To solve this problem, we need to calculate the base dissociation constant ($K_b$) for diethylamine, $(C_2H_5)_2NH$, using the given information. Let's go through the steps:

Given:

1. Molarity of the solution ($[B]$) = $0.423 \, \text{M}$
2. pH of the solution = $12.218$
3. $K_b$ formula: [ K_b = \frac{[OH^-]^2}{[B] - [OH^-]} ]

Steps:

1. Find $pOH$: Using the relationship: $pH + pOH = 14$ $pOH = 14 - 12.218 = 1.782$

2. Calculate $[OH^-]$: Use the formula: $[OH^-] = 10^{-\text{pOH}}$ Substituting $pOH = 1.782$: $[OH^-] = 10^{-1.782} \approx 1.65 \times 10^{-2} \, \text{M}$

3. Determine $K_b$: Plugging values into the $K_b$ formula: [ K_b = \frac{[OH^-]^2}{[B] - [OH^-]} ] Substituting: $K_b = \frac{(1.65 \times 10^{-2})^2}{0.423 - 1.65 \times 10^{-2}}$ Calculate step by step:

   • $(1.65 \times 10^{-2})^2 = 2.72 \times 10^{-4}$
   • $0.423 - 1.65 \times 10^{-2} = 0.406$
   • $K_b = \frac{2.72 \times 10^{-4}}{0.406} \approx 6.70 \times 10^{-4}$

Final Answer:

$K_b (\text{experiment}) \approx 6.70 \times 10^{-4}$

Would you like further clarification or detailed steps?

Related Questions:

1. What is the significance of $K_b$ in determining the strength of a base?
2. How does the pH relate to the dissociation of a base in water?
3. How does temperature affect the value of $K_b$?
4. What other experimental methods could determine $K_b$?
5. Can $K_b$ be used to calculate the pKa of the conjugate acid of diethylamine?

Tip:

When solving problems involving $K_b$, always ensure that the concentration of the base after dissociation ($[B] - [OH^-]$) remains non-negative for meaningful results.