Which of the following electronic configurations have spin multiplicity equal to zero ?

Options

A.
B.
C.
D.

Answer

To determine which electronic configurations have a spin multiplicity equal to zero, we will follow these steps:

### Step 1: Understand the Spin Multiplicity Formula
The spin multiplicity (M) is given by the formula:
\[ M = 2s + 1 \]
where \( s \) is the total spin of the electrons in the configuration.

### Step 2: Analyze Each Configuration
We will analyze the spin of the electrons in each configuration provided in the options.

#### Option 1:
Assuming the configuration has three electrons with spins all pointing up:
- Spins: \( +\frac{1}{2}, +\frac{1}{2}, +\frac{1}{2} \)
- Total Spin \( s = +\frac{3}{2} \)
- Spin Multiplicity:
\[ M = 2 \times \frac{3}{2} + 1 = 4 \]

#### Option 2:
Assuming the configuration has two spins up and one spin down:
- Spins: \( +\frac{1}{2}, +\frac{1}{2}, -\frac{1}{2} \)
- Total Spin \( s = +\frac{1}{2} \)
- Spin Multiplicity:
\[ M = 2 \times \frac{1}{2} + 1 = 2 \]

#### Option 3:
Assuming the configuration has one spin up and two spins down:
- Spins: \( +\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2} \)
- Total Spin \( s = -\frac{1}{2} \)
- Spin Multiplicity:
\[ M = 2 \times \left(-\frac{1}{2}\right) + 1 = 0 \]

#### Option 4:
Assuming the configuration has three spins down:
- Spins: \( -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2} \)
- Total Spin \( s = -\frac{3}{2} \)
- Spin Multiplicity:
\[ M = 2 \times \left(-\frac{3}{2}\right) + 1 = -2 \]

### Step 3: Identify the Configuration with Spin Multiplicity Zero
From the calculations:
- Option 1: Spin Multiplicity = 4
- Option 2: Spin Multiplicity = 2
- Option 3: Spin Multiplicity = 0 (This is the correct option)
- Option 4: Spin Multiplicity = -2

Thus, the configuration with spin multiplicity equal to zero is **Option 3**.

### Summary of the Solution
The electronic configuration that has a spin multiplicity equal to zero is found in **Option 3**.