Question

The ionization energy of hydrogen atom is 13.6 eV and the first ionization energy of sodium atom is 5.1 eV. The effective nuclear charge experienced by the valence electron of sodium atom is ................. (Round off to one decimal place)

Hint:

The effective nuclear charge is a measure of the net attractive force exerted by the nucleus on an electron, considering the shielding effect of other electrons.

Answer 1

Correct Answer: 1.7

Step 1: Formula for effective nuclear charge.
The effective nuclear charge \(Z_{\text{eff}}\) can be calculated using the following relation: \[ Z_{\text{eff}} = Z - S \] where \(Z\) is the atomic number (number of protons) and \(S\) is the shielding constant.
Step 2: Ionization energy relation.
Ionization energy (\(I_1\)) is related to the effective nuclear charge by the following equation: \[ I_1 = 13.6 \, \text{eV} \times \left(\frac{Z_{\text{eff}}^2}{n^2}\right) \] where \(n\) is the principal quantum number. For sodium, \(n = 3\).
Step 3: Calculate the effective nuclear charge.
From the given data: For sodium, \(I_1 = 5.1 \, \text{eV}\). Using the relation: \[ 5.1 = 13.6 \times \left(\frac{Z_{\text{eff}}^2}{3^2}\right) \] \[ Z_{\text{eff}}^2 = \frac{5.1 \times 9}{13.6} \] \[ Z_{\text{eff}}^2 = 3.39 \] \[ Z_{\text{eff}} = 1.84 \] Thus, the effective nuclear charge is 3.1.