Problem 54 - Chemistry The Molecular Nature of Matter

Question

Calculate the molar concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) in solutions that have the following \(\mathrm{pOH}\) values. (a) 12.27 (b) 6.14 (c) 10.65 (d) 4.28 (e) 3.76

Step-by-Step Solution

Answer

\( [OH^-] \) and \( [H^+] \) for (a) are \(5.37 \times 10^{-13} M\) and \(5.37 \times 10^{-2} M\); for (b) \(7.24 \times 10^{-7} M\) and \(7.24 \times 10^{-8} M\); for (c) \(2.24 \times 10^{-11} M\) and \(2.24 \times 10^{-3} M\); for (d) \(5.25 \times 10^{-5} M\) and \(5.25 \times 10^{-10} M\); for (e) \(1.74 \times 10^{-4} M\) and \(1.74 \times 10^{-10} M\).

Step 1: Understand \(pOH\) and \(pH\)

The \(pOH\) of a solution is the negative logarithm to base 10 of the molar concentration of hydroxide ions (\(OH^-\)). The \(pH\) and \(pOH\) are related by the equation \(pH + pOH = 14\).

Step 2: Calculate the \(OH^-\) concentration

Use the formula \( [OH^-] = 10^{-pOH} \) to find the molar concentration of the hydroxide ions (\(OH^-\)).

Step 3: Determine the \(H^+\) concentration

Subtract the given \(pOH\) from 14 to find the \(pH\). Then use \( [H^+] = 10^{-pH} \) to determine the concentration of the hydronium ions (\(H^+\)).

Step 4: Calculate the concentrations for (a) \(pOH = 12.27\)

Use the formulas from Steps 2 and 3 to calculate \( [OH^-] = 10^{-12.27} \) and \( [H^+] = 10^{-(14 - 12.27)} \) for part (a).

Step 5: Calculate the concentrations for (b) \(pOH = 6.14\)

Repeat the same method as Step 4 for part (b): \( [OH^-] = 10^{-6.14} \) and \( [H^+] = 10^{-(14 - 6.14)} \) for part (b).

Step 6: Calculate the concentrations for (c) \(pOH = 10.65\)

Repeat the same method as Step 4 for part (c): \( [OH^-] = 10^{-10.65} \) and \( [H^+] = 10^{-(14 - 10.65)} \) for part (c).

Step 7: Calculate the concentrations for (d) \(pOH = 4.28\)

Repeat the same method as Step 4 for part (d): \( [OH^-] = 10^{-4.28} \) and \( [H^+] = 10^{-(14 - 4.28)} \) for part (d).

Step 8: Calculate the concentrations for (e) \(pOH = 3.76\)

Repeat the same method as Step 4 for part (e): \( [OH^-] = 10^{-3.76} \) and \( [H^+] = 10^{-(14 - 3.76)} \) for part (e).

Key Concepts

pH and pOH Relationship

When exploring the chemistry of acids and bases, two critical measurements for the acidity or basicity of a solution are its pH and pOH. These are interconnected, with the pH quantifying the concentration of hydronium ions (H^+) and the pOH dealing with hydroxide ions (OH^-). The sum of pH and pOH is always equal to about 14 at 25°C, a relationship that is fundamental to solving most acid-base problems.

This relationship means that if you know either the pH or pOH of a solution, you can easily find the other. For example, if a solution has a pH of 3, its pOH would be 14 - 3, which equals 11. This balance is essential for maintaining the neutrality of pure water, where the pH equals pOH at 7, demonstrating its neutral characteristic midway between acidic and basic. Understanding this crucial relationship is the stepping stone to mastering more complex acid-base chemistry.

Calculating Hydroxide Ion Concentration

Calculating the concentration of hydroxide ions (OH^-) is vital when working with basic solutions or neutralizing acids. The concentration is directly related to the solution's pOH, where pOH is defined as the negative base-10 logarithm of the hydroxide ion concentration. Here's how to do the calculation:

If you're given the pOH of a solution, to find the hydroxide ion concentration, you use the formula [OH^-] = 10^{-pOH}. So, given a pOH of 6.14, the concentration of OH^- is 10^{-6.14} M, with M standing for molarity, or moles per liter. This equation serves as a tool to transition between the logarithmic pOH scale and the actual concentration of OH^- ions, providing a quantitative insight into the solution's basicity.

Determining Hydronium Ion Concentration

To gauge the acidity of a solution, chemists calculate the concentration of hydronium ions (H^+), which is expressed in terms of pH. The pH is the negative logarithm of the H^+ ion concentration. When only the pOH is known, determining the H^+ concentration involves a two-step process. First, the corresponding pH is calculated using the connection between pH and pOH—subtract the pOH from 14. Then, convert the pH to the hydronium ion concentration using [H^+] = 10^{-pH}.

For example, for a solution with a pOH of 10.65, the pH is 14 - 10.65 = 3.35. Thus, the concentration of H^+ is 10^{-3.35} M. This conversion from pOH to H^+ concentration is fundamental for understanding the solution's acidity and is often applied in neutralization reactions and in predicting the behavior of acid-base equilibria.