The electrons, identified by quantum number n and l: 
(I) `n = 3, l =0 ` (II) `n = 5, l = 2`
  (III) `n = 2,l  = 1` (IV) `n=4,l = 3`
  can be placed in order of increasing energy from the lowest to highest as: 

Options

A. `III lt I lt IV lt II`
B. `I lt III lt IV lt II`
C. `II lt III lt IV lt I`
D. `II lt IV lt I lt III`

Answer

To solve the problem of arranging the electrons identified by the quantum numbers \( n \) and \( l \) in order of increasing energy, we will use the \( n + l \) rule. This rule states that the energy of an electron in an atom can be approximated by the sum of its principal quantum number \( n \) and its azimuthal quantum number \( l \). The lower the \( n + l \) value, the lower the energy of the electron.

### Step-by-Step Solution:

1. **Identify the Quantum Numbers:**
- (I) \( n = 3, l = 0 \)
- (II) \( n = 5, l = 2 \)
- (III) \( n = 2, l = 1 \)
- (IV) \( n = 4, l = 3 \)

2. **Calculate \( n + l \) for Each Electron:**
- For (I): \( n + l = 3 + 0 = 3 \)
- For (II): \( n + l = 5 + 2 = 7 \)
- For (III): \( n + l = 2 + 1 = 3 \)
- For (IV): \( n + l = 4 + 3 = 7 \)

3. **List the \( n + l \) Values:**
- (I) \( n + l = 3 \)
- (II) \( n + l = 7 \)
- (III) \( n + l = 3 \)
- (IV) \( n + l = 7 \)

4. **Order by \( n + l \) Values:**
- The lowest \( n + l \) value is 3 (for both (I) and (III)).
- The next \( n + l \) value is 7 (for both (II) and (IV)).

5. **Resolve Ties:**
- For (I) and (III) both having \( n + l = 3 \):
- Compare \( n \):
- (I) has \( n = 3 \)
- (III) has \( n = 2 \)
- Since \( n = 2 \) is lower, (III) has lower energy than (I).

- For (II) and (IV) both having \( n + l = 7 \):
- Compare \( n \):
- (II) has \( n = 5 \)
- (IV) has \( n = 4 \)
- Since \( n = 4 \) is lower, (IV) has lower energy than (II).

6. **Final Order of Increasing Energy:**
- (III) \( n = 2, l = 1 \) (Energy 1)
- (I) \( n = 3, l = 0 \) (Energy 2)
- (IV) \( n = 4, l = 3 \) (Energy 3)
- (II) \( n = 5, l = 2 \) (Energy 4)

Thus, the order of increasing energy from lowest to highest is:
**(III), (I), (IV), (II)**.