The rapid change of `pH` near the stoichiometric point of an acid-base titration is the basic of indicator detection. `pH` of the solution is related to the ratio of the concentration of conjugate acid `(Hin)` and base `(In^(-))` forms of the indicator by the expression

Options

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

Explanation

To derive the expression relating the pH of a solution to the concentrations of the conjugate acid (HIn) and base (In^(-)) forms of an indicator, we follow these steps:

### Step-by-Step Solution:

1. **Dissociation of the Indicator**:
The indicator (HIn) dissociates in solution as follows:
\[ \text{HIn} \rightleftharpoons \text{H}^+ + \text{In}^- \]

2. **Equilibrium Constant Expression**:
The equilibrium constant (K_in) for this dissociation can be expressed as:
\[ K_{in} = \frac{[\text{H}^+][\text{In}^-]}{[\text{HIn}]} \]

3. **Taking Logarithm**:
To relate this to pH, we take the logarithm of both sides:
\[ \log K_{in} = \log \left( \frac{[\text{H}^+][\text{In}^-]}{[\text{HIn}]} \right) \]

4. **Applying Logarithmic Properties**:
Using the properties of logarithms, we can separate the terms:
\[ \log K_{in} = \log [\text{H}^+] + \log [\text{In}^-] - \log [\text{HIn}] \]

5. **Substituting pH**:
Since \(-\log [\text{H}^+] = \text{pH}\), we can rewrite the equation:
\[ \log K_{in} = -\text{pH} + \log [\text{In}^-] - \log [\text{HIn}] \]

6. **Rearranging the Equation**:
Rearranging gives us:
\[ \text{pH} = \log [\text{In}^-] - \log [\text{HIn}] + \log K_{in} \]

7. **Final Expression**:
This can be simplified to:
\[ \text{pH} = \text{pK}_{in} + \log \left( \frac{[\text{In}^-]}{[\text{HIn}]} \right) \]

### Conclusion:
Thus, the expression relating the pH of the solution to the ratio of the concentrations of the conjugate acid and base forms of the indicator is:
\[ \log \left( \frac{[\text{In}^-]}{[\text{HIn}]} \right) = \text{pH} - \text{pK}_{in} \]