Problem 31 - Chemistry: The Central Science

Question

Calculate \(\left[\mathrm{H}^{+}\right]\) for each of the following solutions, and indicate whether the solution is acidic, basic, or neutral: (a) \(\left[\mathrm{OH}^{-}\right]=0.00045 M ;\) (b) \(\left[\mathrm{OH}^{-}\right]=8.8 \times 10^{-9} \mathrm{M} ;(\mathrm{c}) \mathrm{a}\) solution in which \(\left[\mathrm{OH}^{-}\right]\) is 100 times greater than \(\left[\mathrm{H}^{+}\right]\).

Step-by-Step Solution

Answer

(a) \([\mathrm{H}^{+}]\) = 2.22 x 10^{-11} M, basic; (b) \([\mathrm{H}^{+}]\) = 1.14 x 10^{-6} M, acidic; (c) \([\mathrm{H}^{+}]\) = 1 x 10^{-8} M, basic.

Step 1: Recall the ion product constant of water

Recall the ion product constant of water equation: \[K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}]\] At 25 degrees Celsius, the ion product constant \(K_w=1 \times 10^{-14}\).

Step 2: Calculate \([\mathrm{H}^{+}]\) for solution (a)

Given, \([\mathrm{OH}^{-}] = 0.00045\) M. We can use the ion product constant equation to find \([\mathrm{H}^{+}]\): \[[\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1 \times 10^{-14}}{0.00045} = 2.22 \times 10^{-11} M\]

Step 3: Determine the nature of solution (a)

Since \([\mathrm{H}^{+}] < [\mathrm{OH}^{-}]\), solution (a) is basic.

Step 4: Calculate \([\mathrm{H}^{+}]\) for solution (b)

Given, \([\mathrm{OH}^{-}] = 8.8 \times 10^{-9}\) M. We can use the ion product constant equation to find \([\mathrm{H}^{+}]\): \[[\mathrm{H}^{+}] = \frac{K_w}{[\mathrm{OH}^{-}]} = \frac{1 \times 10^{-14}}{8.8 \times 10^{-9}} = 1.14 \times 10^{-6} M\]

Step 5: Determine the nature of solution (b)

Since \([\mathrm{H}^{+}] > [\mathrm{OH}^{-}]\), solution (b) is acidic.

Step 6: Calculate \([\mathrm{H}^{+}]\) for solution (c)

Given, \([\mathrm{OH}^{-}]\) is 100 times greater than \([\mathrm{H}^{+}]\). Let \(x\) represent the concentration of \([\mathrm{H}^{+}]\): \[[\mathrm{OH}^{-}] = 100[\mathrm{H}^{+}] = 100x\] Substitute this value into the ion product constant equation: \[K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] \Rightarrow 1 \times 10^{-14} = x(100x) \Rightarrow x^2 = 1 \times 10^{-16}\] Solve for \(x\): \[x = [\mathrm{H}^{+}] = \sqrt{1 \times 10^{-16}} = 1 \times 10^{-8} M\]

Step 7: Determine the nature of solution (c)

Since \([\mathrm{H}^{+}] < [\mathrm{OH}^{-}]\), solution (c) is basic. In conclusion, the solution (a) is basic with \([\mathrm{H}^{+}] = 2.22 \times 10^{-11} M\), solution (b) is acidic with \([\mathrm{H}^{+}] = 1.14 \times 10^{-6} M\), and solution (c) is basic with \([\mathrm{H}^{+}] = 1 \times 10^{-8} M\).

Key Concepts

pH Calculation

Understanding pH is crucial for identifying the acidity or basicity of a solution. The pH scale ranges from 0 to 14, where a pH less than 7 indicates an acidic solution, a pH of 7 denotes a neutral solution, and a pH higher than 7 signifies a basic solution.

The pH can be calculated using the formula:
\[\text{pH} = -\log([\text{H}^{+}])\]
In this formula, \( [\text{H}^{+}] \) is the concentration of hydrogen ions in moles per liter (M). It's important to note that the pH scale is logarithmic, which means each whole pH value below 7 is ten times more acidic than the next higher value. For instance, a solution with a pH of 3 is ten times more acidic than one with a pH of 4.

To incorporate exercise improvement advice, let's ensure when we calculate the pH, we maintain the correct number of significant figures and use a scientific calculator to handle logarithmic calculations, as these are prone to rounding errors.

Acidic and Basic Solutions

The nature of a solution, whether acidic or basic, is determined by the relative concentrations of hydrogen ions (\( [\text{H}^{+}] \) ) and hydroxide ions (\( [\text{OH}^{-}] \) ). In a neutral solution, the concentrations of \( [\text{H}^{+}] \) and \( [\text{OH}^{-}] \) are equal. In an acidic solution, the concentration of \( [\text{H}^{+}] \) exceeds that of \( [\text{OH}^{-}] \), while in a basic solution, the concentration of \( [\text{OH}^{-}] \) exceeds that of \( [\text{H}^{+}] \).

The ion product constant of water (\( K_w \) ) is a key value used to interrelate these two concentrations. At 25 degrees Celsius, \( K_w \) is \(1 \times 10^{-14}\), which is the product of the concentrations of \( [\text{H}^{+}] \) and \( [\text{OH}^{-}] \) in pure water.

By understanding these basic principles, students can determine the nature of a solution by comparing the concentrations of \( [\text{H}^{+}] \) and \( [\text{OH}^{-}] \), aiding in characterizing the solution correctly.

Hydrogen Ion Concentration

Hydrogen ion concentration (\( [\text{H}^{+}] \) ) is a measure of the amount of hydrogen ions present in a solution. It not only defines the solution's acidity but also is integral to the pH calculation. The concentration of hydrogen ions is expressed in molarity, or moles per liter (\( M \) ), and it influences the chemical properties and reactions of the solution.

Enriching the step-by-step approach with practical tips, it's worth emphasizing that when conducting calculations involving \( [\text{H}^{+}] \) , attention must be paid to the fact that very small changes in hydrogen ion concentration can lead to significant changes in pH due to the logarithmic nature of the pH scale.

Additionally, when performing calculations using the ion product constant (\( K_w = [\text{H}^{+}][\text{OH}^{-}] \) ), be mindful of units and maintain consistency throughout the calculation to avoid errors. Verifying the final \( [\text{H}^{+}] \) against the nature of the solution, whether acidic or basic, serves as a check to ensure that the calculations have been executed correctly.