To determine the effective nuclear charge (Z_eff) for the various electrons in chromium (Cr) and its ion (Cr³⁺), we need to understand how Z_eff is calculated and the factors that influence it. The effective nuclear charge is a measure of the net positive charge experienced by an electron in an atom, taking into account both the actual nuclear charge and the shielding effect of other electrons. Let's break down the Z_eff values for the specified electrons in chromium and its ion.
Understanding Effective Nuclear Charge (Z_eff)
The effective nuclear charge can be calculated using the formula:
Z_eff = Z - S
Where:
- Z is the atomic number (total number of protons).
- S is the shielding constant, which accounts for the repulsion from other electrons that reduces the full nuclear charge felt by an electron.
Electrons in Chromium
Chromium has the electron configuration of [Ar] 3d5 4s1. In its neutral state, the atomic number of chromium is 24. The electrons we are considering are:
- 3d electron of Cr
- 4s electron of Cr
- 3d electron of Cr³⁺
- 3s electron of Cr³⁺
Calculating Z_eff for Each Electron
Now, let's analyze the Z_eff for each of these electrons:
1. 3d Electron of Cr
For the 3d electron in neutral Cr:
- Z = 24 (total protons)
- S = 19.4 (approximate shielding from 4s and 3d electrons)
Thus, Z_eff = 24 - 19.4 = 4.6.
2. 4s Electron of Cr
For the 4s electron in neutral Cr:
- Z = 24
- S = 21.05 (more shielding due to the presence of 3d electrons)
Therefore, Z_eff = 24 - 21.05 = 2.95.
3. 3d Electron of Cr³⁺
In the Cr³⁺ ion, which has lost three electrons (two from 4s and one from 3d):
- Z = 24
- S = 18.95 (less shielding since there are fewer electrons)
So, Z_eff = 24 - 18.95 = 5.05.
4. 3s Electron of Cr³⁺
For the 3s electron in Cr³⁺:
- Z = 24
- S = 11.95 (considering the remaining electrons)
This gives us Z_eff = 24 - 11.95 = 12.05.
Final Order of Z_eff Values
Based on our calculations, the Z_eff values for the specified electrons are:
- 3d electron of Cr: 4.6
- 4s electron of Cr: 2.95
- 3d electron of Cr³⁺: 5.05
- 3s electron of Cr³⁺: 12.05
Now, comparing these values with the options provided:
- (1) 4.6, 2.95, 4.95, 8.05
- (2) 4.95, 2.95, 4.6, 8.08
- (3) 4.6, 2.95, 5.3, 12.75
- (4) 4.6, 2.95, 5.3, 12.65
None of the options perfectly match our calculated values, but the closest one is option (4) 4.6, 2.95, 5.3, 12.65, considering slight variations in the shielding constants used in different sources. This highlights the importance of understanding the underlying principles of Z_eff and how they can vary based on different interpretations of electron shielding.
