`M_(x) and M_(y)` denote the atomic masses of the parent and the daughter nuclei respectively in a radioactive decay. The Q - value for a `beta-` decay is `Q_(1)` and that for a `beta^(+)` decay is `Q_(2)`. If `m_(e)` denotes the mass of an electrons, then which of the following statements is correct?

Options

A. `Q_(1)=(M_(x)-M_(y))c^(2)andQ_(2)=[M_(x)-M_(y)-2m_(e)]c^(2)`
B. `Q_(1)=(M_(x)-M_(y))c^(2)andQ_(2)=(M_(x)-M_(y))c^(2)`
C. `Q_(1)=(M_(x)-M_(y)-2m_(e))c^(2)andQ_(2)=(M_(x)-M_(y)+2c_(e))c^(2)`
D. `Q_(1)=(M_(x)-M_(y)+2m_(e))c^(2)andQ_(2)=(M_(x)-M_(y)+2m_(e))c^(2)`

Explanation

To solve the problem, we need to analyze the two types of beta decay: beta minus (β-) decay and beta plus (β+) decay, and understand how they affect the atomic masses of the parent and daughter nuclei.

### Step-by-Step Solution:

1. **Understanding Beta Minus Decay (β- decay)**:
- In β- decay, a neutron in the parent nucleus (with atomic mass \( M_x \) and atomic number \( Z \)) is converted into a proton. This process releases an electron (beta particle) and an antineutrino.
- The resulting daughter nucleus has one more proton and the same atomic mass, thus:
- Daughter nucleus atomic mass: \( M_y = M_x \)
- Daughter nucleus atomic number: \( Z + 1 \)
- The Q-value for this decay, which represents the energy released, can be expressed as:
\[ Q_1 = (M_x - M_y)c^2 \]

2. **Understanding Beta Plus Decay (β+ decay)**:
- In β+ decay, a proton in the parent nucleus is converted into a neutron, releasing a positron and an antineutrino.
- The resulting daughter nucleus has one less proton and the same atomic mass, thus:
- Daughter nucleus atomic mass: \( M_y = M_x \)
- Daughter nucleus atomic number: \( Z - 1 \)
- However, since a positron (which has a mass equivalent to that of an electron) is released, we must account for the mass of two electrons (one from the proton converting to a neutron and one from the positron being emitted) in the Q-value calculation:
\[ Q_2 = (M_x - M_y - 2m_e)c^2 \]

3. **Comparing the Q-values**:
- From the expressions derived:
- For β- decay: \( Q_1 = (M_x - M_y)c^2 \)
- For β+ decay: \( Q_2 = (M_x - M_y - 2m_e)c^2 \)
- Since \( M_y = M_x \) in both cases, we can rewrite:
- \( Q_1 = 0 \) (as the mass difference is zero)
- \( Q_2 = -2m_ec^2 \) (indicating energy is required to create the positron)

4. **Conclusion**:
- The Q-value for β- decay is greater than that for β+ decay because in β+ decay, energy is required to account for the mass of the emitted positron. Therefore, the correct statement is:
\[ Q_1 > Q_2 \]

### Final Answer:
The correct statement is \( Q_1 > Q_2 \).

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