The position and energy of an electron is specified with the help of four quantum numbers namely, principal quantum number (n), azimuthal quantum number (l), magnetic quantum number `(m_l)` and spin quantum number `(m_s)` . The permissible values of these are :
`n = 1,2.....`
`l = 0,1,.....(n-1)`
`m_l = -l,......0,......+l`
`m_s = +1/2 and -1/2` for each value of `m_l`.
The angular momentum of electron is given as `sqrt(l(l + 1)) cdot h/(2pi)`
While spin angular momentum is given as `sqrt(s(s+1)) cdot (h/(2pi))` where `s = 1/2`
The electrons having the same value of `n, l and m_l` are said to belong to the same orbital. According to Pauli's exclusion principle, an orbital can have maximum of two electrons and these two must have opposite spin.
The maximum number of electrons having `n + l = 5` in an atom is

Options

A. 32
B. 18
C. 10
D. 8

Explanation

To solve the problem of finding the maximum number of electrons in an atom with the condition \( n + l = 5 \), we will follow these steps:

### Step 1: Understand the Quantum Numbers
The quantum numbers are:
- Principal quantum number \( n \): Can take values \( 1, 2, 3, \ldots \)
- Azimuthal quantum number \( l \): Can take values \( 0, 1, \ldots, (n-1) \)
- Magnetic quantum number \( m_l \): Can take values \( -l, \ldots, 0, \ldots, +l \)
- Spin quantum number \( m_s \): Can take values \( +\frac{1}{2} \) and \( -\frac{1}{2} \)

### Step 2: Determine Possible Values of \( n \) and \( l \)
Given \( n + l = 5 \), we can explore the possible values of \( n \) and \( l \):
- If \( n = 1 \), then \( l = 4 \) (not possible since \( l \) can only be \( 0 \)).
- If \( n = 2 \), then \( l = 3 \) (not possible since \( l \) can only be \( 0, 1 \)).
- If \( n = 3 \), then \( l = 2 \) (possible).
- If \( n = 4 \), then \( l = 1 \) (possible).
- If \( n = 5 \), then \( l = 0 \) (possible).

### Step 3: Calculate Maximum Electrons for Each Valid \( (n, l) \) Pair
1. **For \( n = 3, l = 2 \)** (3d subshell):
- Number of orbitals = 5 (from \( m_l = -2, -1, 0, +1, +2 \))
- Maximum electrons = \( 5 \text{ orbitals} \times 2 \text{ electrons/orbital} = 10 \)

2. **For \( n = 4, l = 1 \)** (4p subshell):
- Number of orbitals = 3 (from \( m_l = -1, 0, +1 \))
- Maximum electrons = \( 3 \text{ orbitals} \times 2 \text{ electrons/orbital} = 6 \)

3. **For \( n = 5, l = 0 \)** (5s subshell):
- Number of orbitals = 1 (from \( m_l = 0 \))
- Maximum electrons = \( 1 \text{ orbital} \times 2 \text{ electrons/orbital} = 2 \)

### Step 4: Sum the Maximum Electrons
Now, we sum the maximum electrons from each valid pair:
- From \( n = 3, l = 2 \): 10 electrons
- From \( n = 4, l = 1 \): 6 electrons
- From \( n = 5, l = 0 \): 2 electrons

Total maximum electrons = \( 10 + 6 + 2 = 18 \)

### Conclusion
The maximum number of electrons having \( n + l = 5 \) in an atom is **18**.

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