Arrange the following in increasing order of their `pK_a` values.
(x) `CH_3-underset(O)underset(||)overset(O)overset(||)S-O-H` (y) `CH_3-overset(O)overset(||)C-O-H` (z) `CH_3-OH`

Options

A. `y lt x lt z`
B. `x lt y lt z`
C. `y lt z lt x`
D. `x lt z lt y`

Explanation

To arrange the compounds in increasing order of their pKa values, we first need to understand the acidic strength of each compound. The pKa value is inversely related to the acidity of a compound; a lower pKa value indicates a stronger acid.

### Step-by-Step Solution:

1. **Identify the Compounds**:
- (x) `CH₃-CO-S-O-H`
- (y) `CH₃-COH`
- (z) `CH₃-OH`

2. **Analyze the Acidic Strength**:
- For each compound, we need to determine the stability of the conjugate base formed when the acidic hydrogen (H⁺) is removed.

3. **Compound (x) Analysis**:
- When H⁺ is removed from (x), it forms a conjugate base that can be resonance stabilized. The negative charge on the conjugate base can be delocalized over multiple atoms, leading to three resonance structures. This stabilization increases the acidic strength of (x).

4. **Compound (y) Analysis**:
- When H⁺ is removed from (y), it also forms a conjugate base that is resonance stabilized, but it can only form two resonance structures. This means that while it is still stabilized by resonance, it is less stable than the conjugate base of (x).

5. **Compound (z) Analysis**:
- When H⁺ is removed from (z), it forms a conjugate base that is not resonance stabilized. Additionally, the presence of the electron-donating methyl group (CH₃) destabilizes the negative charge further. Therefore, (z) is the least acidic of the three.

6. **Rank the Acidity**:
- Based on the analysis:
- (z) `CH₃-OH` is the least acidic (highest pKa).
- (y) `CH₃-COH` is more acidic than (z) but less than (x).
- (x) `CH₃-CO-S-O-H` is the most acidic (lowest pKa).

7. **Order of Increasing pKa Values**:
- Therefore, the increasing order of pKa values is:
- pKa(z) > pKa(y) > pKa(x)
- In terms of the compounds, this can be written as:
- (z) > (y) > (x)

### Final Answer:
The increasing order of pKa values is: **(z) `CH₃-OH` > (y) `CH₃-COH` > (x) `CH₃-CO-S-O-H`**.