How many electrons in an atom can have n = 3, l = 1, m =-1 and `s = + 1//2`

Options

A. 1
B. 2
C. 4
D. 6

Answer

To solve the question of how many electrons in an atom can have the quantum numbers \( n = 3 \), \( l = 1 \), \( m = -1 \), and \( s = +\frac{1}{2} \), we can follow these steps:

### Step 1: Identify the Quantum Numbers
The quantum numbers provided are:
- Principal quantum number \( n = 3 \)
- Azimuthal quantum number \( l = 1 \)
- Magnetic quantum number \( m = -1 \)
- Spin quantum number \( s = +\frac{1}{2} \)

### Step 2: Determine the Subshell
The azimuthal quantum number \( l \) indicates the subshell:
- \( l = 0 \) corresponds to the s subshell
- \( l = 1 \) corresponds to the p subshell
- \( l = 2 \) corresponds to the d subshell

Since \( l = 1 \), we are dealing with the 3p subshell.

### Step 3: Identify Possible \( m_l \) Values
The magnetic quantum number \( m \) can take values from \( -l \) to \( +l \):
- For \( l = 1 \), the possible \( m_l \) values are \( -1, 0, +1 \).

### Step 4: Analyze the Given \( m \) Value
The question specifies \( m = -1 \). This means we are focusing on the specific orbital within the 3p subshell that corresponds to \( m = -1 \).

### Step 5: Determine the Spin Quantum Number
The spin quantum number \( s \) can have two possible values for each orbital:
- \( s = +\frac{1}{2} \)
- \( s = -\frac{1}{2} \)

Since the question specifies \( s = +\frac{1}{2} \), we are looking for electrons with this specific spin in the \( m = -1 \) orbital.

### Step 6: Count the Electrons
In any given orbital (in this case, the 3p orbital with \( m = -1 \)), there can be a maximum of 2 electrons, one with spin \( +\frac{1}{2} \) and one with spin \( -\frac{1}{2} \). However, since we are only given \( s = +\frac{1}{2} \), we can only have one electron in this orbital.

### Final Answer
Thus, the number of electrons in an atom that can have the specified quantum numbers \( n = 3 \), \( l = 1 \), \( m = -1 \), and \( s = +\frac{1}{2} \) is **1**.

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