Which of the following pairs of species would you expect to have largest difference in spin only magnetic moment ?

Options

A. `O_(2), O_(2)^(+)`
B. `O_(2)O_(2)^(2-)`
C. `O_(2)^(+), O_(2)^(2-)`
D. `O_(2)^(-),O_(2)^(+)`

Explanation

To determine which pair of species has the largest difference in spin-only magnetic moment, we will follow these steps:

### Step 1: Understand the Magnetic Moment Formula
The spin-only magnetic moment (\( \mu \)) is calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Identify the Species
The species we need to analyze are:
1. \( O_2 \)
2. \( O_2^+ \)
3. \( O_2^- \)
4. \( O_2^{2-} \)

### Step 3: Determine the Number of Unpaired Electrons
Using molecular orbital theory (MOT), we can determine the number of unpaired electrons for each species.

1. **For \( O_2 \)**:
- Total electrons = 16 (8 from each oxygen atom)
- Molecular orbital configuration: \( (1\sigma)^2 (1\sigma^*)^2 (2\sigma)^2 (2\sigma^*)^2 (1\pi)^4 (1\pi^*)^2 \)
- Unpaired electrons = 2 (in the \( \pi^* \) orbitals)

2. **For \( O_2^+ \)**:
- It has one less electron than \( O_2 \).
- Unpaired electrons = 1 (removal from one of the \( \pi^* \) orbitals)

3. **For \( O_2^- \)**:
- It has one more electron than \( O_2 \).
- Unpaired electrons = 3 (adding to the \( \pi^* \) orbitals)

4. **For \( O_2^{2-} \)**:
- It has two more electrons than \( O_2 \).
- Unpaired electrons = 2 (both \( \pi^* \) orbitals are filled)

### Step 4: Calculate the Magnetic Moments
Now we can calculate the magnetic moments for each species:

1. **For \( O_2 \)**:
\[ \mu_{O_2} = \sqrt{2(2 + 2)} = \sqrt{8} = 2\sqrt{2} \]

2. **For \( O_2^+ \)**:
\[ \mu_{O_2^+} = \sqrt{1(1 + 2)} = \sqrt{3} \]

3. **For \( O_2^- \)**:
\[ \mu_{O_2^-} = \sqrt{3(3 + 2)} = \sqrt{15} \]

4. **For \( O_2^{2-} \)**:
\[ \mu_{O_2^{2-}} = \sqrt{0(0 + 2)} = 0 \]

### Step 5: Determine the Differences in Magnetic Moments
Now we will find the differences in magnetic moments between the pairs:

- Difference between \( O_2 \) and \( O_2^{2-} \):
\[ \Delta \mu = 2\sqrt{2} - 0 = 2\sqrt{2} \]

- Difference between \( O_2 \) and \( O_2^+ \):
\[ \Delta \mu = 2\sqrt{2} - \sqrt{3} \]

- Difference between \( O_2^- \) and \( O_2^{2-} \):
\[ \Delta \mu = \sqrt{15} - 0 = \sqrt{15} \]

- Difference between \( O_2 \) and \( O_2^- \):
\[ \Delta \mu = 2\sqrt{2} - \sqrt{15} \]

### Step 6: Conclusion
The largest difference in magnetic moments occurs between \( O_2 \) and \( O_2^{2-} \), which is \( 2\sqrt{2} \).

Thus, the answer is:
**The pair with the largest difference in spin-only magnetic moment is \( O_2 \) and \( O_2^{2-} \).**