Calculate the \(\mathrm{pH}\) of a \(2.0-M \mathrm{H}_{2} \mathrm{SO}_{4}\) solution.

Understanding the behavior of diprotic acids is crucial when learning about acidity and pH calculations. A diprotic acid is one that has two hydrogen ions (H⁺) to donate. These acids undergo a two-step dissociation process. The first dissociation involves the release of the first hydrogen ion, while in the second step, the acid releases the second hydrogen ion. However, with strong diprotic acids such as sulfuric acid (\(H_{2}SO_{4}\)), the first dissociation is often complete because of its high tendency to lose the first hydrogen ion.

As we observed in the exercise, sulfuric acid dissociates fully in water to produce two moles of hydrogen ions for each mole of acid. This dissociation is represented by the chemical equation: \[H_{2}SO_{4} \rightarrow 2H^{+} + SO_{4}^{2-}\] This complete dissociation results in a high concentration of hydrogen ions in the solution, which is directly related to the acidity and ultimately the pH value of the solution.

The concentration of hydrogen ions (\(H^+\) ions) in a solution is a key factor in determining its acidity. The higher the concentration of these ions, the more acidic the solution is. In the context of a strong diprotic acid like sulfuric acid, the complete dissociation means that the concentration of hydrogen ions is directly proportional to the initial concentration of the acid itself. Essentially, for every mole of H₂SO₄ that dissociates, we get two moles of hydrogen ions.

Therefore, if sulfuric acid has an initial concentration of 2.0 M, the resulting concentration of hydrogen ions is: \[2.0 \, M \times 2 = 4.0 \, M\] Being able to calculate the concentration of these ions is foundational to finding the pH of the solution, as pH is a measure of the hydrogen ion concentration in a solution.

The pH scale is a logarithmic measurement that expresses the acidity or basicity of a solution. It ranges typically from 0 to 14, with 7 being neutral, values less than 7 indicating acidity, and values greater than 7 indicating alkalinity. The relationship between the hydrogen ion concentration and pH is inversely logarithmic; meaning as the concentration of hydrogen ions increases, the pH number decreases.

The formula for pH is as follows: \[pH = -\log_{10} [H^+]\] Here, \(\log_{10}\) represents the base-10 logarithm. To illustrate, using the exercise's calculated concentration of hydrogen ions (4.0 M), the pH is computed in this manner: \[pH = -\log_{10} (4.0)\] Using a basic scientific calculator, you would find that the pH of the solution is approximately 0.40. It's an essential skill to be able to compute pH since it's often used in chemistry to determine the suitability of a substance for various applications, and can indicate the strength of acidic or basic solutions.