A nucleus undergoes a series of decay according to the scheme `A overset(alpha)(rarr) B overset(beta)(rarr)C overset(alpha)(rarr)D overset(gamma)(rarr) E`
Atomic number and mass number of E are 69 and 172

Options

A. Atomic number of A is 72
B. Mass number of B is 176
C. Atomic number of D is 69
D. Atomic number of C is 69

Answer

To solve the problem, we need to analyze the decay chain step by step, keeping track of the atomic number (Z) and mass number (A) at each stage of the decay process.

### Step-by-Step Solution:

1. **Identify the Decay Process:**
- The nucleus A undergoes a series of decays:
- A → B (alpha decay)
- B → C (beta decay)
- C → D (alpha decay)
- D → E (gamma decay)

2. **Understand the Changes in Atomic and Mass Numbers:**
- **Alpha Decay:**
- Mass number decreases by 4 (A → A-4)
- Atomic number decreases by 2 (Z → Z-2)
- **Beta Decay:**
- Mass number remains unchanged (A → A)
- Atomic number increases by 1 (Z → Z+1)
- **Gamma Decay:**
- No change in mass number or atomic number.

3. **Determine the Properties of E:**
- Given:
- Atomic number of E (Z_E) = 69
- Mass number of E (A_E) = 172

4. **Work Backwards to Find A, B, C, and D:**
- **From D to E (Gamma Decay):**
- Since gamma decay does not change A or Z, we have:
- A_D = A_E = 172
- Z_D = Z_E = 69

- **From C to D (Alpha Decay):**
- For alpha decay:
- A_C = A_D + 4 = 172 + 4 = 176
- Z_C = Z_D + 2 = 69 + 2 = 71

- **From B to C (Beta Decay):**
- For beta decay:
- A_B = A_C = 176 (no change in mass number)
- Z_B = Z_C - 1 = 71 - 1 = 70

- **From A to B (Alpha Decay):**
- For alpha decay:
- A_A = A_B + 4 = 176 + 4 = 180
- Z_A = Z_B + 2 = 70 + 2 = 72

5. **Summary of Results:**
- Atomic number of A (Z_A) = 72
- Mass number of B (A_B) = 176
- Atomic number of C (Z_C) = 71
- Atomic number of D (Z_D) = 69

### Conclusion:
- The atomic number of A is 72.
- The mass number of B is 176.
- The atomic number of D is 69.
- The atomic number of C is 71 (which is incorrect as per the question).

### Final Answers:
- Atomic number of A = 72 (Correct)
- Mass number of B = 176 (Correct)
- Atomic number of C = 71 (Incorrect)
- Atomic number of D = 69 (Correct)