Which of the following compounds would have the smallest value for `pK_(a)`?

Options

A. `CHF_(2)CH_(2)CH_(2)COOH`
B. `CH_(3)CH_(2)CF_(2)COOH`
C. `CH_(2)FCHFCH_(2)COOH`
D. `CH_(3)CF_(2)CH_(2)COOH`

Explanation

To determine which of the given compounds has the smallest value for \( pK_a \), we need to analyze the acidic strength of each compound based on the position of the electron-withdrawing groups relative to the functional group.

### Step-by-Step Solution:

1. **Understanding \( pK_a \)**:
- The \( pK_a \) value is a measure of the acidity of a compound. A lower \( pK_a \) value indicates a stronger acid. Therefore, we are looking for the compound with the highest acidity.

2. **Identifying the Functional Group**:
- In this case, the functional group we are focusing on is the carboxylic acid group (-COOH).

3. **Analyzing the Compounds**:
- We need to look at the compounds provided in the options and identify where the electron-withdrawing groups (EWGs) are located in relation to the carboxylic acid group.
- The presence of electron-withdrawing groups increases the acidity of carboxylic acids by stabilizing the negative charge on the conjugate base after deprotonation.

4. **Position of the Electron-Withdrawing Groups**:
- For each option, identify the position of the electron-withdrawing groups (like -F or -Cl) relative to the carboxylic acid group.
- The closer the EWG is to the carboxylic acid group, the stronger the acid will be.

5. **Comparing the Options**:
- **Option 1**: EWG at the 3rd position from the -COOH group.
- **Option 2**: EWG at the 1st position from the -COOH group (closest).
- **Option 3**: EWG at the 2nd position from the -COOH group.
- **Option 4**: EWG at the 2nd position from the -COOH group (similar to option 3 but different EWG).

6. **Determining the Strongest Acid**:
- Since option 2 has the EWG closest to the carboxylic acid group, it will have the highest acidic strength.
- Therefore, option 2 will have the smallest \( pK_a \) value.

### Conclusion:
The compound with the smallest value for \( pK_a \) is **Option 2**.