Question

The ${{Z}_{eff}}(approx)$ for 4s-electron of Ni - atom according to Slater's rule is ________.

Answer 1

Hint: Write down the electronic configuration for the ground state of Ni atom. Now make a note of the electrons present in the subshell 4s other than the electron for which we are going to calculate the ${{Z}_{eff}}$. Apply Slater's rule for d and f block elements to find the ${{Z}_{eff}}$ for an electron present in the 4s subshell.

Complete step by step answer:
Slater's rules provide numerical values for the calculation of effective nuclear charge in a many-electron atom system. Each electron is said to experience less than the actual nuclear charge, because of shielding or screening effect by the other electrons. For each electron in an atom, Slater's rules provide a value for the screening constant, denoted by $\sigma $, which relates the actual nuclear charge and effective nuclear charge as:
${{Z}_{eff}}\text{ = }Z\text{ - }\sigma $
Rules: The electrons are firstly arranged into a series of groups in the order of increasing principal quantum number n, and for equal n in order of increasing azimuthal quantum number l, however s and p orbitals are kept together. Example:
$[1s][2s,2p][3s,3p][3d][4s,4p]......$

Each group is given a different shielding constant that depends on the types and number of electrons in those groups preceding it.
The shielding constant for each group is formed as a sum of the following contributions:
- An amount of 0.35 from every other electron within the same group except for the $[1s]$ group, where the other electron contributes only 0.30.
- If the group is of the \[\left[ ns,\text{ }np \right]\] type, an amount of 0.85 from each electron with principal quantum number (n–1), and an amount of 1.00 for each electron with principal quantum number (n–2) or less.
- If the group is of the $[d]\text{ or }[f]$ type, an amount of 1.00 for each electron "closer" to the nucleus than the group.

We will now calculate the effective nuclear charge for 4s-electron of Ni atom:
Atomic number(Z): 28
Electronic configuration: $1{{s}^{2}}2{{s}^{2}}2{{p}^{6}}3{{s}^{2}}3{{p}^{6}}3{{d}^{8}}4{{s}^{2}}$
Calculation of $\sigma $:
$\sigma $ = 0.35 x 1 + 0.85 x 16 + 1 x 10
$\sigma $ = 23.95

Substituting the value of $\sigma $ we get,
${{Z}_{eff}}\text{ = }Z\text{ - }\sigma $
${{Z}_{eff}}\text{ = 28 - 23}\text{.95}$
${{Z}_{eff}}\text{ = 4}\text{.05}$
Therefore, the ${{Z}_{eff}}(approx)$ for 4s-electrons of Ni - atoms according to Slater's rule is 4.

Note: The rules were developed by John C. Slater in order to make an attempt in constructing a simple analytic expression for the atomic orbital of an arbitrary electron present in an atom.
Slater wished to determine the exact values of shielding constants and effective quantum numbers such that the equation provides a reasonable approximation to a single-electron wave function.