The spin only magnetic moment of a divalent ion in aqueous solution (atomic number 29) is _______ BM

Answer

To solve the problem of finding the spin-only magnetic moment of a divalent ion of the element with atomic number 29 (Copper), we can follow these steps:

### Step-by-Step Solution

1. **Identify the Element**:
The atomic number 29 corresponds to Copper (Cu).

2. **Determine the Electron Configuration**:
The electron configuration of a neutral Copper atom (Cu) is:
\[ \text{Cu: } 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \]
However, Copper is known to have a stable configuration of:
\[ \text{Cu: } [Ar] \, 4s^1 \, 3d^{10} \]
This configuration arises because one electron from the 4s subshell is promoted to the 3d subshell, resulting in a half-filled 4s subshell and a fully filled 3d subshell.

3. **Determine the Divalent Ion**:
A divalent ion means that Copper has lost two electrons. The two electrons lost will be from the 4s and 3d subshells:
\[ \text{Cu}^{2+}: [Ar] \, 3d^9 \]

4. **Count the Unpaired Electrons**:
In the \(3d^9\) configuration, there are 9 electrons in the d subshell. The distribution of electrons in the five d orbitals (each can hold 2 electrons) will be:
- 4 orbitals will have 2 electrons each (4 x 2 = 8 electrons)
- 1 orbital will have 1 electron (1 x 1 = 1 electron)

Therefore, there is **1 unpaired electron** in the \(3d^9\) configuration.

5. **Use the Spin-Only Magnetic Moment Formula**:
The formula for the spin-only magnetic moment (\(\mu\)) is given by:
\[ \mu = \sqrt{n(n + 2)} \]
where \(n\) is the number of unpaired electrons. Here, \(n = 1\).

6. **Calculate the Magnetic Moment**:
Substitute \(n = 1\) into the formula:
\[ \mu = \sqrt{1(1 + 2)} = \sqrt{1 \times 3} = \sqrt{3} \]

7. **Convert to Numerical Value**:
The value of \(\sqrt{3}\) is approximately \(1.732\).

8. **State the Final Answer**:
The spin-only magnetic moment of the divalent ion of Copper in aqueous solution is:
\[ \text{Answer: } 1.732 \, \text{BM} \]