An acid-base indicator has a `K_(a)` of `3.0 xx 10^(-5)`. The acid form of the indicator is red and the basic form is blue. (a) By how much must the `pH` change in order to change the indicator from `75%` red to `75%` blue?

Options

A. `8xx10^(-5)`M
B. `9xx10^(-5)`M
C. `1xx10^(-5)`M
D. `3xx10^(-4)`M

Answer

To solve the problem of determining how much the pH must change to switch the indicator from 75% red (acidic form) to 75% blue (basic form), we will follow these steps:

### Step 1: Understand the Initial and Final Conditions
- **Initial Condition**: 75% red (acid form) and 25% blue (basic form).
- **Final Condition**: 25% red (acid form) and 75% blue (basic form).

### Step 2: Calculate the Initial pH
Using the formula for the pH of a weak acid:
\[ \text{pH}_1 = \text{pK}_a + \log\left(\frac{[\text{Base}]}{[\text{Acid}]}\right) \]

Where:
- \( K_a = 3.0 \times 10^{-5} \)
- \( \text{pK}_a = -\log(K_a) = -\log(3.0 \times 10^{-5}) \)

Calculating \( \text{pK}_a \):
\[ \text{pK}_a \approx 4.52 \]

Now substituting the concentrations:
- Base = 25% = 0.25
- Acid = 75% = 0.75

Calculating \( \text{pH}_1 \):
\[ \text{pH}_1 = 4.52 + \log\left(\frac{0.25}{0.75}\right) \]
\[ \text{pH}_1 = 4.52 + \log(0.3333) \approx 4.52 - 0.477 = 4.043 \]

### Step 3: Calculate the Final pH
Now, we calculate the final pH when the indicator is 25% red and 75% blue:
- Base = 75% = 0.75
- Acid = 25% = 0.25

Calculating \( \text{pH}_2 \):
\[ \text{pH}_2 = 4.52 + \log\left(\frac{0.75}{0.25}\right) \]
\[ \text{pH}_2 = 4.52 + \log(3) \approx 4.52 + 0.477 = 4.997 \]

### Step 4: Calculate the Change in pH
Now, we find the change in pH:
\[ \Delta \text{pH} = \text{pH}_2 - \text{pH}_1 \]
\[ \Delta \text{pH} = 4.997 - 4.043 = 0.954 \]

### Final Answer
The pH must change by approximately **0.95** to switch the indicator from 75% red to 75% blue.

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