An acid base indicator has `K_(a)=1.0xx10^(-5)` the acid form of the indicator is red and the basic form is blue. Calculate the pH change required to change the colour of the indicator from 80% red to 80% blue.

Options

A. 1.2
B. 0.8
C. 0.2
D. 1.4

Explanation

To solve the problem of calculating the pH change required to change the color of the indicator from 80% red to 80% blue, we will follow these steps:

### Step 1: Understand the Indicator System
The indicator can be represented as:
- Acid form (red): \( \text{HIn} \)
- Base form (blue): \( \text{In}^- \)

Given the dissociation:
\[ \text{HIn} \rightleftharpoons \text{H}^+ + \text{In}^- \]

### Step 2: Set Up the Initial Conditions
When the indicator is 80% red (acid form), this means:
- 80% of the total concentration is \( \text{HIn} \)
- 20% of the total concentration is \( \text{In}^- \)

Let’s assume the total concentration of the indicator is \( C \). Therefore:
- \( [\text{HIn}] = 0.8C \)
- \( [\text{In}^-] = 0.2C \)

### Step 3: Calculate the H\(^+\) Concentration for 80% Red
Using the equilibrium expression for \( K_a \):
\[ K_a = \frac{[\text{H}^+][\text{In}^-]}{[\text{HIn}]} \]

Substituting the known values:
\[ 1.0 \times 10^{-5} = \frac{[\text{H}^+](0.2C)}{(0.8C)} \]

This simplifies to:
\[ 1.0 \times 10^{-5} = \frac{[\text{H}^+]}{4} \]
\[ [\text{H}^+] = 4.0 \times 10^{-5} \, \text{M} \]

### Step 4: Calculate the pH for 80% Red
Using the formula for pH:
\[ \text{pH} = -\log[\text{H}^+] \]
\[ \text{pH} = -\log(4.0 \times 10^{-5}) \]
Calculating this gives:
\[ \text{pH} \approx 4.4 \]

### Step 5: Set Up the Conditions for 80% Blue
When the indicator is 80% blue (basic form), this means:
- 80% of the total concentration is \( \text{In}^- \)
- 20% of the total concentration is \( \text{HIn} \)

Thus:
- \( [\text{In}^-] = 0.8C \)
- \( [\text{HIn}] = 0.2C \)

### Step 6: Calculate the H\(^+\) Concentration for 80% Blue
Using the same equilibrium expression:
\[ 1.0 \times 10^{-5} = \frac{[\text{H}^+](0.8C)}{(0.2C)} \]

This simplifies to:
\[ 1.0 \times 10^{-5} = 4[\text{H}^+] \]
\[ [\text{H}^+] = 2.5 \times 10^{-6} \, \text{M} \]

### Step 7: Calculate the pH for 80% Blue
Using the formula for pH:
\[ \text{pH} = -\log(2.5 \times 10^{-6}) \]
Calculating this gives:
\[ \text{pH} \approx 5.6 \]

### Step 8: Calculate the pH Change
The change in pH is:
\[ \Delta \text{pH} = \text{pH}_{\text{blue}} - \text{pH}_{\text{red}} \]
\[ \Delta \text{pH} = 5.6 - 4.4 = 1.2 \]

### Final Answer
The pH change required to change the color of the indicator from 80% red to 80% blue is **1.2**.

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