How many electrons in an atom have the following quantum numbers?
n=4, `m_(s)= -1//2`

Options

A. 32
B. 18
C. 8
D. 16

Explanation

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

### Step-by-Step Solution:

1. **Understanding Quantum Numbers**:
- The principal quantum number \( n \) indicates the energy level of the electron. Here, \( n = 4 \) means we are looking at the fourth energy level.
- The spin quantum number \( m_s \) can take values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \). In this case, we are interested in \( m_s = -\frac{1}{2} \).

2. **Calculating the Maximum Number of Electrons for \( n = 4 \)**:
- The maximum number of electrons in a given energy level can be calculated using the formula \( 2n^2 \).
- For \( n = 4 \):
\[ \text{Maximum electrons} = 2n^2 = 2 \times (4^2) = 2 \times 16 = 32 \]
- Therefore, there can be a maximum of 32 electrons in the fourth energy level.

3. **Determining the Distribution of Electrons**:
- Each orbital can hold a maximum of 2 electrons, one with \( m_s = +\frac{1}{2} \) and the other with \( m_s = -\frac{1}{2} \).
- Since we have 32 electrons in total at \( n = 4 \), they will be distributed across various orbitals.

4. **Finding Electrons with \( m_s = -\frac{1}{2} \)**:
- Since each of the orbitals can accommodate one electron with \( m_s = -\frac{1}{2} \), and there are 32 electrons total, half of these will have \( m_s = -\frac{1}{2} \) and half will have \( m_s = +\frac{1}{2} \).
- Therefore, the number of electrons with \( m_s = -\frac{1}{2} \) is:
\[ \text{Electrons with } m_s = -\frac{1}{2} = \frac{32}{2} = 16 \]

### Final Answer:
The number of electrons in an atom that have the quantum numbers \( n = 4 \) and \( m_s = -\frac{1}{2} \) is **16**.

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