The rapid change of pH near the stoichiometric point of an acid-base titration is the basis of indicator detection. pH of the solution is related to ratio of the concentration of the conjugate acid (Hin) and base `[H^(+)] = 10^(-7) + 10^(-8) = 10^(-7)[1+0.1]` forms of the indicator by the expression

Options

A. `log.([HIn])/([In^(-)]) = pH - pK_(In)`
B. `log.([In^(-)])/([HIn]) = pH - pK_(In)`
C. `log.([In^(-)])/([HIn]) = pK_(In) - pH`
D. `log.([HIn])/([In^(-)]) = pK_(In) - pH`

Answer

To solve the problem regarding the relationship between pH and the ratio of concentrations of the conjugate acid (HIn) and base (In⁻) forms of an indicator during an acid-base titration, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the pH Relationship**:
The pH of the solution can be related to the concentrations of the conjugate acid (HIn) and the conjugate base (In⁻) using the Henderson-Hasselbalch equation:
\[ \text{pH} = \text{pK}_a + \log\left(\frac{[\text{In}^-]}{[\text{HIn}]}\right) \]

2. **Identify the Components**:
- Here, \([\text{In}^-]\) is the concentration of the base form of the indicator.
- \([\text{HIn}]\) is the concentration of the acid form of the indicator.
- \(\text{pK}_a\) is the negative logarithm of the acid dissociation constant for the indicator.

3. **Substituting Known Values**:
- In the problem, we are given that \([H^+] = 10^{-7} + 10^{-8} = 1.1 \times 10^{-7}\).
- This implies that the pH can be calculated as:
\[ \text{pH} = -\log(1.1 \times 10^{-7}) \]

4. **Calculating pH**:
- To find the pH, first calculate:
\[ \text{pH} \approx 7 + \log(1.1) \approx 7 + 0.041 = 7.041 \]

5. **Rearranging the Henderson-Hasselbalch Equation**:
- Rearranging the equation gives us:
\[ \log\left(\frac{[\text{In}^-]}{[\text{HIn}]}\right) = \text{pH} - \text{pK}_a \]

6. **Final Expression**:
- Thus, we can express the relationship as:
\[ \frac{[\text{In}^-]}{[\text{HIn}]} = 10^{(\text{pH} - \text{pK}_a)} \]

### Conclusion
The final expression relates the pH of the solution to the ratio of the concentrations of the conjugate base and acid forms of the indicator, which is crucial for understanding how indicators function in acid-base titrations.