An electron in an atom can be completely designated with the help of four quantum numbers. Out of these, the first three i.e., principal (n), azimuthal (l) and magnetic (m) quantum number are obtained from the solution of Shrodinger wave equation while the spin(s) quantum number arises from the spin of the electron around its axis clockwise or anticlockwise. Of these principal quantum number tells about the size, azimuthal quantum number about the shape and magnetic quantum signifies the orientation of the electron orbital.
How many electrons in a given atom have the following set of quantum numbers?
`n = 3, l =2, m = +2, s = -1//2`

Options

A. 1
B. 18
C. 14
D. cannot be known

Explanation

To determine how many electrons in a given atom have the specified set of quantum numbers \( n = 3 \), \( l = 2 \), \( m = +2 \), and \( s = -\frac{1}{2} \), we will analyze each quantum number step by step.

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n = 3 \) indicates that the electron is in the third energy level or shell of the atom.

2. **Identify the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l = 2 \) corresponds to the d subshell. This means we are dealing with the 3d subshell.

3. **Identify the Magnetic Quantum Number (m)**:
- The magnetic quantum number \( m = +2 \) specifies the orientation of the orbital within the d subshell. For \( l = 2 \), the possible values of \( m \) range from \( -2 \) to \( +2 \) (i.e., \( m = -2, -1, 0, +1, +2 \)). Thus, \( m = +2 \) refers to one specific orbital, which is typically denoted as \( d_{xy} \).

4. **Identify the Spin Quantum Number (s)**:
- The spin quantum number \( s = -\frac{1}{2} \) indicates the spin orientation of the electron. According to the Pauli Exclusion Principle, each orbital can hold a maximum of two electrons: one with spin \( +\frac{1}{2} \) and the other with spin \( -\frac{1}{2} \).

5. **Determine the Number of Electrons**:
- Since we have specified \( m = +2 \) for the 3d orbital, and the spin is \( -\frac{1}{2} \), there can only be one electron in that specific orbital with the given spin orientation. Therefore, there is only **one electron** that can have the quantum numbers \( n = 3, l = 2, m = +2, s = -\frac{1}{2} \).

### Final Answer:
**There is only 1 electron in the given atom with the specified set of quantum numbers.**

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