When 1.0 \(\mathrm{mol}\) of a weak acid was dissolved in 10.0 L of water, the pH was found to be \(3.90 .\) What is \(K_{a}\) for the acid?

Understanding the pH of a solution is essential in chemistry, especially when dealing with acids and bases. The pH scale is used to determine the acidity or basicity of a solution, which is fundamentally rooted in the concentration of hydrogen ions, denoted as \( [H^+] \). The scale ranges from 0 to 14, where pH less than 7 indicates acidity, pH greater than 7 indicates alkalinity, and a pH of 7 is neutral.

To calculate the pH, which indirectly measures the concentration of hydrogen ions in a solution, we use the formula: \[ pH = -\log_{10} [H^+] \]. Conversely, if we know the pH, we can determine the hydrogen ion concentration with the formula \( [H^+] = 10^{-pH} \). For instance, with a given pH of 3.90, the hydrogen ion concentration becomes \( [H^+] = 10^{-3.90} = 1.26 \times 10^{-4} \, \text{M} \), where \( \text{M} \) represents molarity, the unit for concentration in moles per liter.

A weak acid does not fully dissociate in water, creating an equilibrium between the non-dissociated acid and the ions produced by its partial dissociation. The chemical equation for a weak acid \( HA \) can be written as \( HA \rightleftharpoons H^+ + A^- \). The equilibrium constant for this reaction is the acid dissociation constant, denoted as \( K_a \).

The formula for \( K_a \) is expressed as \( K_a = \frac{[H^+][A^-]}{[HA]} \), where \( [H^+] \) is the concentration of hydrogen ions, \( [A^-] \) is the concentration of the conjugate base, and \( [HA] \) is the concentration of the undisassociated acid. In the case of the provided exercise, where a 1.0 molar solution of a weak acid results in a pH of 3.90, the concentration of \( [H^+] \) and \( [A^-] \) can be considered equal due to the 1:1 stoichiometry of hydrogen ion to conjugate base production.

The concept of chemical concentration pertains to the amount of a substance within a certain volume of solvent. It is a fundamental measure in chemistry because reactions, equilibrium, and pH are all affected by the concentrations of the reactants and products involved. Concentration is typically expressed in molarity, denoted as M, which is moles of solute per liter of solution.

For a solution created by dissolving 1.0 mole of a substance in 10.0 liters of water, the concentration would be \( 1.0 \, \text{mol} \div 10.0 \, \text{L} = 0.1 \, \text{M} \). This value is crucial when computing the \( K_a \) of a weak acid, as the equilibrium concentration of the undissociated acid \( [HA] \) in the \( K_a \) formula is essentially the initial molarity of the acid before any dissociation begins. Understanding how to convert between moles, volume, and molarity enables students to tackle a variety of chemical problems, including those involving weak acid equilibria and pH calculations.