Question

Calculate the \(\mathrm{pH}\) of an aqueous solution at \(25^{\circ} \mathrm{C}\) that is (a) \(1.02 \mathrm{M}\) in \(\mathrm{HI}\), (b) \(0.035 \mathrm{M}\) in \(\mathrm{HClO}_{4}\), and (c) \(1.5 \times 10^{-6} \mathrm{M}\) in \(\mathrm{HCl}\).

Short answer

The pH values are approximately 0.0 for HI, 1.46 for HClO4, and 5.80 for HCl.

Step-by-step solution

Understanding the Problem

The problem requires us to calculate the pH of three different aqueous solutions. Each solution contains a strong acid: HI, HClO4, and HCl. It's important to remember that strong acids fully dissociate in water.

Calculating pH for 1.02 M HI

Since HI is a strong acid, it completely dissociates in water. Thus, the concentration of H+ ions is equal to the concentration of HI.The pH is calculated using the formula: \[ \text{pH} = -\log[\text{H}^+] \]Substitute the concentration of H+ ions:\[ \text{pH} = -\log(1.02) = -0.009 \]Since pH is usually expressed with one decimal, pH ≈ 0.0.

Calculating pH for 0.035 M HClO4

HClO4 is also a strong acid and fully dissociates in water. This means that the concentration of H+ ions is equal to the concentration of HClO4.Calculate the pH:\[ \text{pH} = -\log[0.035] \approx 1.46 \]

Calculating pH for 1.5 x 10^-6 M HCl

Despite being a strong acid, the concentration of HCl (1.5 x 10^-6 M) is very low, which means we must also account for the autoinization of water which contributes \(1 \times 10^{-7}\) M of \(\text{H}^+\). The final \([\text{H}^+]\) is:\[ [\text{H}^+]_{\text{total}} = 1.5 \times 10^{-6} + 1 \times 10^{-7} = 1.6 \times 10^{-6} \]Calculate the pH:\[ \text{pH} = -\log[1.6 \times 10^{-6}] \approx 5.80 \]

Conclusion

For the given solutions, the calculated pH values are: - The pH of the 1.02 M HI solution is approximately 0.0. - The pH of the 0.035 M HClO4 solution is approximately 1.46. - The pH of the 1.5 x 10^-6 M HCl solution is approximately 5.80.

Key concepts

Strong Acids

When we talk about strong acids, it's essential to understand what makes an acid "strong." A strong acid is one that fully dissociates in water. This means that when you dissolve a strong acid in water, it breaks apart completely into its ions.
This complete dissociation allows for a straightforward pH calculation.
For the acids mentioned in the exercise:

• HI (Hydroiodic acid)
• HClO₄ (Perchloric acid)
• HCl (Hydrochloric acid)

They all behave similarly under typical conditions because of their strong nature. Breaking fully into

• H⁺ ions
• The corresponding negative ion

results in a direct relationship between the acid's initial concentration and the H⁺ ions in the solution.
Therefore, calculating the pH involves simply using this concentration.

Autoinization of Water

Water is a fascinating substance due to its ability to autoionize. Autoinization of water refers to the process where water molecules dissociate into

• H⁺ ions
• OH⁻ ions

even without any added substances. At 25°C, this process provides a constant H⁺ concentration of approximately

• \(1 imes 10^{-7}\) M

under neutral conditions.
When dealing with extremely diluted strong acids, like

• a concentration close to or less than \(1 imes 10^{-7}\) M

this autoionization must be considered.
For example, in the case of HCl with concentration being \(1.5 imes 10^{-6}\) M, the autoionization of water adds an additional \(1 imes 10^{-7}\) M to the total H⁺ concentration.
Ignoring this would slightly alter the accuracy of pH calculations.

Concentration of H+ Ions

One of the critical factors in pH calculations is understanding the concentration of H⁺ ions. The pH of a solution is directly related to the ul>

[H⁺]

through the formula:
\[ ext{pH} = -\log[ ext{H}^+] \]
When strong acids are present, calculating [H⁺] becomes much more direct. This is because the concentration of the acids themselves almost exactly equals the concentration of H⁺ ions.
However, with very low concentrations, such as in diluted strong acid solutions, we must also account for the additional H⁺ ions from the water's autoionization. Adding these concentrations provides a more complete picture of [H⁺], ensuring our pH calculations are precise.
The adjusted concentration is then used to find the pH, reflecting the total potential of hydrogen in the solution accurately.