If the dissociation constant of `5 xx 10^(-4)` M aqueous solution of diethylamine is `2.5 xx 10^(-5)`, its pH value is

Options

A. 8
B. 3.95
C. 10.05
D. 2

Answer

To find the pH of a 5 x 10^(-4) M aqueous solution of diethylamine with a dissociation constant (Kb) of 2.5 x 10^(-5), we can follow these steps:

### Step 1: Write the dissociation equation
Diethylamine (C2H5)2NH can be represented as a weak base that dissociates in water:
\[ (C_2H_5)_2NH + H_2O \rightleftharpoons (C_2H_5)_2NH_2^+ + OH^- \]

### Step 2: Set up the expression for Kb
The dissociation constant (Kb) for the reaction is given by:
\[ K_b = \frac{[(C_2H_5)_2NH_2^+][OH^-]}{[(C_2H_5)_2NH]} \]
Let the initial concentration of diethylamine be \(C = 5 \times 10^{-4}\) M. If \(\alpha\) is the degree of dissociation, then at equilibrium:
- Concentration of \((C_2H_5)_2NH_2^+\) = \(\alpha C\)
- Concentration of \(OH^-\) = \(\alpha C\)
- Concentration of \((C_2H_5)_2NH\) = \(C - \alpha C = C(1 - \alpha)\)

Substituting these into the Kb expression gives:
\[ K_b = \frac{(\alpha C)(\alpha C)}{C(1 - \alpha)} = \frac{\alpha^2 C}{1 - \alpha} \]

### Step 3: Substitute known values
We know \(K_b = 2.5 \times 10^{-5}\) and \(C = 5 \times 10^{-4}\):
\[ 2.5 \times 10^{-5} = \frac{\alpha^2 (5 \times 10^{-4})}{1 - \alpha} \]

### Step 4: Assume \(\alpha\) is small
Since \(K_b\) is small, we can assume \(\alpha\) is small compared to 1, thus \(1 - \alpha \approx 1\):
\[ 2.5 \times 10^{-5} \approx \alpha^2 (5 \times 10^{-4}) \]

### Step 5: Solve for \(\alpha\)
Rearranging the equation gives:
\[ \alpha^2 = \frac{2.5 \times 10^{-5}}{5 \times 10^{-4}} = 0.05 \]
Taking the square root:
\[ \alpha = \sqrt{0.05} \approx 0.2236 \]

### Step 6: Calculate the concentration of OH⁻ ions
Now, we can find the concentration of \(OH^-\):
\[ [OH^-] = \alpha C = 0.2236 \times 5 \times 10^{-4} \approx 1.118 \times 10^{-4} \text{ M} \]

### Step 7: Calculate pOH
Using the concentration of \(OH^-\), we find pOH:
\[ pOH = -\log[OH^-] = -\log(1.118 \times 10^{-4}) \approx 3.95 \]

### Step 8: Calculate pH
Finally, we can find pH using the relationship:
\[ pH + pOH = 14 \]
Thus,
\[ pH = 14 - pOH = 14 - 3.95 = 10.05 \]

### Final Answer
The pH of the solution is approximately **10.05**.