A radioactive nucleus undergoes a series of decay according to the scheme. `Aoverset(alpha)toA_(1)overset(beta)toA_(2)overset(alpha)toA_(3)overset(gamma)toA_(4)`. If the mass number and atomic number of A are 180 and 72 respectively, then the mass number (x) and atomic and atomic number (y) of `A_(4)` are

Options

A. x= 172: y = 72
B. x = 174, y = 69
C. x = 176, y = 69
D. x = 172: y = 69

Answer

To solve the problem, we need to analyze the decay process of the radioactive nucleus step by step. The nucleus A has a mass number of 180 and an atomic number of 72. The decay sequence is as follows:

1. **Alpha Decay**: A → A1
2. **Beta Decay**: A1 → A2
3. **Alpha Decay**: A2 → A3
4. **Gamma Decay**: A3 → A4

### Step-by-Step Solution:

**Step 1: Initial Values**
- Mass number of A (A) = 180
- Atomic number of A (Z) = 72

**Step 2: First Decay (Alpha Decay)**
- In alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- Mass number of A1 = 180 - 4 = 176
- Atomic number of A1 = 72 - 2 = 70

**Step 3: Second Decay (Beta Decay)**
- In beta decay, the mass number remains the same, but the atomic number increases by 1.
- Mass number of A2 = 176 (remains the same)
- Atomic number of A2 = 70 + 1 = 71

**Step 4: Third Decay (Alpha Decay)**
- Again, in alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- Mass number of A3 = 176 - 4 = 172
- Atomic number of A3 = 71 - 2 = 69

**Step 5: Fourth Decay (Gamma Decay)**
- In gamma decay, both the mass number and atomic number remain unchanged.
- Mass number of A4 = 172 (remains the same)
- Atomic number of A4 = 69 (remains the same)

### Final Values:
- Mass number of A4 (x) = 172
- Atomic number of A4 (y) = 69

Thus, the mass number (x) and atomic number (y) of A4 are:
- x = 172
- y = 69