Question

What is the \(\mathrm{pH}\) to the correct number of significant figures for solutions with the following concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}\) or \(\mathrm{OH}^{-}\) ? (a) \(\left[\mathrm{OH}^{-}\right]=7.6 \times 10^{-3} \mathrm{M}\) (b) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=10^{-8} \mathrm{M}\) (c) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=5.0 \mathrm{M}\) (d) \(\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-7} \mathrm{M}\) (c) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=2.18 \times 10^{-10} \mathrm{M}\)

Short answer

(a) pH = 11.88, (b) pH = 8.0, (c) pH = -0.70, (d) pH = 7.00, (e) pH = 9.66.

Step-by-step solution

Understand the relationship between pH and pOH

The pH is a measure of the acidity of a solution, calculated using the concentration of \[\text{{pH}} = -\log[\mathrm{H}_3\mathrm{O}^+]\]The pOH is calculated using the concentration of \(\mathrm{OH}^-\):\[\text{{pOH}} = -\log[\mathrm{OH}^-]\]For aqueous solutions, the sum of pH and pOH is:\[\text{{pH}} + \text{{pOH}} = 14\]

Calculate pH for each case separately

We will now calculate the pH values for each given concentration using the above formulas.(a) Given \([\mathrm{OH}^-]=7.6 \times 10^{-3} \mathrm{M}\), first calculate pOH: \[\text{{pOH}} = -\log(7.6 \times 10^{-3}) = 2.12\]Then, calculate pH: \[\text{{pH}} = 14 - 2.12 = 11.88\](b) Given \([\mathrm{H}_3\mathrm{O}^+]=10^{-8} \mathrm{M}\):\[\text{{pH}} = -\log(10^{-8}) = 8 \](c) Given \([\mathrm{H}_3\mathrm{O}^+]=5.0 \mathrm{M}\):\[\text{{pH}} = -\log(5.0) = -0.70 \](d) Given \([\mathrm{OH}^-]=1.0 \times 10^{-7} \mathrm{M}\), first calculate pOH:\[\text{{pOH}} = -\log(1.0 \times 10^{-7}) = 7\]Then, calculate pH:\[\text{{pH}} = 14 - 7 = 7\](e) Given \([\mathrm{H}_3\mathrm{O}^+]=2.18 \times 10^{-10} \mathrm{M}\):\[\text{{pH}} = -\log(2.18 \times 10^{-10}) = 9.66\]

Apply significant figures rule

The pH is reported with as many decimal places as the number of significant figures in the original concentration value. - (a) The concentration \(7.6 \times 10^{-3} \mathrm{M}\) has 2 significant figures, so pH is 11.88. - (b) The concentration \(10^{-8} \mathrm{M}\) has 1 significant figure, so pH is 8.0.- (c) The concentration \(5.0 \mathrm{M}\) has 2 significant figures, so pH is -0.70.- (d) The concentration \(1.0 \times 10^{-7} \mathrm{M}\) has 2 significant figures, so pH is 7.00.- (e) The concentration \(2.18 \times 10^{-10} \mathrm{M}\) has 3 significant figures, so pH is 9.66.

Key concepts

Acid-Base Chemistry

Acid-base chemistry is a fundamental part of understanding chemical reactions in solutions. It involves studying how acids and bases interact in water and how they affect the solution's pH level. In solutions, acids like hydrochloric acid (\( ext{HCl} \)) release hydronium ions (\( ext{H}_3 ext{O}^+ \)), while bases like sodium hydroxide (\( ext{NaOH} \)) release hydroxide ions (\( ext{OH}^- \)). Both ions affect the solution's pH level.

• The pH scale measures how acidic or basic a solution is on a scale from 0 to 14.
• Acids have a pH less than 7, while bases have a pH greater than 7.
• Neutral solutions like pure water have a pH of exactly 7.

In an aqueous solution, the product of the concentrations of hydronium ions and hydroxide ions is always constant at \( 10^{-14} \) at 25°C. By understanding these concepts, you can predict and calculate the pH of a solution, which is critical in many chemical and biological processes.

Significant Figures

Significant figures are essential in accurately reporting scientific measurements. They indicate the precision of a measurement, reflecting the accuracy of the measurement tool or method used. The rules for determining significant figures in calculations are crucial, especially in pH calculations. Let's break down a few points:

• Significant figures in a number include all certain digits plus one uncertain digit.
• For example, the number 7.6 has two significant figures.
• When using logarithms, like in pH calculations, the significant figures in the original concentration determine the number of decimal places in the pH.

This rule is crucial in pH calculations, as it ensures that the pH value reflects the measurement's precision. For instance, if the concentration of \( ext{H}_3 ext{O}^+ \) is \( 2.18 imes 10^{-10} \, ext{M} \), the pH should have three decimal places due to the three significant figures in the concentration value. Understanding these rules helps maintain accuracy in scientific reporting.

Hydronium Ion Concentration

Hydronium ions (\( ext{H}_3 ext{O}^+ \)) play a critical role in determining the acidity of a solution. The concentration of these ions is directly related to how acidic a solution is, making it a valuable aspect of acid-base chemistry.
The connection between hydronium concentration and pH is given by the formula:\[\text{pH} = -\log [\text{H}_3\text{O}^+]\]

• The higher the hydronium ion concentration, the lower the pH, making the solution more acidic.
• Conversely, a lower hydronium ion concentration results in a higher pH, indicating a more basic solution.

The pH provides a quick way to understand the acidic or basic nature of a solution. In real-world applications, this understanding helps in areas like medicine, agriculture, and water treatment, where controlling the pH is vital to ensure health and safety. Knowing how to calculate and interpret hydronium ion concentration is a fundamental skill in acid-base chemistry.