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. `H_(gamma)gt H_(alpha) gt H_(beta)`
B. `H_(gamma)gt H_(beta) gt H_(alpha)`
C. `H_(alpha)gt H_(beta) gt H_(gamma)`
D. `H_(beta)gt H_(alpha) gt H_(gamma)`

Answer

To determine the order of acidic strength of the hydrogen atoms \( H_{\alpha}, H_{\beta}, H_{\gamma} \) in the given molecule, we will analyze the stability of the resulting anions formed when each hydrogen is removed.

### 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 carbon adjacent to the carbonyl group, \( H_{\beta} \) is attached to the carbon between the two carbonyl groups, and \( H_{\gamma} \) is attached to the carbon adjacent to the second carbonyl group.

2. **Remove Each Hydrogen**:
- **Removing \( H_{\alpha} \)**:
When \( H_{\alpha} \) is removed, we form a carbanion:
\[ CH_3 - C^{-} - C(=O) - CH - C(=O) - CH_2 \]
The negative charge is on a carbon adjacent to one carbonyl group (C=O), which is an electron-withdrawing group. This increases the stability of the carbanion.

- **Removing \( H_{\beta} \)**:
When \( H_{\beta} \) is removed, we form:
\[ CH_3 - CH - C(=O) - C^{-} - C(=O) - CH_2 \]
Here, the negative charge is on a carbon between two carbonyl groups. The presence of two electron-withdrawing groups (C=O) stabilizes the carbanion even more than in the case of \( H_{\alpha} \).

- **Removing \( H_{\gamma} \)**:
When \( H_{\gamma} \) is removed, we form:
\[ CH_3 - CH - C(=O) - CH - C^{-} - CH_2 \]
The negative charge is on a carbon adjacent to one carbonyl group, similar to \( H_{\alpha} \), but here the carbon is also adjacent to a carbon that has a positive inductive effect due to the carbonyl group. This makes the carbanion less stable compared to the one formed by removing \( H_{\beta} \).

3. **Compare the Stability of Carbanions**:
- The carbanion formed by removing \( H_{\beta} \) is the most stable due to the presence of two electron-withdrawing carbonyl groups.
- The carbanion formed by removing \( H_{\alpha} \) is less stable than that of \( H_{\beta} \) but more stable than that of \( H_{\gamma} \).
- The carbanion formed by removing \( H_{\gamma} \) is the least stable due to the presence of only one electron-withdrawing group.

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

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