Calculate the value of `Z_("eff")` for 3d electron of `._(21)Sc`.

Options

A. `18.69`
B. `21.69`
C. `3.69`
D. `3.00`

Explanation

To calculate the value of \( Z_{\text{eff}} \) for the 3d electron of Scandium-21 (\( _{21}^{45}Sc \)), we can follow these steps:

### Step 1: Determine the Electron Configuration
The electron configuration of Scandium (Sc) is:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 3d^1 \]
This indicates that Scandium has a total of 21 electrons.

### Step 2: Identify the Atomic Number (Z)
The atomic number \( Z \) of Scandium is 21, which represents the total number of protons and electrons in a neutral atom.

### Step 3: Use Slater's Rules to Calculate \( \sigma \)
According to Slater's rules, we need to calculate the shielding constant \( \sigma \) for the 3d electron. The formula for \( Z_{\text{eff}} \) is:
\[ Z_{\text{eff}} = Z - \sigma \]

### Step 4: Calculate \( \sigma \)
To calculate \( \sigma \) for the 3d electron, we consider the contributions from other electrons:
- Electrons in the same group (3d): 0 (as there is only one 3d electron)
- Electrons in the n-1 shell (3s and 3p): 9 electrons (2 from 3s and 6 from 3p), each contributing 0.85
- Electrons in the n-2 shell (1s and 2s, 2p): 10 electrons (2 from 1s, 2 from 2s, and 6 from 2p), each contributing 1.00

Now, we can calculate \( \sigma \):
\[ \sigma = (0 \times 0.35) + (9 \times 0.85) + (10 \times 1.00) \]
Calculating each term:
- Contribution from 3d: \( 0 \)
- Contribution from 3s and 3p: \( 9 \times 0.85 = 7.65 \)
- Contribution from 1s, 2s, and 2p: \( 10 \times 1.00 = 10 \)

Adding these contributions together:
\[ \sigma = 0 + 7.65 + 10 = 17.65 \approx 18 \]

### Step 5: Calculate \( Z_{\text{eff}} \)
Now we can substitute the values into the equation for \( Z_{\text{eff}} \):
\[ Z_{\text{eff}} = Z - \sigma = 21 - 18 = 3 \]

### Final Answer
The value of \( Z_{\text{eff}} \) for the 3d electron of Scandium-21 is:
\[ \boxed{3} \]