The electrons identified by quantum numbers `n` and `l` :
(a) `n=5, l=1` , (b) `n=5, l=0`
(c ) `n=4, l=2` , (d) `n=4, l=1`
can be placed in order of increasing energy as :

Options

A. `(c )lt(d)lt(b)lt(a)`
B. `(d)lt(b)lt(c )lt(a)`
C. `(b)lt(d)lt(a)lt(c )`
D. `(a)lt(c )lt(b)lt(d)`

Explanation

To determine the order of increasing energy for the electrons identified by the given quantum numbers \( n \) and \( l \), we will follow these steps:

### Step 1: Calculate \( n + l \) for each set of quantum numbers.
The energy of an electron in an atom is primarily determined by the sum of the principal quantum number \( n \) and the azimuthal quantum number \( l \). The higher the value of \( n + l \), the higher the energy level.

1. For (a) \( n = 5, l = 1 \):
\[ n + l = 5 + 1 = 6 \]

2. For (b) \( n = 5, l = 0 \):
\[ n + l = 5 + 0 = 5 \]

3. For (c) \( n = 4, l = 2 \):
\[ n + l = 4 + 2 = 6 \]

4. For (d) \( n = 4, l = 1 \):
\[ n + l = 4 + 1 = 5 \]

### Step 2: List the \( n + l \) values.
- (a) \( n + l = 6 \)
- (b) \( n + l = 5 \)
- (c) \( n + l = 6 \)
- (d) \( n + l = 5 \)

### Step 3: Arrange the \( n + l \) values in increasing order.
From the calculations:
- The lowest \( n + l \) value is 5, which corresponds to options (b) and (d).
- The next \( n + l \) value is 6, which corresponds to options (a) and (c).

### Step 4: Compare the principal quantum numbers for the same \( n + l \) values.
Since (b) and (d) both have \( n + l = 5 \):
- (b) has \( n = 5 \)
- (d) has \( n = 4 \)

Since a higher principal quantum number indicates higher energy, we have:
- (d) \( n = 4 \) will have lower energy than (b) \( n = 5 \).

Since (a) and (c) both have \( n + l = 6 \):
- (a) has \( n = 5 \)
- (c) has \( n = 4 \)

Again, since a higher principal quantum number indicates higher energy, we have:
- (c) \( n = 4 \) will have lower energy than (a) \( n = 5 \).

### Step 5: Final order of increasing energy.
Putting it all together, we have:
1. (d) \( n = 4, l = 1 \) (lowest energy)
2. (b) \( n = 5, l = 0 \)
3. (c) \( n = 4, l = 2 \)
4. (a) \( n = 5, l = 1 \) (highest energy)

Thus, the order of increasing energy is:
**(d), (b), (c), (a)**