`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)` and `Q_(2)=(M_(X)-M_(Y)-2m_(e))c^(2)`
B. `Q_(1)=(M_(X)-M_(Y))c^(2)` and `Q_(2)=(M_(X)-M_(Y))c^(2)`
C. `Q_(1)=(M_(X)-M_(Y)-2m_(e))c^(2)` and `Q_(2)=(M_(X)-M_(Y)+2m_(e))c^(2)`
D. `Q_(1)=(M_(X)-M_(Y)+2m_(e))c^(2)` and `Q_(2)=(M_(X)-M_(Y)+2m_(e))c^(2)`

Explanation

To solve the problem regarding the Q-values for beta decay processes, we need to analyze the relationship between the atomic masses of the parent and daughter nuclei and the mass of the electron involved in the decay processes.

### Step-by-Step Solution:

1. **Understanding Beta Decay**:
- In beta decay, a neutron in the nucleus is converted into a proton, emitting an electron (beta minus decay) and an antineutrino.
- In beta plus decay, a proton is converted into a neutron, emitting a positron (beta plus decay) and a neutrino.

2. **Defining Q-values**:
- The Q-value for a decay process is defined as the difference in mass-energy between the parent and daughter nuclei, which can be expressed as:
\[ Q = (M_{parent} - M_{daughter} - m_{e})c^2 \]
- For beta minus decay, the equation becomes:
\[ Q_1 = (M_x - M_y - m_e)c^2 \]
- For beta plus decay, the equation becomes:
\[ Q_2 = (M_x - M_y + m_e)c^2 \]

3. **Relating Q-values**:
- From the above equations, we can express the Q-values:
- For beta minus decay:
\[ Q_1 = (M_x - M_y - m_e)c^2 \]
- For beta plus decay:
\[ Q_2 = (M_x - M_y + m_e)c^2 \]

4. **Finding the Relationship**:
- To find the relationship between \(Q_1\) and \(Q_2\), we can rearrange both equations:
- From \(Q_1\):
\[ M_x - M_y = \frac{Q_1}{c^2} + m_e \]
- From \(Q_2\):
\[ M_x - M_y = \frac{Q_2}{c^2} - m_e \]

5. **Equating the Two Expressions**:
- Since both expressions equal \(M_x - M_y\), we can set them equal to each other:
\[ \frac{Q_1}{c^2} + m_e = \frac{Q_2}{c^2} - m_e \]

6. **Solving for Q-values**:
- Rearranging gives:
\[ Q_1 + 2m_e c^2 = Q_2 \]
- Thus, we derive:
\[ Q_2 = Q_1 + 2m_e c^2 \]

### Conclusion:
The correct relationship between the Q-values for beta minus and beta plus decay is:
\[ Q_2 = Q_1 + 2m_e c^2 \]