# pH and pOH

The calculations we have just done show that the concentrations of hydronium and hydroxide ions in aqueousDescribing a solution in which the solvent is water. solutionA mixture of one or more substances dissolved in a solvent to give a homogeneous mixture. can vary from about 1 mol dm^{–3} down to about 1 × 10^{–14} mol dm^{–3}, and perhaps over an even wider range. The numbers used to express [H_{3}O^{+}] and [OH^{–}] in the units moleThat chemical amount of a substance containing the same number of units as 12 g of carbon-12. per cubic decimeter will often include large negative powers of 10. Consequently it is convenient to define the following:

Note carefully what these equations tell us to do. To obtain pH, for example, we divide [H_{3}O^{+}] by the units mole per cubic decimeter. This gives a pure number, and so we can take its logarithm. (It does not make sense to take the logarithm of a unit, such as mole per cubic decimeter.) The minus sign insures that we will obtain a positive result most of the time.

The logarithm of a number is the power to which 10 must be raised to give the number itself. Therefore the definitions of pH and pOH mean that we can deal with powers of 10 rather than numerical values. Since the numbers needed to express [H_{3}O^{+}] and [OH^{–}] are usually between 1 and 10^{-14} pH and pOH values are usually between 0 and 14.

**EXAMPLE 1** Calculate the pH and the pOH of each of the following aqueous solutions: (a) 1.00 *M* HNO_{3}; 0.306 *M* Ba(OH)_{2}* .*

**Solution**

**a)** Our previous discussion showed that for this solution [H_{3}O^{+}] = 1.00 mol dm^{–3} and [OH^{–}] = 1.00 x 10^{-14}. Applying the definitions of pH and pOH, we have

**b)**In the example in the section on ionization of water, we found for this solution [H_{3}O^{+}] = 1.63 × 10^{–14} mol dm^{–3} and [OH^{–}] = 6.12 × 10^{–1} mol dm^{–3}. Thus

*b*would commonly be rounded to pH = 13.79 and pOH = 0.21.

Because pH measurements are so easily made, it is essential that you be able to convert from pH to [H_{3}O^{+}]. This is the reverse of finding pH from [H_{3}O^{+}]. Consequently it involves antilogs instead of logs. From the definition

we have

Taking the antilog of both sides, we have

so that

remembering that antilog *x* = 10^{x}, we can write this expression as

or

(1*a*)

An alternative method of writing this equation is

(1*b*)

**EXAMPLE 2** The pH of a solution is found to be 3.40. Find the hydronium-ion concentration of the solution.

**Solution** If you have a calculator which has an antilog or 10^{x} button, the problem is very simple. You enter – 3.40 and hit the button. The number thus obtained, 3.9822 x 10^{–4} is the number of moles of hydronium ion per cubic decimeter. This follows from Eq. (1*a*):

The same result is almost as easy to find using Eq. (1*b*).

Thus

in other words,

There is a very simple relationship between the pH and the pOH of an aqueous solution at 25°C. We know that at this temperatureA physical property that indicates whether one object can transfer thermal energy to another object.

Dividing both sides by mol^{2} dm^{–6}, we obtain

Taking logs and multiplying both by – 1, we then have

or

(2)

This simple relationship is often useful in finding the pH of solutions containing bases, as the following example shows.

**EXAMPLE 3** If 3.53 g of pure NaOH is dissolved in 10 dm^{3} of H_{2}O find the pH of the resulting solution.

**Solution** We first calculate the concentration of the NaOH.

so that

Since NaOH is a strong baseA base that dissociates completely or ionizes completely in a particular solvent., each mole of NaOH dissolved produces 1 mol OH^{–} ions, so that

Thus

From which

While the ability to calculate the pH of a solution from the hydronium-ion concentration and vice versa is useful, it is not the only thing we need to understand about pH. If someone gives you a solution whose pH is 14.74, it is true that the hydronium-ion concentration must be

1.82 × 10^{–15} mol dm^{–3} but it is perhaps more important to know that the solution is corrosively basic. In general, then, we need not only to be able to calculate a pH but also to have some realization of what kind of solutions have what kind of pH, as displayed in the following table. This table is part of our collection of acid-base resources.

**The pH Scale**

Substance
| pH
| [H
_{3}O^{+}] | [OH
^{-}] | pOH
| Strength
| ||

Battery acid | 0 | 1 | 10^{-14}
| 14 | Strongly acidic | ||

| |||||||

1 | 10^{-1}
| 10^{-13}
| 13 | ||||

2 | 10^{-2}
| 10^{-12}
| 12 | ||||

3 | 10^{-3}
| 10^{-11}
| 11 | Weakly acidic | |||

Soda water | 4 | 10^{-4}
| 10^{-10}
| 10 | |||

Black coffee | 5 | 10^{-5}
| 10^{-9}
| 9 | Barely acidic | ||

6 | 10^{-6}
| 10^{-8}
| 8 | ||||

Pure water | 7 | 10^{-7}
| 10^{-7}
| 7 | Neutral | ||

Seawater | 8 | 10^{-8}
| 10^{-6}
| 6 | Barely basic | ||

Baking soda | 9 | 10^{-9}
| 10^{-5}
| 5 | |||

Toilette soapA salt of a fatty acid produced by the saponification of fat. | 10 | 10^{-10}
| 10^{-4}
| 4 | Mildly basic | ||

Laundry water | 11 | 10^{-11}
| 10^{-3}
| 3 | |||

Household ammonia | 12 | 10^{-12}
| 10^{-2}
| 2 | Very basic | ||

13 | 10^{-13}
| 10^{-1}
| 1 | ||||

Drain cleaner | 14 | 10^{-14}
| 1 | 0 |

In pure water at 25°C the hydronium-ion concentration is close to 1.00 × 10^{–7} mol dm^{–3}, so that the pH is 7. In consequence any solution, not only pure water, which has a *pH of 7* is described as being *neutral*. An *acidic* solution, as we know, is one in which the hydronium-ion concentration is greater than that of pure water, i.e., *greater than* 10^{–7} mol dm^{–3}. In pH terms this translates into a pH which is *less than 7* (because the pH is a negative logarithm). Small pH values are thus characteristic of acidic solutions; the smaller the pH, the more acidic the solution.

By contrast, a *basic* solution is one in which the hydroxide-ion concentration is greater than 10^{–7} mol dm^{–3}. In such a solution the hydronium-ion concentration is *less than* 10^{–7} mol dm^{–3}, so that the pH of a basic solution is *greater than 7*. Large pH values are thus characteristic of basic solutions. The larger the pH, the more basic the solution.

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