# 16.3 The pH Scale

The concentration of H+(aq) in an aqueous solution is usually quite small. For convenience, we therefore usually express [H+] in terms of pH, which is defined as the negative logarithm in base 10 of [H+] (because [H+] and [H3O+] are used interchangeably, you might see pH defined as -log [H3O+]):

pH = -log [H+]

[16.6]

If you need to review the use of logs, see Appendix A.

We can use Equation 16.6 to calculate the pH of a neutral solution at 25°C; that is, one in which [H+] = 1.0 × 10-7 M:

pH = -log (1.0 × 10-7) = -(-7.00) = 7.00

The pH of a neutral solution is 7.00.

What happens to the pH of a solution as we make the solution acidic? An acidic solution is one in which [H+] > 1.0 × 10-7 M. Because of the negative sign in Equation 16.6, the pH decreases as [H+] increases. For example, consider an acidic solution in which [H+] = 1.0 × 10-3 M. The pH of this solution is

pH = -log (1.0 × 10-3) = -(-3.00) = 3.00

At 25°C, the pH of an acidic solution is less than 7.

Similarly, let's consider a basic solution, one in which [OH-] > 1.0 × 10-7 M. Suppose [OH-] = 2.0 × 10-3. We can use Equation 16.2 to calculate [H+] for this solution, and Equation 16.6 to calculate the pH:

We see that the pH of a basic solution is greater than 7. The relationships among [H+], [OH-], and pH are summarized in Table 16.1.

The pH values characteristic of several familiar solutions are shown in Figure 16.2. Notice that a change in [H+] by a factor of 10 causes the pH to change by 1. For example, a solution of pH 6 has 10 times the concentration of H+(aq) that a solution of pH 7 has.

You might think that when [H+] is very small, as it is for some of the examples shown in Figure 16.2, it would be unimportant. Nothing is further from the truth. Remember that if [H+] is part of a kinetic rate law, changing its concentration will change the rate. (For more information, see Section 14.2) Thus, if the rate law is first order in [H+], doubling its concentration will double the rate even if the change is merely from 1 × 10-7 M to 2 × 10-7 M. In biological systems many reactions involve proton transfers and have rates dependent on [H+]. Because the speeds of these reactions are crucial, the pH of biological fluids must be maintained within narrow limits. For example, human blood has a normal pH range of 7.35 to 7.45. Illness and even death can result if the pH varies much from this narrow range.

FIGURE 16.2 Values of pH for some common solutions. The pH scale is shown to extend from 0 to 14 because nearly all solutions commonly encountered have pH values in that range. In principle, however, the pH values for strongly acidic solutions can be less than 0, and for strongly basic solutions can be greater than 14.

### Sample Exercise 16.3

Calculate the pH values for the two solutions described in Sample Exercise 16.2.

SOLUTION (a) In the first instance we found [H+] to be 1.0 × 10-12 M. Because log 10x = x, we can calculate the pH quite readily:

pH = -log (1.0 × 10-12) = -(-12.00) = 12.00

The rule for using significant figures with logs is that the number of decimal places in the log equals the number of significant figures in the original number (see Appendix A). Because 1.0 × 10-12 has two significant figures, the pH has two decimal places, 12.00.

(b) For the second solution, [H+] = 5.0 × 10-6 M. Thus the pH equals -log (5.0 × 10-6). Before performing the calculation, it is helpful to make an estimate of the pH. To do so, we bracket the concentration by its closest powers of 10: 10-6 < 5.0 × 10-6 < 10-5. We therefore expect the pH of the solution to be between 5 (the pH for [H+] = 10-5 M) and 6 (the pH for [H+] = 10-6 M). You should use an estimation of pH to check whether your final answer is reasonable.

On most calculators, calculation of the pH can be performed with the following keystrokes:

We see that the pH = -log (5.0 × 10-6) = -(-5.30) = 5.30.

### Practice Exercise

(a) In a sample of lemon juice [H+] is 3.8 × 10-4 M. What is the pH? (b) A commonly available window-cleaning solution has a [H+] of 5.3 × 10-9 M. What is the pH? Answers: (a) 3.42; (b) 8.28

### Sample Exercise 16.4

A sample of freshly pressed apple juice has a pH of 3.76. Calculate [H+].

SOLUTION Because the pH is between 3 and 4, we know immediately that [H+] will be between 10-3 and 10-4 M. From the equation defining pH, we have

Thus

To find [H+] we need to determine the antilog of -3.76. Many calculators have an antilog function (sometimes labeled INV log) that allows us to perform the calculation with the following key sequence:

The result, expressed in standard exponential notation, is 1.7 × 10-4 M. Some calculators rely on 10x or yx functions to find antilogs: antilog (-3.76) = 10-3.76 = 1.7 × 10-4. Consult the user's manual for your calculator to find out how to perform the antilog operation. Notice that the number of significant figures in [H+] equals the number of decimal places in the pH (2).

### Practice Exercise

A solution formed by dissolving an antacid tablet has a pH of 9.18. Calculate [H+]. Answer: [H+] = 6.6 × 10-10 M

## Other "p" Scales

The negative log is also a convenient way of expressing the magnitudes of other small quantities. We use the convention that the negative log of a quantity is labeled p (quantity). For example, one can express the concentration of OH- as pOH:

pOH = -log [OH-]

[16.7]

By taking the log of both sides of Equation 16.2 and multiplying through by -1, we can obtain

pH + pOH = -log Kw = 14.00

[16.8]

This expression is often convenient to use. We will see later (Section 16.8) that the pX notation is also useful in dealing with equilibrium constants.

## Measuring pH

The pH of a solution can be measured quickly and accurately with a pH meter (Figure 16.3). A complete understanding of how these important devices work requires a knowledge of electrochemistry, a subject we take up in Chapter 20. In brief, a pH meter consists of a pair of electrodes connected to a meter capable of measuring small voltages, on the order of millivolts. A voltage, which varies with the pH, is generated when the electrodes are placed in a solution. This voltage is read by the meter, which is calibrated to give pH.

The electrodes used with pH meters come in many shapes and sizes, depending on their intended use. Electrodes have even been developed that are so small that they can be inserted into single living cells in order to monitor the pH of the cell medium. Pocket-size pH meters are also available for use in environmental studies, in monitoring industrial effluents, and in agricultural work.

Although less precise, acid-base indicators are often used to measure pH. An acid-base indicator is a colored substance that itself can exist in either an acid or a base form. The two forms have different colors. Thus, the indicator turns one color in an acid and turns another color if placed in a base. If you know the pH at which the indicator turns from one form to the other, you can determine whether a solution has a higher or lower pH than this value. For example, litmus, one of the most common indicators, changes color in the vicinity of pH 7. However, the color change is not very sharp. Red litmus indicates a pH of about 5 or lower, and blue litmus indicates a pH of about 8 or higher.

Some of the more common indicators are listed in Figure 16.4. We see from this figure that methyl orange, for example, changes color over the pH interval from 3.1 to 4.4. Below pH 3.1 it is in the acid form, which is red. In the interval between 3.1 and 4.4 it is gradually converted to its basic form, which has a yellow color. By pH 4.4 the conversion is complete, and the solution is yellow. Paper tape that is impregnated with various indicators and comes complete with a comparator color scale is widely used for approximate determinations of pH.

FIGURE 16.4 The pH ranges for the color changes of some common acid-base indicators. Most indicators have a useful range of about 2 pH units.