In an atom, the total number of electrons having quantum numbers
`n = 4,|m_(l)| = 1` and `m_(s) =- (1)/(2)` is

Answer

To find the total number of electrons in an atom with the given quantum numbers \( n = 4 \), \( |m_l| = 1 \), and \( m_s = -\frac{1}{2} \), we will follow these steps:

### Step 1: Identify the values of the quantum numbers
- The principal quantum number \( n \) is given as 4.
- The absolute value of the magnetic quantum number \( |m_l| \) is given as 1, which means \( m_l \) can be either +1 or -1.
- The spin quantum number \( m_s \) is given as -\(\frac{1}{2}\).

### Step 2: Determine the azimuthal quantum number \( l \)
- The azimuthal quantum number \( l \) can take values from 0 to \( n-1 \). Therefore, for \( n = 4 \), \( l \) can be 0, 1, 2, or 3.

### Step 3: Determine the possible values of \( m_l \)
- For each value of \( l \):
- If \( l = 0 \), then \( m_l = 0 \).
- If \( l = 1 \), then \( m_l = -1, 0, +1 \).
- If \( l = 2 \), then \( m_l = -2, -1, 0, +1, +2 \).
- If \( l = 3 \), then \( m_l = -3, -2, -1, 0, +1, +2, +3 \).

Since we are interested in \( |m_l| = 1 \), we can see that \( m_l \) can be +1 or -1.

### Step 4: Count the electrons for each \( l \) that allows \( |m_l| = 1 \)
- For \( l = 1 \):
- Possible \( m_l \) values: -1, 0, +1.
- Each \( m_l \) can have 2 electrons (one with \( m_s = +\frac{1}{2} \) and one with \( m_s = -\frac{1}{2} \)).
- Total electrons = 2 (for \( m_l = -1 \)) + 2 (for \( m_l = +1 \)) = 4 electrons.

- For \( l = 2 \):
- Possible \( m_l \) values: -2, -1, 0, +1, +2.
- \( m_l = -1 \) and \( m_l = +1 \) can each have 2 electrons.
- Total electrons = 2 (for \( m_l = -1 \)) + 2 (for \( m_l = +1 \)) = 4 electrons.

- For \( l = 3 \):
- Possible \( m_l \) values: -3, -2, -1, 0, +1, +2, +3.
- \( m_l = -1 \) and \( m_l = +1 \) can each have 2 electrons.
- Total electrons = 2 (for \( m_l = -1 \)) + 2 (for \( m_l = +1 \)) = 4 electrons.

### Step 5: Sum the total number of electrons
- From \( l = 1 \): 4 electrons
- From \( l = 2 \): 4 electrons
- From \( l = 3 \): 4 electrons

Total number of electrons = \( 4 + 4 + 4 = 12 \).

### Final Answer
The total number of electrons having quantum numbers \( n = 4 \), \( |m_l| = 1 \), and \( m_s = -\frac{1}{2} \) is **12**.