J.C. Slater proposed an empirical constant that represents the cumulative extent to which the other electrons of an atom shield (or screen) any particular electron from the nuclear charge. Thus, slater's screening contant `sigma` is as : `Z^(**)=Z-sigma`
Here, Z is the atomic number of the atom, and hence is equal to the actual number of protons in the atom. the parameter `Z^(**)` is the effective nuclear charge, which according to is smaller than Z, since the electron in question is screened (shielded) from Z by an amount `sigma`. Conversely, an electron that is well shielded from the nuclear charge Z experiences a small effective nuclear charge `Z^(**)`.
The value of `sigma` for any one electron in a given electron configuration (i.e., in the presence of the other electrons of the atom in question) is calculated using a set of empirical rules developed by slater. according to these rules, the value of `sigma` for the electron in question is the cumulative total provided by the various other electrons of the atom.
Q. According to Slater's rule, order of effective nuclear charge `(Z^(**))` for last electron in case of Li, Na and K.

Options

A. `Li gt Na gt K`
B. `K gt Na gt Li`
C. `Na gt Li gt K`
D. `K=Na gt Li`

Explanation

To determine the order of effective nuclear charge (Z*) for the last electron in lithium (Li), sodium (Na), and potassium (K) according to Slater's rules, we will follow these steps:

### Step 1: Write the Electronic Configurations
- **Lithium (Li)**: 1s² 2s¹ (Atomic number Z = 3)
- **Sodium (Na)**: 1s² 2s² 2p⁶ 3s¹ (Atomic number Z = 11)
- **Potassium (K)**: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ (Atomic number Z = 19)

### Step 2: Calculate the Screening Constant (σ)
Using Slater's rules, we calculate the screening constant (σ) for the outermost electron in each case.

#### For Lithium (Li):
- The last electron is in the 2s orbital.
- The contribution to σ from the 1s electrons (2 electrons) is:
- Each 1s electron contributes 0.85, so total contribution = 2 * 0.85 = 1.7.
- There are no other electrons in the same shell (2s).
- Therefore, σ = 1.7.

#### For Sodium (Na):
- The last electron is in the 3s orbital.
- The contribution to σ from the 1s², 2s², and 2p⁶ electrons (8 electrons) is:
- Each contributes 0.85, so total contribution = 8 * 0.85 = 6.8.
- The contribution from the 3s electron itself is 0 (it does not shield itself).
- Therefore, σ = 6.8.

#### For Potassium (K):
- The last electron is in the 4s orbital.
- The contribution to σ from the 1s², 2s², 2p⁶, 3s², and 3p⁶ electrons (18 electrons) is:
- Each contributes 0.85, so total contribution = 18 * 0.85 = 15.3.
- The contribution from the 3s and 3p electrons (which are in the same shell) is:
- Each contributes 0.35, so total contribution from 8 electrons = 8 * 0.35 = 2.8.
- Therefore, σ = 15.3 + 2.8 = 18.1.

### Step 3: Calculate Effective Nuclear Charge (Z*)
Using the formula Z* = Z - σ for each element:
- **For Lithium (Li)**:
- Z* = 3 - 1.7 = 1.3
- **For Sodium (Na)**:
- Z* = 11 - 6.8 = 4.2
- **For Potassium (K)**:
- Z* = 19 - 18.1 = 0.9

### Step 4: Order of Effective Nuclear Charge
Now we can order the effective nuclear charges calculated:
- **Li**: Z* = 1.3
- **Na**: Z* = 4.2
- **K**: Z* = 0.9

Thus, the order of effective nuclear charge (Z*) for the last electron in Li, Na, and K is:
- **Na > Li > K**

### Summary
The effective nuclear charge for the last electron in lithium, sodium, and potassium according to Slater's rules is ordered as follows:
- Sodium (Na) has the highest effective nuclear charge, followed by Lithium (Li), and then Potassium (K).