A nuclear decay is possible if the mass of the parent nucleus exceeds the total mass of the decay particles. If M(A, Z) denotes the mass of a single neutral atom of an element with mass number A and atomic number Z , then the minimal condition that the `beta`- decay `X_(Z)^(A) rarr Y_(Z+ 1)^(A) + beta^(-) + bar(V)e` will occur is ( `m_(e)` denotes the mass of the `beta`− particle and the neutrino mass mv can be neglected)

Options

A. `M(A, Z) gt M(A, Z+1) + m_(e)`
B. `M(A, Z) gt M(A, Z+1)`
C. `M(A, Z) gt M(A, Z+1)+ Zm_(e)`
D. `M(A, Z) gt M(A, Z +1)- m_(e)`

Explanation

To determine the minimal condition for the beta decay process \( X_{Z}^{A} \rightarrow Y_{Z+1}^{A} + \beta^{-} + \bar{V} \), we need to analyze the mass-energy relationship involved in nuclear decay.

### Step-by-Step Solution:

1. **Identify the Masses Involved:**
- Let \( M(X) \) be the mass of the parent nucleus \( X \) with atomic number \( Z \) and mass number \( A \).
- The mass of the daughter nucleus \( Y \) will be \( M(Y) \), which has atomic number \( Z+1 \) and the same mass number \( A \).
- The mass of the emitted beta particle (electron) is denoted as \( m_e \).
- The mass of the neutrino \( \bar{V} \) is negligible, so we can ignore it in our calculations.

2. **Write the Mass-Energy Condition:**
- For the decay to occur, the mass of the parent nucleus must be greater than the total mass of the decay products:
\[ M(X) > M(Y) + m_e \]

3. **Substituting the Masses:**
- We know that the mass of the daughter nucleus \( Y \) is \( M(Y) \) and the mass of the beta particle is \( m_e \). Therefore, we can rewrite the inequality:
\[ M(A, Z) > M(A, Z+1) + m_e \]

4. **Analyzing the Mass Numbers:**
- Since the mass number \( A \) remains the same before and after the decay, we can denote the masses as follows:
\[ M(A, Z) \text{ (mass of parent)} \quad \text{and} \quad M(A, Z+1) \text{ (mass of daughter)} \]

5. **Conclusion:**
- The minimal condition for beta decay to occur is that the mass of the parent nucleus \( M(A, Z) \) must exceed the sum of the mass of the daughter nucleus \( M(A, Z+1) \) and the mass of the beta particle \( m_e \):
\[ M(A, Z) > M(A, Z+1) + m_e \]