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 antiaclockwise. Ot 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 quantium 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 can have the specified set of quantum numbers \( n = 3, l = 2, m = +2, s = -\frac{1}{2} \), we can follow these steps:

### Step 1: Understand the Quantum Numbers
- **Principal Quantum Number (n)**: Indicates the energy level and size of the orbital. Here, \( n = 3 \) means the electron is in the third energy level.
- **Azimuthal Quantum Number (l)**: Indicates the shape of the orbital. For \( l = 2 \), the orbital is a d-orbital.
- **Magnetic Quantum Number (m)**: Indicates the orientation of the orbital. For \( m = +2 \), it corresponds to one of the d-orbitals (specifically, the \( d_{xy} \) orbital).
- **Spin Quantum Number (s)**: Indicates the spin of the electron. Here, \( s = -\frac{1}{2} \) indicates that the electron is spinning in one direction (usually considered as "down").

### Step 2: Apply the Pauli Exclusion Principle
According to the Pauli Exclusion Principle, no two electrons in an atom can have the same set of four quantum numbers. This means that for any given set of quantum numbers, there can be at most one electron.

### Step 3: Determine the Number of Electrons
Given the specific set of quantum numbers \( n = 3, l = 2, m = +2, s = -\frac{1}{2} \):
- Since this set of quantum numbers is unique, there can only be **one electron** that has this exact combination.

### Conclusion
Thus, the answer is that there is **1 electron** in a given atom that can have the quantum numbers \( n = 3, l = 2, m = +2, s = -\frac{1}{2} \).

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