When 0.05 M dimethyl amine is dissolve in 0.1 M NaOH solution then the percentage dissociation of dimethyl amine is : `(K_(b))_((CH_(3))_(2)NH)=5xx10^(-4)`

Options

A. `5xx10^(-5)`
B. `5xx10^(-3)`
C. `5xx10^(-1)`
D. `5xx10^(-2)`

Explanation

To solve the problem of calculating the percentage dissociation of dimethylamine when it is dissolved in a NaOH solution, we will follow these steps:

### Step 1: Understand the given information
- Concentration of dimethylamine (C) = 0.05 M
- Concentration of NaOH = 0.1 M
- \( K_b \) for dimethylamine = \( 5 \times 10^{-4} \)

### Step 2: Identify the dissociation of dimethylamine
Dimethylamine (\( (CH_3)_2NH \)) is a weak base and will dissociate in water as follows:
\[ (CH_3)_2NH + H_2O \rightleftharpoons (CH_3)_2NH_2^+ + OH^- \]

### Step 3: Set up the equilibrium expression using \( K_b \)
The expression for the base dissociation constant \( K_b \) is given by:
\[ K_b = \frac{[OH^-][(CH_3)_2NH_2^+]}{[(CH_3)_2NH]} \]

### Step 4: Determine the concentrations at equilibrium
Let \( \alpha \) be the degree of dissociation of dimethylamine. At equilibrium:
- The concentration of \( (CH_3)_2NH \) will be \( C - C\alpha = 0.05 - 0.05\alpha \)
- The concentration of \( OH^- \) from NaOH is 0.1 M, and the contribution from dimethylamine is negligible compared to this.
- The concentration of \( (CH_3)_2NH_2^+ \) will be \( C\alpha = 0.05\alpha \)

Substituting these into the \( K_b \) expression:
\[ K_b = \frac{(0.1)(0.05\alpha)}{(0.05 - 0.05\alpha)} \]

### Step 5: Simplify the equation
Since \( \alpha \) is small compared to 1, we can approximate:
\[ 0.05 - 0.05\alpha \approx 0.05 \]
Thus, the equation simplifies to:
\[ K_b = \frac{(0.1)(0.05\alpha)}{0.05} \]
This simplifies further to:
\[ K_b = 0.1\alpha \]

### Step 6: Substitute the value of \( K_b \) and solve for \( \alpha \)
Substituting \( K_b = 5 \times 10^{-4} \):
\[ 5 \times 10^{-4} = 0.1\alpha \]
\[ \alpha = \frac{5 \times 10^{-4}}{0.1} = 5 \times 10^{-3} \]

### Step 7: Calculate the percentage dissociation
The percentage dissociation is given by:
\[ \text{Percentage dissociation} = \alpha \times 100 = (5 \times 10^{-3}) \times 100 = 0.5\% \]

### Final Answer
The percentage dissociation of dimethylamine is **0.5%**.

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