Question

For each hydrogen ion concentration listed, calculate the \(\mathrm{pH}\) of the solution as well as the concentration of hydroxide ion in the solution. Indicate whether the solutions are acidic or basic. a. \(\left[\mathrm{H}^{+}\right]=4.76 \times 10^{-8} \mathrm{M}\) b. \(\left[\mathrm{H}^{+}\right]=8.92 \times 10^{-3} \mathrm{M}\) c. \(\left[\mathrm{H}^{+}\right]=7.00 \times 10^{-5} \mathrm{M}\) d. \(\left[\mathrm{H}^{+}\right]=1.25 \times 10^{-12} \mathrm{M}\)

Short answer

a. \(\mathrm{pH} = 7.32\), \(\mathrm{[OH^{-}]} = 2.10 \times 10^{-7} \mathrm{M}\), acidic. b. \(\mathrm{pH} = 2.05\), \(\mathrm{[OH^{-}]} = 1.12 \times 10^{-12} \mathrm{M}\), acidic. c. \(\mathrm{pH} = 4.15\), \(\mathrm{[OH^{-}]} = 1.43 \times 10^{-10} \mathrm{M}\), acidic. d. \(\mathrm{pH} = 11.90\), \(\mathrm{[OH^{-}]} = 8.00 \times 10^{-3} \mathrm{M}\), basic.

Step-by-step solution

Calculate the pH

The \(\mathrm{pH}\) of a solution is given by the negative logarithm of the hydrogen ion concentration. So to calculate the pH, use the formula: \(\mathrm{pH} = -\log_{10}(\mathrm{[H^{+}]})\) For this solution: \(\mathrm{pH} = -\log_{10}(4.76 \times 10^{-8})\)

Find the concentration of hydroxide ions

Since the relationship between \(\mathrm{[H^{+}]}\) and \(\mathrm{[OH^{-}]}\) is given by the ion product of water (\(\mathrm{K_{w}}\)): \(\mathrm{K_{w} = [H^{+}][OH^{-}]}\) At 25°C, the value of \(\mathrm{K_{w}}\) is \(1.00 \times 10^{-14}\), so we have: \(\mathrm{[OH^{-}]} = \frac{\mathrm{K_{w}}}{\mathrm{[H^{+}]}}\) For this solution: \(\mathrm{[OH^{-}]} = \frac{1.00 \times 10^{-14}}{4.76 \times 10^{-8}}\)

Determine if the solution is acidic or basic

To determine if the solution is acidic or basic, compare the \(\mathrm{pH}\) value to 7. If \(\mathrm{pH} > 7\), the solution is basic, and if \(\mathrm{pH} < 7\), the solution is acidic. Since we calculated the \(\mathrm{pH}\) value in the first step, we just need to compare it to 7. Repeat the steps above for the other three solutions: b. \(\left[\mathrm{H}^{+}\right]=8.92 \times 10^{-3} \mathrm{M}\) c. \(\left[\mathrm{H}^{+}\right]=7.00 \times 10^{-5} \mathrm{M}\) d. \(\left[\mathrm{H}^{+}\right]=1.25 \times 10^{-12} \mathrm{M}\)

Key concepts

Understanding Hydrogen Ion Concentration

Hydrogen ion concentration, represented as \([H^{+}]\), is a key indicator of a solution's acidity. It informs how many hydrogen ions are present in a given volume of solution. The higher the concentration of \([H^{+}]\), the more acidic the solution is. The pH scale, which ranges from 0 to 14, quantifies this concentration. A low pH value indicates a higher level of hydrogen ions, suggesting an acidic nature.

To calculate the pH, use the formula:

• \(\text{pH} = -\log_{10}([H^{+}])\)

Plugging the concentration into this formula will yield the pH value, revealing whether the solution is acidic or not.
For instance, if \([H^{+}] = 4.76 \times 10^{-8} \text{ M}\), applying the formula gives us the pH. This calculation is crucial for determining how acidic a solution is.

Examining Hydroxide Ion Concentration

Hydroxide ion concentration, represented as \([OH^{-}]\), helps in understanding the basicity of a solution. In pure water at 25°C, the product of hydrogen ions and hydroxide ions is constant and equal to \(1.00 \times 10^{-14}\). This relationship is captured by the ion product constant of water, \(K_w\).

• \(K_w = [H^{+}][OH^{-}]\)
• To find \([OH^{-}]\), use: \([OH^{-}] = \frac{K_w}{[H^{+}]}\)

So, when you have the \([H^{+}]\), finding \([OH^{-}]\) is straightforward. This concentration tells us if the solution leans towards being basic. The higher the \([OH^{-}]\), the more basic it is. For example, given that \([H^{+}] = 4.76 \times 10^{-8} \text{ M}\), you can calculate \([OH^{-}]\) to see if the balance is tipped towards basicity.

Differentiating Acidic and Basic Solutions

Solutions are often described as acidic or basic based on the pH value.

• Acidic Solutions: These have a pH less than 7, meaning there is a higher concentration of hydrogen ions compared to hydroxide ions.
• Neutral Solutions: A pH of 7 indicates a perfect balance, typical of pure water.
• Basic Solutions: A pH greater than 7 means a higher concentration of hydroxide ions, making the solution basic.

To determine the nature of a solution, compare its calculated pH to 7. For example, if the pH is calculated as 5.3, the solution is acidic. If the pH is 9.2, it’s basic. Understanding this helps in predicting the behavior of solutions in different chemical reactions and real-life applications.