Which of the following is least stable?

Options

A. a. `._(20)Co^(40)`
B. b. `._(13)Al^(30)`
C. c. `._(50)Sn^(119)`
D. d. `._(25)Mn^(55)`

Explanation

To determine which of the given elements is the least stable, we will analyze the neutron to proton (n/p) ratio for each element and check for magic numbers. The magic numbers for nuclear stability are 2, 8, 20, 28, 50, 82, and 126.

### Step-by-Step Solution:

1. **Identify the Elements and Their Atomic Numbers:**
- Calcium (Ca): Atomic number = 20
- Aluminium (Al): Atomic number = 13
- Tin (Sn): Atomic number = 50
- Manganese (Mn): Atomic number = 25

2. **Calculate Neutron Numbers:**
- For Calcium (Ca):
- Mass number = 40 (assumed)
- Neutrons = Mass number - Atomic number = 40 - 20 = 20
- For Aluminium (Al):
- Mass number = 27 (assumed)
- Neutrons = 27 - 13 = 14
- For Tin (Sn):
- Mass number = 119 (given)
- Neutrons = 119 - 50 = 69
- For Manganese (Mn):
- Mass number = 55 (given)
- Neutrons = 55 - 25 = 30

3. **Calculate Neutron to Proton Ratios (n/p):**
- Calcium (Ca): n/p = 20/20 = 1.0
- Aluminium (Al): n/p = 14/13 ≈ 1.08
- Tin (Sn): n/p = 69/50 = 1.38
- Manganese (Mn): n/p = 30/25 = 1.2

4. **Check for Magic Numbers:**
- Calcium: Both n and p are magic numbers (20).
- Aluminium: Neither n nor p is a magic number.
- Tin: Proton (50) is a magic number.
- Manganese: Neither n nor p is a magic number.

5. **Determine Stability:**
- A stable nucleus has an n/p ratio close to 1 and ideally has magic numbers.
- Calcium is extremely stable due to both n and p being magic numbers.
- Tin is also stable due to the magic number of protons.
- Manganese has a ratio of 1.2, indicating instability.
- Aluminium has the highest n/p ratio of 1.08, indicating it is the least stable.

### Conclusion:
The element that is least stable among the given options is **Aluminium (Al)** with an atomic number of 13.

### Final Answer:
**Aluminium (Al)** is the least stable element.

---