An acid-base indicator has `K_(a) = 10^(-5)`. The acid form of the indicator is red and basic form is blue. Which of the following is//are correct?

Options

A. At `pH = 4.52`, solution is red
B. At `pH - 5.47`, solution is blue.
C. At `pH =6`, solution is `75%` red
D. At `pH = 8`, solution is `75%` blue.

Explanation

To solve the question regarding the acid-base indicator with a given \( K_a = 10^{-5} \), we will follow these steps:

### Step 1: Write the dissociation reaction of the indicator
The indicator can be represented as:
\[ \text{H-I-N} \rightleftharpoons \text{H}^+ + \text{I-N}^- \]
Where:
- \( \text{H-I-N} \) is the acidic form (red)
- \( \text{I-N}^- \) is the basic form (blue)

### Step 2: Calculate the \( pK_a \)
Using the formula:
\[ pK_a = -\log(K_a) \]
Substituting the given \( K_a \):
\[ pK_a = -\log(10^{-5}) = 5 \]

### Step 3: Analyze the color of the solution based on pH
- If \( \text{pH} < pK_a \) (i.e., pH < 5), the acidic form predominates, and the solution will be red.
- If \( \text{pH} > pK_a \) (i.e., pH > 5), the basic form predominates, and the solution will be blue.

### Step 4: Evaluate the options given in the question

#### Option A: pH 4.52 (Solution is red)
- \( 4.52 < 5 \) (pH < pK_a)
- Correct: The solution is red.

#### Option B: pH 5.47 (Solution is blue)
- \( 5.47 > 5 \) (pH > pK_a)
- Correct: The solution is blue.

#### Option C: pH 6 (Solution is 75% red)
- 75% red means 75% acidic and 25% basic.
- Using the Henderson-Hasselbalch equation:
\[ \text{pH} = pK_a + \log\left(\frac{\text{base}}{\text{acid}}\right) \]
\[ \text{pH} = 5 + \log\left(\frac{0.25}{0.75}\right) \]
Calculating:
\[ \text{pH} = 5 + \log\left(\frac{1}{3}\right) \approx 5 - 0.477 = 4.52 \]
- However, the question states pH is 6, which is incorrect.

#### Option D: pH 8 (Solution is 75% blue)
- 75% blue means 75% basic and 25% acidic.
- Using the Henderson-Hasselbalch equation:
\[ \text{pH} = 5 + \log\left(\frac{0.75}{0.25}\right) \]
Calculating:
\[ \text{pH} = 5 + \log(3) \approx 5 + 0.477 = 5.477 \]
- However, the question states pH is 8, which is incorrect.

### Conclusion
The correct options are:
- Option A: Correct
- Option B: Correct
- Option C: Incorrect
- Option D: Incorrect

### Final Answer
The correct statements are Option A and Option B.

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