral pH Values

Classification	Relative Ion Concentrations	pH at 25 °C
acidic	[H^+] > [OH^−]	pH < 7
neutral	[H^+] = [OH^−]	pH = 7
basic	[H^+] < [OH^−]	pH > 7

Example \(\PageIndex{3}\)

Find the pH, given the \([H^+]\) of the following:

1. 1 ×10^-3 M
2. 2.5 ×10^-11 M
3. 4.7 ×10^-9 M

Solution

pH = - log [H₃O⁺]

Substitute the known quantity into the equation and solve. Use a scientific calculator for b and c.

1. pH = - log [1 × 10⁻³ ] = 3.0 (1 decimal place since 1 has 1 significant figure)
2. pH = - log [2.5 ×10^-11] = 10.60 (2 decimal places since 2.5 has 2 significant figures)
3. pH = - log [4.7 ×10^-9] = 8.33 (2 decimal places since 4.7 has 2 significant figures)

Note on significant figures:

Because the number(s) before the decimal point in the pH value relate to the power on 10, the number of digits after the decimal point (underlined) is what determines the number of significant figures in the final answer.

Exercise \(\PageIndex{3}\)

Find the pH, given [H⁺] of the following:

1. 5.8 ×10^-4 M
2. 1.0×10^-7 M

Answer

a. 3.24

b. 7.00

Table \(\PageIndex{5}\) lists the pH of several common solutions. The most acidic among the listed solutions is battery acid with the lowest pH value (0.3). The most basic is 1M NaOH solution with the highest pH value of 14.0. Notice that some biological fluids (stomach acid and urine) are not neutral. You may also notice that many food products are slightly acidic because they contain solutions of weak acids. pH depends on both the identity of the substance and its concentration, so concentrations are listed for pure substances.  

Table \(\PageIndex{5}\): The pH Values of Some Common Solutions

Solution	pH
battery acid	0.3
stomach acid	1–2
lemon or lime juice	2.1
vinegar	2.8–3.0
Coca-Cola	3
wine	2.8–3.8
beer	4–5
coffee	5
milk	6
urine	6
pure H2O	7
(human) blood	7.3–7.5
sea water	8
antacid (milk of magnesia)	10.5
NH3 (1 M)	11.6
bleach	12.6
NaOH (1 M)	14.0

Example \(\PageIndex{4}\)

Label each solution as acidic, basic, or neutral based only on the stated \(pH\).

1. milk of magnesia, pH = 10.5
2. pure water, pH = 7
3. wine, pH = 3.0

Solution

1. With a pH greater than 7, milk of magnesia is basic. (Milk of magnesia is largely Mg(OH)₂.)
2. Pure water, with a pH of 7, is neutral.
3. With a pH of less than 7, wine is acidic.

Exercise \(\PageIndex{4}\)

Identify each substance as acidic, basic, or neutral based only on the stated \(pH\).

1. human blood with \(pH\) = 7.4
2. household ammonia with \(pH\) = 11.0
3. cherries with \(pH\) = 3.6

Answer

a. slightly basic

b. basic

c. acidic

Calculating [H₃O⁺] and [OH^-]

pH can be determined experimentally using chemical indicator or a pH meter. The pH can then be used to calculate \([H_{3}O^+]\) by rearranging the pH equation. 

\[pH = -log(H_{3}O^+) \]

rearranges to

\[[H_{3}O^+] = 10^{-pH}\]

in order to get \([H_{3}O^+]\) by itself. If the pH of human blood is 7.4, then the concentration of \([H_{3}O^+]\) is

\[[H_{3}O^+] =  10^{-pH} = 10^{-7.4} = 4 x 10^{-8} M \]

A concentration of 4 x 10^-8 M makes sense for a pH of 7.4 because it is close to 1 x 10^-7 M which we would expect for a pH of 7. 

If the pH is known and the concentration of hydroxide ions is desired, then a two step calculation is needed. You could use \[[H_{3}O^+] = 10^{-pH}\] to use the pH to calculate \([H_{3}O^+]\) followed by \[K_w = [\ce{H_{3}O^{+}}][\ce{OH^-}]\] in which the \([H_{3}O^+]\) is used to calculate \([OH^-]).

Another option is to calculate pOH which is the -log[OH-]. Just as pH is a more convenient way to express \([H_{3}O^+]\), the pOH is a more convenient way to express \([OH^-]). While there are not usually instruments available that measure pOH directly, there is a simple relationship between pH and pOH.

\[14 = pH + pOH\]

For example, the antacid milk of magnesia has a pH of 10.5. We can use that to calculate pOH

\[14 = pH + pOH\]

\[pOH = 14 - pH = 14 - 10.5 = 3.5\]

Then the concentration of \([OH^-]\) can be calculated using an equation analogous to \([H_{3}O^+] = 10^{-pH}\)

\[[OH^-] = 10^{-pOH} = 10^{-3.5} = 3 x 1-^{-4} M\]

A concentration of hydroxide ions greater than \(1 x 10^{-7}\) M makes sense since milk of magnesia is basic. (Remember that -4 is greater than -7).

Acid Rain

Normal rainwater has a pH between 5 and 6 due to the presence of dissolved CO₂ which forms carbonic acid:

\[\ce{H2O (l) + CO2(g) ⟶ H2CO3(aq)} \label{14}\]

\[\ce{H2CO3(aq) \rightleftharpoons H^+(aq) + HCO3^- (aq)} \label{15}\]

Acid rain is rainwater that has a pH of less than 5, due to a variety of nonmetal oxides, including CO₂, SO₂, SO₃, NO, and NO₂ being dissolved in the water and reacting with it to form not only carbonic acid, but sulfuric acid and nitric acid. The formation and subsequent ionization of sulfuric acid are shown here:

\[\ce{H2O (l) + SO3(g) ⟶ H2SO4(aq)} \label{16}\]

\[\ce{H2SO4(aq) ⟶ H^+(aq) + HSO4^- (aq)} \label{17}\]

Carbon dioxide is naturally present in the atmosphere because we and most other organisms produce it as a waste product of metabolism. Carbon dioxide is also formed when fires release carbon stored in vegetation or when we burn wood or fossil fuels. Sulfur trioxide in the atmosphere is naturally produced by volcanic activity, but it also stems from burning fossil fuels, which have traces of sulfur, and from the process of “roasting” ores of metal sulfides in metal-refining processes. Oxides of nitrogen are formed in internal combustion engines where the high temperatures make it possible for the nitrogen and oxygen in air to chemically combine.

Acid rain is a particular problem in industrial areas where the products of combustion and smelting are released into the air without being stripped of sulfur and nitrogen oxides. In North America and Europe until the 1980s, it was responsible for the destruction of forests and freshwater lakes, when the acidity of the rain actually killed trees, damaged soil, and made lakes uninhabitable for all but the most acid-tolerant species. Acid rain also corrodes statuary and building facades that are made of marble and limestone (Figure \(\PageIndex{4}\)). Regulations limiting the amount of sulfur and nitrogen oxides that can be released into the atmosphere by industry and automobiles have reduced the severity of acid damage to both natural and manmade environments in North America and Europe. It is now a growing problem in industrial areas of China and India.

Figure \(\PageIndex{4}\): (a) Acid rain makes trees more susceptible to drought and insect infestation, and depletes nutrients in the soil. (b) It also is corrodes statues that are carved from marble or limestone. (credit a: modification of work by Chris M Morris; credit b: modification of work by “Eden, Janine and Jim”/Flickr)

Key Takeaways

• The ion-product of [H⁺][OH⁻] in an aqueous solution remains constant.
• A pH value is simply the negative of the logarithm of the H⁺ ion concentration (-log[H⁺]).
• The pH scale is used to succinctly communicate the acidity or basicity of a solution.
• A solution is acidic if pH < 7.
• A solution is basic if pH > 7.
• A solution is neutral if pH = 7.

Concept Review Exercises

1. Define pH.

Answers

1. pH is a measure of the hydrogen ion concentration.

Exercises

Indicate whether solutions with the following pH values are acidic, basic, or neutral:

1. pH = 9.4
2. pH = 7.0
3. pH = 1.2
4. pH = 6.5

Answers

1. basic
2. neutral
3. acidic (strongly)
4. acidic (mildly)