In an orbital electrons are filled according to Aufbau principle, Pauli's exclusion principle and Hund's rule of maximum muliplicity .
According to Aufbau principal the orbital are filled in order to their increasing energies . In the order of increase of energy of orbitals can be calculated from `( n + l )` rule. Lower the value of `( n + l )` for an orbital the lowest energy hence orbital are filled in order of increasing `( n + l ) ` value. If two orbitals have same `( n + l ) ` value, the orbital with lower value of 'n' has lower energy hence it is filled first.
According to Pauli's exclusion principle, an orbital can have maximum two electrons with opposite spin.
According to Hunds rule pairing of electron in degenerate orbitals of the same sub shell does not take place until each orbital belonging to that sub shell has got one electron each i.e., singly occupied.
The electron identified by quantum number n & l
(i) n = 4, l = 1 (ii) n = 4, l = 0
( iii) n = 3, l = 2 (iv) n = 3, l = 1
can be placed in order of increasing energy from the lowest to the highest as

Options

A. `(iv) lt (ii) lt (iii) lt (i)`
B. `(ii) lt (iv) lt (i) lt (iii) `
C. `(i) lt (iii) lt (ii) lt (iv)`
D. `(iii) lt (i) lt (iv) lt (ii) `

Answer

To determine the order of increasing energy for the given orbitals based on the quantum numbers \( n \) and \( l \), we will apply the principles of the Aufbau principle, Pauli's exclusion principle, and Hund's rule of maximum multiplicity as described.

### Step-by-Step Solution:

1. **Identify the Quantum Numbers**:
We have four sets of quantum numbers:
- (i) \( n = 4, l = 1 \)
- (ii) \( n = 4, l = 0 \)
- (iii) \( n = 3, l = 2 \)
- (iv) \( n = 3, l = 1 \)

2. **Calculate \( n + l \) for Each Orbital**:
- For (i): \( n + l = 4 + 1 = 5 \)
- For (ii): \( n + l = 4 + 0 = 4 \)
- For (iii): \( n + l = 3 + 2 = 5 \)
- For (iv): \( n + l = 3 + 1 = 4 \)

3. **Compare \( n + l \) Values**:
- The lowest \( n + l \) value indicates the lowest energy.
- From our calculations:
- (ii) \( n + l = 4 \) (lowest)
- (iv) \( n + l = 4 \)
- (i) \( n + l = 5 \)
- (iii) \( n + l = 5 \)

4. **Determine the Order of Energy**:
- For (ii) and (iv), both have the same \( n + l \) value of 4. We compare their \( n \) values:
- (ii) \( n = 4 \)
- (iv) \( n = 3 \)
- Since \( n = 3 \) is lower, (iv) has lower energy than (ii).
- For (i) and (iii), both have the same \( n + l \) value of 5. We compare their \( n \) values:
- (i) \( n = 4 \)
- (iii) \( n = 3 \)
- Again, since \( n = 3 \) is lower, (iii) has lower energy than (i).

5. **Final Order of Increasing Energy**:
- Lowest to highest energy order:
- (iv) \( n = 3, l = 1 \) (4)
- (ii) \( n = 4, l = 0 \) (4)
- (iii) \( n = 3, l = 2 \) (5)
- (i) \( n = 4, l = 1 \) (5)

Thus, the order of increasing energy from lowest to highest is:
**(iv) < (ii) < (iii) < (i)**