How many electrons in an atom can have n = 4, l =2 ,m= -2 and `s = +1/2`?

Options

A. 1
B. 2
C. 5
D. 10

Answer

To determine how many electrons in an atom can have the quantum numbers \( n = 4 \), \( l = 2 \), \( m = -2 \), and \( s = +\frac{1}{2} \), we can follow these steps:

### Step 1: Identify the Quantum Numbers
- The principal quantum number \( n = 4 \) indicates the energy level.
- The azimuthal quantum number \( l = 2 \) indicates the type of subshell (in this case, it corresponds to the d subshell).
- The magnetic quantum number \( m = -2 \) specifies one of the five d orbitals.
- The spin quantum number \( s = +\frac{1}{2} \) indicates the spin orientation of the electron.

### Step 2: Determine the Number of Orbitals
The number of orbitals in a subshell is calculated using the formula:
\[ \text{Number of orbitals} = 2l + 1 \]
Substituting \( l = 2 \):
\[ \text{Number of orbitals} = 2(2) + 1 = 5 \]
This means there are 5 d orbitals in the 4th energy level.

### Step 3: Calculate the Total Number of Electrons
Each orbital can hold a maximum of 2 electrons (one with spin \( +\frac{1}{2} \) and one with spin \( -\frac{1}{2} \)). Therefore, the total number of electrons that can be accommodated in the 5 d orbitals is:
\[ \text{Total electrons} = \text{Number of orbitals} \times 2 = 5 \times 2 = 10 \]

### Conclusion
Thus, the total number of electrons in an atom that can have the given quantum numbers \( n = 4 \), \( l = 2 \), \( m = -2 \), and \( s = +\frac{1}{2} \) is **10 electrons**.

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