# 14.8: The pH and pOH Scales - Ways to Express Acidity and Basicity

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As we have seen, \([H_3O^+]\) and \([OH^−]\) values can be markedly different from one aqueous solution to another. So chemists defined a new scale that succinctly indicates the concentrations of either of these two ions.

\(pH\) is a logarithmic function of \([H_3O^+]\):

\[pH = −\log[H_3O^+] \label{pH}\]

\(pH\) is usually (but not always) between 0 and 14. Knowing the dependence of \(pH\) on \([H_3O^+]\), we can summarize as follows:

- If pH < 7, then the solution is acidic.
- If pH = 7, then the solution is neutral.
- If pH > 7, then the solution is basic.

This is known as the \(pH\) scale. The range of values from 0 to 14 that describes the acidity or basicity of a solution. You can use \(pH\) to make a quick determination whether a given aqueous solution is acidic, basic, or neutral. Figure \(\PageIndex{1}\) 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.

## Calculating pH from Hydronium Concentration

The pH of solutions can be determined by using logarithms as illustrated in the next example for stomach acid. Stomach acid is a solution of \(HCl\) with a hydronium ion concentration of \(1.2 \times 10^{−3}\; M\), what is the \(pH\) of the solution?

\[ \begin{align} \mathrm{pH} &= \mathrm{-\log [H_3O^+]} \nonumber \\ &=-\log(1.2 \times 10^{−3}) \nonumber \\ &=−(−2.92)=2.92 \nonumber \end{align}\]

### Calculating Hydronium Concentration from pH

Sometimes you need to work "backwards" - you know the pH of a solution and need to find \([H_3O^+]\), or even the concentration of the acid solution. How do you do that? To convert pH into \([H_3O^+]\) we solve Equation \ref{pH} for \([H_3O^+]\). This involves taking the antilog (or inverse log) of the negative value of pH .

\[[\ce{H3O^{+}}] = \text{antilog} (-pH)\]

or

\[[\ce{H_3O^+}] = 10^{-pH} \label{ph1}\]

As mentioned above, different calculators work slightly differently - make sure you can do the following calculations using * your* calculator.

### The pOH scale

As with the hydrogen-ion concentration, the concentration of the hydroxide ion can be expressed logarithmically by the pOH. The **pOH** of a solution is the negative logarithm of the hydroxide-ion concentration.

\[\text{pOH} = -\text{log} \left[ \ce{OH^-} \right]\]

The relation between the hydronium and hydroxide ion concentrations expressed as p-functions is easily derived from the \(K_w\) expression:

\[K_\ce{w}=\ce{[H_3O^+][OH^- ]} \label{\(\PageIndex{6}\)}\]

\[-\log K_\ce{w}=\mathrm{-\log([H_3O^+][OH^−])=-\log[H_3O^+] + -\log[OH^-]}\label{\(\PageIndex{7}\)}\]

\[\mathrm{p\mathit{K}_w=pH + pOH} \label{\(\PageIndex{8}\)}\]

At 25 °C, the value of \(K_w\) is \(1.0 \times 10^{−14}\), and so:

\[\mathrm{14.00=pH + pOH} \label{\(\PageIndex{9}\)}\]

The hydronium ion molarity in pure water (or any neutral solution) is \( 1.0 \times 10^{-7}\; M\) at 25 °C. The pH and pOH of a neutral solution at this temperature are therefore:

\[\mathrm{pH=-\log[H_3O^+]=-\log(1.0\times 10^{−7}) = 7.00} \label{\(\PageIndex{1}\)0}\]

\[\mathrm{pOH=-\log[OH^−]=-\log(1.0\times 10^{−7}) = 7.00} \label{\(\PageIndex{1}\)1}\]

And so, at this temperature, acidic solutions are those with hydronium ion molarities greater than \( 1.0 \times 10^{-7}\; M\) and hydroxide ion molarities less than \( 1.0 \times 10^{-7}\; M\) (corresponding to pH values less than 7.00 and pOH values greater than 7.00). Basic solutions are those with hydronium ion molarities less than \( 1.0 \times 10^{-7}\; M\) and hydroxide ion molarities greater than \( 1.0 \times 10^{-7}\; M\) (corresponding to pH values greater than 7.00 and pOH values less than 7.00).

The diagram below shows all of the interrelationships between [H3O+][H3O+], [OH−][OH−], pH, and pOH.

## Contributions & Attributions

This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:

Peggy Lawson (Oxbow Prairie Heights School). Funded by Saskatchewan Educational Technology Consortium.

CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.

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Henry Agnew (UC Davis)