Hin is an acidic indicator `(K_(Ind) =10^(-7))` which dissociates into aqueous acidic solution of `30mL` of `0.05M H_(3)PO_(4) (K_(1) = 10^(-3), K_(2) = 10^(-7), K_(3) = 10^(-13))`
If Hin and `Ind^(Theta)` posses colour `P` and `Q`, respectively, and concentration of `HIn` is `120` times than that of `Ind^(Theta)`. colour `Q` predominates over `P` when concnetration of `Ind^(Theta)` is `127` times of `HIn`.
What is the `pH` range of the indicator.

Options

A. `4.896 to 9.0792`
B. `4.896 to 8.0792`
C. `4.896 to 7.0792`
D. `4.896 to 6.0792`

Explanation

To find the pH range of the indicator Hin, we will follow these steps:

### Step 1: Understand the dissociation of the indicator
The indicator Hin dissociates according to the following equilibrium:
\[ \text{Hin} \rightleftharpoons \text{H}^+ + \text{Ind}^- \]
The dissociation constant \( K_{Ind} \) is given as \( 10^{-7} \).

### Step 2: Set up the equations for the two cases
We have two scenarios based on the dominance of colors P and Q:
1. **Color P dominates**: This occurs when the concentration of Hin is 120 times that of Ind⁻.
\[ \frac{[\text{Ind}^-]}{[\text{Hin}]} = \frac{1}{120} \]

2. **Color Q dominates**: This occurs when the concentration of Ind⁻ is 127 times that of Hin.
\[ \frac{[\text{Ind}^-]}{[\text{Hin}]} = 127 \]

### Step 3: Calculate the pH for the first case (Color P dominates)
Using the Henderson-Hasselbalch equation:
\[ \text{pH} = \text{p}K_{Ind} + \log \left( \frac{[\text{Ind}^-]}{[\text{Hin}]} \right) \]
Substituting the ratio for the first case:
\[ \text{pH}_1 = \text{p}K_{Ind} + \log \left( \frac{1}{120} \right) \]
Given \( K_{Ind} = 10^{-7} \), we find:
\[ \text{p}K_{Ind} = -\log(10^{-7}) = 7 \]
Thus,
\[ \text{pH}_1 = 7 + \log \left( \frac{1}{120} \right) = 7 - 2.08 \approx 4.92 \]

### Step 4: Calculate the pH for the second case (Color Q dominates)
Using the same equation:
\[ \text{pH}_2 = \text{p}K_{Ind} + \log \left( 127 \right) \]
Calculating:
\[ \text{pH}_2 = 7 + \log(127) \approx 7 + 2.103 = 9.10 \]

### Step 5: Determine the pH range of the indicator
The pH range of the indicator Hin is between the two calculated pH values:
\[ \text{pH range} = [4.92, 9.10] \]

### Final Answer
The pH range of the indicator is approximately **4.92 to 9.10**.

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