# 3.4: The pH Scale

- Page ID
- 175022

Because of its* amphoteric *nature (i.e., acts as both an acid or a base), water does not always remain as \(H_2O\) molecules. In fact, two water molecules react to form hydronium and hydroxide ions:

\[ 2\, H_2O \;(l) \rightleftharpoons H_3O^+ \;(aq) + OH^− \; (aq) \label{1}\]

This is also called the self-ionization of water. The concentration of \(H_3O^+\) and \(OH^-\) are equal in pure water because of the 1:1 stoichiometric ratio of Equation 1. The molarity of H_{3}O^{+} and OH^{-} in water are also both \(1.0 \times 10^{-7} \,M\) at 25° C. Therefore, a constant of water (\(K_w\)) is created to show the equilibrium condition for the self-ionization of water. The product of the molarity of hydronium and hydroxide ion is always \(1.0 \times 10^{-14}\).

\[K_w= [H_3O^+][OH^-] = 1.0 \times 10^{-14} \label{2}\]

This equations also applies to all aqueous solutions. However, \(K_w\) does change at different temperatures, which affects the pH range discussed below.

\(H^+\) and \(H_3O^+\) is often used interchangeably to represent the hydrated proton, commonly call the hydronium ion.

The equation for water equilibrium is:

\[ H_2O \rightleftharpoons H^+ + OH^- \label{3}\]

- If an acid (\(H^+\)) is added to the water, the equilibrium shifts to the left and the \(OH^-\) ion concentration decreases
- If base ( \(OH^-\)) is added to water, the equilibrium shifts to left and the \(H^+\) concentration decreases.

## pH and pOH

The constant of water determines the range of the pH scale. To understand what the pK_{w} is, it is important to understand first what the "p" means in pOH, and pH. The Danish biochemist Søren Sørenson proposed the term pH to refer to the "potential of hydrogen ion." He defined the "p" as the negative of the logarithm, -log, of [H^{+}]. Therefore the pH is the negative logarithm of the molarity of H. The pOH is the negative logarithm of the molarity of OH^{-} and the pK_{w}_{ }is the negative logarithm of the constant of water. These definitions give the following equations:

\[pH= -\log [H^+] \label{4a}\]

\[pOH= -\log [OH^-] \label{4b}\]

\[pK_w= -\log [K_w] \label{4c}\]

At room temperature (25 °C),

\[K_w =1.0 \times 10^{-14} \label{4d}\]

So

\[pK_w=-\log [1.0 \times 10^{-14}] \label{4e}\]

Using the properties of logarithms, Equation \(\ref{4e}\) can be rewritten as

\[10^{-pK_w}=10^{-14}. \label{4f}\]

By substituting, we see that pK_{w} is 14. The equation also shows that each increasing unit on the scale decreases by the factor of ten on the concentration of \(H^+\). For example, a pH of 1 has a molarity ten times more concentrated than a solution of pH 2.

Since

\[pK_w\ = 14 \label{5a}\]

\[pK_w= pH + pOH = 14 \label{5b}\]

Since the autoionization constant \(K_w\) is temperature dependent, these correlations between pH values and the acidic/neutral/basic adjectives will be different at temperatures other than 25 °C. For example, the hydronium molarity of pure water at 80 °C is 4.9 × 10^{−7} *M*, which corresponds to pH and pOH values of:

\[\mathrm{pH=-\log[H_3O^+]=-\log(4.9\times 10^{−7})=6.31}\label{12}\]

\[\mathrm{pOH=-\log[OH^-]=-\log(4.9\times 10^{−7})=6.31}\label{13}\]

At this temperature, then, neutral solutions exhibit pH = pOH = 6.31, acidic solutions exhibit pH less than 6.31 and pOH greater than 6.31, whereas basic solutions exhibit pH greater than 6.31 and pOH less than 6.31. This distinction can be important when studying certain processes that occur at nonstandard temperatures, such as enzyme reactions in warm-blooded organisms. Unless otherwise noted, references to pH values are presumed to be those at standard temperature (25 °C) (Table \(\PageIndex{1}\)).

The pH scale is logarithmic, meaning that an increase or decrease of an integer value changes the concentration by a tenfold. For example, a pH of 3 is ten times more acidic than a pH of 4. Likewise, a pH of 3 is one hundred times more acidic than a pH of 5. Similarly a pH of 11 is ten times more basic than a pH of 10.

Figure \(\PageIndex{1}\) depicts the pH scale with common solutions and where they are on the scale.

## Summary

The concentration of hydronium ion in a solution of an acid in water is greater than \( 1.0 \times 10^{-7}\; M\) at 25 °C. The concentration of hydroxide ion in a solution of a base in water is greater than \( 1.0 \times 10^{-7}\; M\) at 25 °C. The concentration of H_{3}O^{+} in a solution can be expressed as the pH of the solution; \(\ce{pH} = -\log \ce{H3O+}\). The concentration of OH^{−} can be expressed as the pOH of the solution: \(\ce{pOH} = -\log[\ce{OH-}]\). In pure water, pH = 7.00 and pOH = 7.00.

## References

- Petrucci, et al. "Self-Ionization of Water and the pH Scale."
*General Chemistry:**Principles & Modern Applications*. 7th ed. Upper Saddle River: Pearson Prentice Hall, 2007. 669-71. - Segel, Irwin H. "Acid and Base."
*Biochemical Calculations*. 2nd ed. Wiley: BK Book, 1976. 12. - Christopher G. McCarty and Ed Vitz, Journal of Chemical Education, 83(5), 752 (2006)

## Contributors

- Emmellin Tung (UCD), Sharon Tsao (UCD), Divya Singh (UCD), Patrick Gormley (Lapeer Community School District)
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110).