The order of acidic strength of the hydrogen atom `(H_(alpha), H_(beta), H_(gamma))` in the given molecule is :
`CH_(3)-underset(H_(beta))underset(|)(CH)-overset(O)overset(||)(C )-underset(H_(gamma))underset(|)(CH)-overset(O)overset(||)(C )-underset(H_(alpha))underset(|)(CH_(2))`

Options

A. (a)`H_(gamma)gt H_(alpha) gt H_(beta)`
B. (b)0`H_(gamma)gt H_(beta) gt H_(alpha)`
C. (c)`H_(alpha)gt H_(beta) gt H_(gamma)`
D. (d)`H_(beta)gt H_(alpha) gt H_(gamma)`

Answer

To determine the order of acidic strength of the hydrogen atoms \( H_{\alpha} \), \( H_{\beta} \), and \( H_{\gamma} \) in the given molecule, we will analyze the stability of the corresponding conjugate bases formed after the removal of each hydrogen atom.

### Step-by-Step Solution:

1. **Identify the Structure**:
The molecule can be represented as:
\[ CH_3 - CH - C(=O) - CH - C(=O) - CH_2 \]
Here, \( H_{\alpha} \) is attached to the terminal \( CH_2 \), \( H_{\beta} \) is attached to the \( CH \) adjacent to the first carbonyl group, and \( H_{\gamma} \) is attached to the \( CH \) adjacent to the second carbonyl group.

2. **Remove Each Hydrogen Atom**:
- **For \( H_{\alpha} \)**: Removing \( H_{\alpha} \) forms a negative charge on the \( CH_2 \) group:
\[ CH_3 - CH - C(=O) - CH - C(=O) - CH_2^{-} \]
The negative charge is stabilized by the adjacent carbonyl groups, which are electron-withdrawing.

- **For \( H_{\beta} \)**: Removing \( H_{\beta} \) forms a negative charge on the \( CH \) group:
\[ CH_3 - CH^{-} - C(=O) - CH - C(=O) - CH_2 \]
This conjugate base has one carbonyl group that stabilizes the negative charge through resonance.

- **For \( H_{\gamma} \)**: Removing \( H_{\gamma} \) forms a negative charge on the \( CH \) group:
\[ CH_3 - CH - C(=O) - CH^{-} - C(=O) - CH_2 \]
Similar to \( H_{\beta} \), this conjugate base has two carbonyl groups that stabilize the negative charge through resonance.

3. **Evaluate Stability of Conjugate Bases**:
- The stability of the conjugate base is crucial in determining acidity. The more stable the conjugate base, the stronger the acid.
- **Conjugate Base from \( H_{\gamma} \)**: Most stable due to resonance stabilization from two carbonyl groups.
- **Conjugate Base from \( H_{\alpha} \)**: Less stable than \( H_{\gamma} \) but more stable than \( H_{\beta} \) due to one carbonyl group.
- **Conjugate Base from \( H_{\beta} \)**: Least stable due to the presence of the methyl group which donates electrons, counteracting the electron-withdrawing effect of the carbonyl group.

4. **Order of Acidity**:
Based on the stability of the conjugate bases:
\[ H_{\gamma} > H_{\alpha} > H_{\beta} \]
Therefore, the order of acidic strength is:
\[ H_{\gamma} > H_{\alpha} > H_{\beta} \]

### Final Answer:
The order of acidic strength of the hydrogen atoms is:
\[ H_{\gamma} > H_{\alpha} > H_{\beta} \]