Which one of the following sets of quantum number is possible for electron in a 4f- orbital?

Options

A. `{:(n,l,m,s),(4,3,+4,+1//2):}`
B. `{:(n,l,m,s),(4,3,+1,-1//2):}`
C. `{:(n,l,m,s),(4,4,+4,-1//2):}`
D. `{:(n,l,m,s),(4,2,+4,+1//2):}`

Answer

To determine which set of quantum numbers is possible for an electron in a 4f orbital, we need to analyze the quantum numbers step by step.

### Step 1: Identify the Principal Quantum Number (n)
The principal quantum number (n) indicates the energy level of the electron. For a 4f orbital, n is equal to 4.

**Hint:** The principal quantum number corresponds to the energy level of the electron.

### Step 2: Determine the Azimuthal Quantum Number (l)
The azimuthal quantum number (l) defines the shape of the orbital. For f orbitals, the value of l is always 3. This is because l can take values from 0 to n-1. Therefore, for n = 4, l = 3.

**Hint:** The azimuthal quantum number for f orbitals is always 3.

### Step 3: Find the Magnetic Quantum Number (m)
The magnetic quantum number (m) can take values from -l to +l. Since l = 3 for the 4f orbital, m can take values from -3 to +3, which includes -3, -2, -1, 0, +1, +2, and +3.

**Hint:** The magnetic quantum number ranges from -l to +l.

### Step 4: Determine the Spin Quantum Number (s)
The spin quantum number (s) can have two possible values: +1/2 or -1/2. This indicates the two possible spin states of an electron.

**Hint:** The spin quantum number can be either +1/2 or -1/2.

### Step 5: Evaluate the Given Options
Now we will evaluate the provided options based on the quantum numbers we have determined:

1. **Option 1:** n = 4, l = 3, m = +4, s = +1/2
- Here, m = +4 is not valid since m can only range from -3 to +3. **Not possible.**

2. **Option 2:** n = 4, l = 3, m = +1, s = -1/2
- Here, all values are valid: n = 4, l = 3, m = +1 (which is within the range), and s = -1/2. **Possible.**

3. **Option 3:** n = 4, l = 4, m = +0, s = +1/2
- Here, l = 4 is not valid since l must be less than n. **Not possible.**

4. **Option 4:** n = 4, l = 2, m = +4, s = +1/2
- Here, l = 2 is not valid for a 4f orbital (it should be 3), and m = +4 is also not valid. **Not possible.**

### Conclusion
The only valid set of quantum numbers for an electron in a 4f orbital is from **Option 2:** n = 4, l = 3, m = +1, s = -1/2.

**Final Answer:** Option 2 is the correct choice.