What is the precent dissociation `(alpha)` of a `0.01` M HA solution? `(K_(a)=10^(-4))`

Options

A. `9.5%`
B. `1%`
C. `10.5%`
D. `17%`

Answer

To find the percent dissociation (α) of a 0.01 M HA solution given that \( K_a = 10^{-4} \), we can follow these steps:

### Step 1: Write the dissociation equation
The dissociation of the weak acid HA can be represented as:
\[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \]

### Step 2: Set up the initial concentrations
Initially, we have:
- \([HA] = 0.01 \, \text{M}\)
- \([H^+] = 0 \, \text{M}\)
- \([A^-] = 0 \, \text{M}\)

At equilibrium, if α is the degree of dissociation, the concentrations will be:
- \([HA] = 0.01 - \alpha\)
- \([H^+] = \alpha\)
- \([A^-] = \alpha\)

### Step 3: Write the expression for the dissociation constant \( K_a \)
The expression for the dissociation constant \( K_a \) is given by:
\[ K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]
Substituting the equilibrium concentrations, we get:
\[ K_a = \frac{\alpha \cdot \alpha}{0.01 - \alpha} = \frac{\alpha^2}{0.01 - \alpha} \]

### Step 4: Substitute the value of \( K_a \)
Given \( K_a = 10^{-4} \), we can substitute this into the equation:
\[ 10^{-4} = \frac{\alpha^2}{0.01 - \alpha} \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 10^{-4}(0.01 - \alpha) = \alpha^2 \]
Expanding this, we have:
\[ 10^{-6} - 10^{-4} \alpha = \alpha^2 \]
Rearranging further, we get:
\[ \alpha^2 + 10^{-4} \alpha - 10^{-6} = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( \alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 10^{-4}, c = -10^{-6} \):
1. Calculate the discriminant:
\[ b^2 - 4ac = (10^{-4})^2 - 4(1)(-10^{-6}) = 10^{-8} + 4 \times 10^{-6} = 4.01 \times 10^{-6} \]
2. Substitute into the quadratic formula:
\[ \alpha = \frac{-10^{-4} \pm \sqrt{4.01 \times 10^{-6}}}{2} \]
\[ \alpha = \frac{-10^{-4} \pm 0.0020025}{2} \]

Calculating the positive root:
\[ \alpha = \frac{-10^{-4} + 0.0020025}{2} \]
\[ \alpha \approx 0.00095 \]

### Step 7: Calculate percent dissociation
To find the percent dissociation, we multiply α by 100:
\[ \text{Percent dissociation} = \alpha \times 100 = 0.00095 \times 100 = 9.5\% \]

### Final Answer
The percent dissociation of the 0.01 M HA solution is **9.5%**.

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