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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding Indicators**:
- Indicators are weak acids or bases that change color depending on their dissociated or undissociated forms. For example, the undissociated form (HIn) may be red, while the dissociated form (In^(-)) may be blue.

2. **Dissociation Constant**:
- The dissociation of the indicator can be represented as:
\[ HIn \rightleftharpoons H^+ + In^- \]
- The dissociation constant (K_in) for this equilibrium is given by:
\[ K_{in} = \frac{[H^+][In^-]}{[HIn]} \]

3. **Rearranging the Equation**:
- To express [H^+] in terms of the concentrations of HIn and In^(-), we rearrange the equation:
\[ [H^+] = K_{in} \cdot \frac{[HIn]}{[In^-]} \]

4. **Taking Logarithms**:
- Taking the logarithm of both sides gives:
\[ \log[H^+] = \log\left(K_{in} \cdot \frac{[HIn]}{[In^-]}\right) \]
- Using the properties of logarithms, this can be expanded to:
\[ \log[H^+] = \log K_{in} + \log[HIn] - \log[In^-] \]

5. **Relating pH and pK**:
- Recall that pH is defined as:
\[ pH = -\log[H^+] \]
- Thus, we can substitute this into our equation:
\[ -pH = \log K_{in} + \log[HIn] - \log[In^-] \]

6. **Rearranging to Find pH**:
- Rearranging gives:
\[ pH = -\log K_{in} + \log[In^-] - \log[HIn] \]
- This can be rewritten as:
\[ pH = pK_{in} + \log\left(\frac{[In^-]}{[HIn]}\right) \]

7. **Final Expression**:
- The final expression relating the pH of the solution to the ratio of the concentrations of the conjugate base and acid forms of the indicator is:
\[ pH = pK_{in} + \log\left(\frac{[In^-]}{[HIn]}\right) \]