When the concentration of a strong acid is not substantially higher than \(1.0 \times 10^{-7} M,\) the ionization of water must be taken into account in the calculation of the solution's pH. (a) Derive an expression for the \(\mathrm{pH}\) of a strong acid solution, including the contribution to \(\left[\mathrm{H}^{+}\right]\) from \(\mathrm{H}_{2} \mathrm{O}\). (b) Calculate the \(\mathrm{pH}\) of a \(1.0 \times 10^{-7} M \mathrm{HCl}\) solution.

Understanding the ion product of water is crucial for pH calculations, especially for very dilute solutions of strong acids. At 25°C, the ion product constant of water, denoted as \( K_w \), is \(1.0 \times 10^{-14} \). This ion product implies that water naturally dissociates to some extent into \( H^+ \) (hydrogen ions) and \( OH^- \) (hydroxide ions) with the equilibrium constant reflecting their product.

For instance, in pure water, because the concentrations of hydrogen ions and hydroxide ions are equal, we can deduce that each concentration is \( \[H^+\] = \[OH^-\] = \( \text{\( K_w \)}^{1/2} \) = 1.0 \times 10^{-7} M \). When considering a strong acid solution, we cannot neglect this natural ionization of water, especially when the acid concentration comes close to the ion concentrations from water itself.

A strong acid is characterized by its ability to completely dissociate in water, contributing to the total \( [H^+] \) in solution. This means when a strong acid, HA, dissolves in water it releases hydrogen ions and leaves behind the conjugate base, A-.

The dissociation can be represented by the following chemical equation: \( HA \rightarrow H^+ + A^- \). If we have a known concentration of a strong acid, such as hydrochloric acid (HCl), we can assume that its molarity directly indicates the concentration of hydrogen ions contributed by the acid in the solution, which significantly simplifies pH calculation for more concentrated solutions of strong acids.

To derive the pH expression for a strong acid solution that includes the ionization of water, we mesh the foundational knowledge of strong acid ionization with the ion product of water. For very dilute strong acid solutions, the hydrogen ion concentration is not only from the acid itself but also includes a significant contribution from the water's ionization.

The derived expression for pH takes the sum of the acid's hydrogen ion concentration and the hydrogen ions from water ionization, squared, and then applies the negative logarithm to obtain the pH value. Deriving and applying this adjusted formula ensures a more accurate pH value for such solutions, as it does not neglect the hydrogen ions that originate from water itself.

The hydrogen ion concentration \( [H^+] \) in solution is a central component in determining the acidity and thus the pH level of a solution. For strong acid solutions, it was traditionally thought that the hydrogen ion concentration equaled the molarity of the acid. However, when working with dilute solutions, where the acid concentration approaches \(1.0 \times 10^{-7} M\), it's imperative to include the contribution of hydrogen ions from the ionization of water.

This comprehensive consideration leads to a more nuanced understanding and calculation of pH in such cases. The exercise example explicitly accounts for both sources of hydrogen ions, reflecting the importance of assessing all contributors to the hydrogen ion pool in a solution for accurate pH calculation.