Problem 46 - Chemistry and Chemical Reactivity

Question

A 0.015 M solution of a base has a pH of 10.09 (a) What are the hydronium and hydroxide ion concentrations of this solution? (b) Is the base a strong base, a moderately weak base \(\left(K_{\mathrm{b}}\right.\) of about \(10^{-5}\) ), or a very weak base ( \(K_{\mathrm{b}}\) of about \(\left.10^{-10}\right) ?\)

Step-by-Step Solution

Answer

Hydronium: \(8.13 \times 10^{-11} \text{ M}\). Hydroxide: \(1.23 \times 10^{-4} \text{ M}\). Moderately weak base.

Step 1: Calculate the pOH

First, note that the pH of the solution is given as 10.09. The relationship between pH and pOH is given by the equation:\[ \text{pH} + \text{pOH} = 14 \]So, rearrange the equation to solve for pOH:\[ \text{pOH} = 14 - \text{pH} = 14 - 10.09 = 3.91 \]

Step 2: Determine Hydroxide Ion Concentration

Using the definition of pOH, calculate the concentration of hydroxide ions, \([\text{OH}^-]\), using the formula:\[ [\text{OH}^-] = 10^{- ext{pOH}} \]Substituting the value of pOH:\[ [\text{OH}^-] = 10^{-3.91} \approx 1.23 \times 10^{-4} \, \text{M} \]

Step 3: Determine Hydronium Ion Concentration

To find the hydronium ion concentration, \([\text{H}_3\text{O}^+]\), use the relation:\[ [\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14} \]We can rearrange to find \([\text{H}_3\text{O}^+]\):\[ [\text{H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{[\text{OH}^-]} = \frac{1.0 \times 10^{-14}}{1.23 \times 10^{-4}} \approx 8.13 \times 10^{-11} \, \text{M} \]

Step 4: Assess the Base Strength

Given the hydroxide ion concentration \([\text{OH}^-] \approx 1.23 \times 10^{-4} \, \text{M}\) and initial base concentration (0.015 M), we will calculate the value of \(K_b\). Use the formula for the base dissociation constant:\[ K_b = \frac{[\text{OH}^-]^2}{[\text{Base initial}] - [\text{OH}^-]} \]Substitute the concentrations into the equation:\[ K_b \approx \frac{(1.23 \times 10^{-4})^2}{0.015 - 1.23 \times 10^{-4}} \approx \frac{1.51 \times 10^{-8}}{0.01488} \approx 1.01 \times 10^{-6} \]This value corresponds to a moderately weak base because it is closer to \(10^{-5}\).

Key Concepts

pH Calculation

Understanding the concept of pH is essential in acid-base chemistry. The pH scale measures how acidic or basic a solution is, ranging from 0 to 14. A pH value below 7 indicates an acidic solution, while a pH value above 7 signifies a basic solution. Neutral solutions, like pure water, have a pH of 7.
Calculating pH involves understanding the negative logarithm of the hydronium ion concentration, \( [\text{H}_3\text{O}^+] \), in the solution. The formula to calculate pH is given by:

• \( \text{pH} = -\log [\text{H}_3\text{O}^+] \)

For the given problem, the pH of 10.09 indicates a basic solution since it is greater than 7. This also helps in the calculation of pOH, since pH and pOH sum up to 14, allowing us to find the concentration of hydroxide ions.

Hydronium Ion Concentration

Once we understand pH, the next step in acid-base chemistry is determining the hydronium ion concentration. Hydronium ions, \( \text{H}_3\text{O}^+ \), tell us the acidic nature of a solution.
For a basic solution with a given pH, you can calculate the concentration of these ions using the formula:

• \( [\text{H}_3\text{O}^+] = 10^{- \text{pH}} \)

In scenarios where you also know the hydroxide ion concentration, you can use the equation:

• \( [\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14} \)

From this, if \( [\text{OH}^-] = 1.23 \times 10^{-4} \), then \( [\text{H}_3\text{O}^+] \approx 8.13 \times 10^{-11} \), signifying a very low concentration in a basic solution.

Hydroxide Ion Concentration

The hydroxide ion concentration, represented as \( [\text{OH}^-] \), is vital for determining how basic a solution is. Solutions with higher concentrations of hydroxide ions have lower pOH values and are more basic.
Use the relationship:

• \( \text{pOH} = 14 - \text{pH} \)
• Calculate \( [\text{OH}^-] \) using: \( [\text{OH}^-] = 10^{-\text{pOH}} \)

For example, for a solution with a pH of 10.09, pOH is 3.91, making \( [\text{OH}^-] \approx 1.23 \times 10^{-4} \). Knowing this helps assess the solution's basicity and potential classification of the base strength.

Base Strength

Determining the strength of a base involves looking at its ability to dissociate in solution. Base strength is often characterized as strong, moderately weak, or very weak based on its dissociation constant \( K_b \).
A strong base dissociates completely in solution, while a weak base only partially dissociates. The base strength can be quantified by calculating \( K_b \):

• \( K_b = \frac{[\text{OH}^-]^2}{[\text{Base initial}] - [\text{OH}^-]} \)

In the given example, \( K_b \approx 1.01 \times 10^{-6} \), suggesting a moderately weak base since the \( K_b \) value is closer to \( 10^{-5} \) than \( 10^{-10} \). Understanding base strength helps in predicting a base's behavior in chemical reactions.

Dissociation Constant

The dissociation constant, denoted as \( K_b \) for bases, is a crucial parameter in acid-base chemistry. It measures the extent to which a base dissociates into hydroxide ions and its conjugate acid in solution.
If you know the concentration of all species in the solution, \( K_b \) can be calculated using:

• \( K_b = \frac{[\text{OH}^-] [\text{Conjugate Acid}]}{[\text{Base}]} \)

This formula provides a bigger insight:

• The higher the \( K_b \), the stronger the base.
• For a moderately weak base, \( K_b \) values often fall around \( 10^{-5} \).
• For very weak bases, these values tend to be closer to \( 10^{-10} \).

Thus, calculating \( K_b \) for a base like the one in this exercise is essential to identify its position on the strength spectrum.