The electrons, identified by quantum numbers n and 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 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 electrons identified by their quantum numbers (n, l), we will use the n + l rule. According to this rule, the energy of an electron in an atom can be determined by the sum of its principal quantum number (n) and its azimuthal quantum number (l). The lower the value of n + l, the lower the energy of the electron. If two electrons have the same n + l value, the one with the lower n value will have lower energy.

### Step-by-Step Solution:

1. **Calculate n + l for each set of quantum numbers:**
- (i) n = 4, l = 1:
- n + l = 4 + 1 = 5
- (ii) n = 4, l = 0:
- n + l = 4 + 0 = 4
- (iii) n = 3, l = 2:
- n + l = 3 + 2 = 5
- (iv) n = 3, l = 1:
- n + l = 3 + 1 = 4

2. **List the n + l values:**
- (i) n + l = 5
- (ii) n + l = 4
- (iii) n + l = 5
- (iv) n + l = 4

3. **Order the n + l values from lowest to highest:**
- The lowest n + l value is 4 (from (ii) and (iv)).
- The next n + l value is 5 (from (i) and (iii)).

4. **Determine the order of increasing energy:**
- For (ii) n = 4, l = 0 (n + l = 4) has lower energy than (iv) n = 3, l = 1 (n + l = 4) because n is lower in (iv).
- For (i) n = 4, l = 1 (n + l = 5) and (iii) n = 3, l = 2 (n + l = 5), since both have the same n + l value, we compare n values. (iii) has lower n (3) than (i) (4), so (iii) has lower energy.

5. **Final order from lowest to highest energy:**
- (ii) n = 4, l = 0 (lowest energy)
- (iv) n = 3, l = 1
- (iii) n = 3, l = 2
- (i) n = 4, l = 1 (highest energy)

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