A radioactive nucleus undergoes a series of decay according to the scheme
`A oversetalphato A_1 oversetbetato A_2 oversetalpha to A_3 oversetgammato A_4`
if the mass number and atomic number of A are 180 and 72 respectively, then what are these number for `A_3`

Options

A. 174 and 70
B. 172 and 69
C. 176 and 69
D. 176 and 70

Answer

To solve the problem, we need to track the changes in mass number and atomic number through each decay step of the radioactive nucleus A.

### Step-by-Step Solution:

1. **Initial Values**:
- The mass number (A) of the nucleus is 180.
- The atomic number (Z) of the nucleus is 72.

2. **Decay Step 1: Alpha Decay**:
- In alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- New mass number (A1) = 180 - 4 = 176.
- New atomic number (Z1) = 72 - 2 = 70.

3. **Decay Step 2: Beta Decay**:
- In beta decay, the mass number remains unchanged, but the atomic number increases by 1.
- Mass number (A2) = A1 = 176.
- Atomic number (Z2) = Z1 + 1 = 70 + 1 = 71.

4. **Decay Step 3: Alpha Decay**:
- Again, in alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- New mass number (A3) = A2 - 4 = 176 - 4 = 172.
- New atomic number (Z3) = Z2 - 2 = 71 - 2 = 69.

5. **Decay Step 4: Gamma Decay**:
- In gamma decay, there is no change in mass number or atomic number.
- Therefore, the values for A4 remain the same as A3:
- Mass number (A4) = A3 = 172.
- Atomic number (Z4) = Z3 = 69.

### Final Answer for A3:
- Mass number of A3 = 172
- Atomic number of A3 = 69

### Summary:
The mass number and atomic number for A3 are 172 and 69, respectively.

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