Question

Calculate \(\left[\mathrm{H}^{+}\right]\) in each of the following solutions, and indicate whether the solution is acidic, basic, or neutral. a. \(\left[\mathrm{OH}^{-}\right]=4.22 \times 10^{-3} M\) b. \(\left[\mathrm{OH}^{-}\right]=1.01 \times 10^{-13} M\) c. \(\left[\mathrm{OH}^{-}\right]=3.05 \times 10^{-7} M\) d. \(\left[\mathrm{OH}^{-}\right]=6.02 \times 10^{-6} M\)

Short answer

a. \([\mathrm{H}^{+}] = 2.37 \times 10^{-12} M\), basic. b. \([\mathrm{H}^{+}] = 0.99 \times 10^{-1} M\), acidic. c. \([\mathrm{H}^{+}] = 3.28 \times 10^{-8} M\), basic. d. \([\mathrm{H}^{+}] = 1.66 \times 10^{-9} M\), basic.

Step-by-step solution

Calculate \(\left[\mathrm{H}^{+}\right]\) for each solution

We will use the ion product constant of water to calculate the concentration of hydrogen ions in each solution: a. \(K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}]\) \([\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1.0 \times 10^{-14}}{4.22 \times 10^{-3}} = 2.37 \times 10^{-12} M\) b. \([\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1.0 \times 10^{-14}}{1.01 \times 10^{-13}} = 0.99 \times 10^{-1} M\) c. \([\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1.0 \times 10^{-14}}{3.05 \times 10^{-7}} = 3.28 \times 10^{-8} M\) d. \([\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1.0 \times 10^{-14}}{6.02 \times 10^{-6}} = 1.66 \times 10^{-9} M\)

Determine the acidity or basicity of each solution

Now that we have the concentration of hydrogen ions, we can determine whether each solution is acidic, basic, or neutral: a. Since \([\mathrm{H}^{+}] = 2.37 \times 10^{-12} M\) and \([\mathrm{OH}^{-}] = 4.22 \times 10^{-3} M\), the solution is basic. b. Since \([\mathrm{H}^{+}] = 0.99 \times 10^{-1} M\) and \([\mathrm{OH}^{-}] = 1.01 \times 10^{-13} M\), the solution is acidic. c. Since \([\mathrm{H}^{+}] = 3.28 \times 10^{-8} M\) and \([\mathrm{OH}^{-}] = 3.05 \times 10^{-7} M\), the solution is basic. d. Since \([\mathrm{H}^{+}] = 1.66 \times 10^{-9} M\) and \([\mathrm{OH}^{-}] = 6.02 \times 10^{-6} M\), the solution is basic.

Key concepts

Ion Product Constant of Water

In chemistry, the ion product constant of water, denoted as \(K_w\), is a fundamental concept to understanding the relationship between hydrogen ions (\(\mathrm{H}^{+}\)) and hydroxide ions (\(\mathrm{OH}^{-}\)) in aqueous solutions. This constant represents the product of the molar concentrations of these ions in pure water at a particular temperature, typically 25°C. The value of \(K_w\) is \(1.0 \times 10^{-14}\) at this standard temperature.

When you dissolve substances in water, they can either increase the concentration of hydrogen ions, making the solution acidic, or increase the concentration of hydroxide ions, making it basic. However, no matter the solution, the product of the concentrations of these ions will still be equal to \(K_w\). If we know the concentration of one type of ion, we can easily calculate the concentration of the other using the formula:

Hydrogen Ion Concentration

The hydrogen ion concentration in a solution, indicated by \([\mathrm{H}^{+}]\), is a measure of its acidity. In a neutral solution, such as pure water, the concentration of hydrogen ions is equal to \(1.0 \times 10^{-7} M\) because \(\sqrt{K_w} = \sqrt{1.0 \times 10^{-14}}\).

When solving problems involving \([\mathrm{H}^{+}]\), it's essential to know that if \([\mathrm{H}^{+}]\) is greater than \(1.0 \times 10^{-7} M\), the solution is acidic, and if it is less, then it's basic. The step-by-step calculation of \([\mathrm{H}^{+}]\) from a known \([\mathrm{OH}^{-}]\) involves using the ion product constant of water as shown above, which is vital for students to understand the concept of chemical equilibrium in aqueous solutions.

Hydroxide Ion Concentration

Similarly, hydroxide ion concentration, indicated by \([\mathrm{OH}^{-}]\), is a measure of the solution's basicity. It ranges inversely with the hydrogen ion concentration, given their relationship through the ion product constant of water, \(K_w\).

To determine the \([\mathrm{OH}^{-}]\) from a known \([\mathrm{H}^{+}]\), you can use the formula \([\mathrm{OH}^{-}] =\frac{K_w}{[\mathrm{H}^{+}]}\). This is particularly useful when adjusting the pH of a solution, for example, when you are trying to neutralize an acidic solution by adding a base. Understanding how these two concentrations work in tandem is crucial for pH calculation.

Acidity and Basicity of Solutions

The concepts of acidity and basicity are central to the pH calculation and understanding the nature of different solutions. Acidity is primarily determined by the hydrogen ion concentration, while basicity is determined by the hydroxide ion concentration. A neutral solution has equal concentrations of both ions, with a pH of 7.

pH is a logarithmic scale used to quantify the acidity or basicity of an aqueous solution. It is calculated as \(-\log([\mathrm{H}^{+}])\). A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic solution. By using the pH scale and understanding the relationship between \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) ions, you can easily categorize any aqueous solution as acidic, basic, or neutral, enhancing your grasp of chemical properties.