Detailed calculations show that the value of \(Z_{\text {eff }}\) for the outermost electrons in \(\mathrm{Si}\) and \(\mathrm{Cl}\) atoms is \(4.29+\) and \(6.12+\), respectively. (a) What value do you estimate for \(Z_{\text {eff }}\) experienced by the outermost electron in both Si and Cl by assuming core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant? (b) What values do you estimate for \(Z_{\text {eff }}\) using Slater's rules? (c) Which approach gives a more accurate estimate of \(Z_{\text {eff }} ?\) (d) Which method of approximation more accurately accounts for the steady increase in \(Z_{\text {eff }}\) that occurs upon moving left to right across a period? (e) Predict \(Z_{\text {eff }}\) for a valence electron in P, phosphorus, based on the calculations for \(\mathrm{Si}\) and \(\mathrm{Cl}\).

Step by step solution

01

Estimated \(Z_{\text {eff}}\) using core and valence electron contributions

For Si (Silicon), we know that there are 10 core electrons and 4 valence electrons with an atomic number (Z) of 14. Assuming core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant, the \(Z_{\text {eff}}\) for Si = 14 - 10 = 4. For Cl (Chlorine), there are 18 core electrons and 7 valence electrons with an atomic number (Z) of 17. Using the same assumption, the \(Z_{\text {eff}}\) for Cl = 17 - 18 =(-1)+7 = 6.

02

Estimate \(Z_{\text {eff}}\) using Slater's rules

For Si, Slater's rule formula for \(Z_{\text {eff}}\) is: \( Z_{\text{eff}} = Z - S \), where Z is the atomic number and S is the shielding constant. We will use the following formula derived from Slater's rules to calculate S: S = 0.85(2) + 0.85(2) + 0.35(2) S = 1.70 + 1.70 + 0.70 = 4.10 Now, we will calculate the \(Z_{\text {eff}}\) for Si: \( Z_{\text{eff}} = 14 - 4.10 = 9.90\) Similar calculation can be done for Cl: S = 1.0(2) + 0.85(8) + 0.35(7) = 2 + 6.80 + 2.45 = 11.25 Now, we will calculate the \(Z_{\text {eff}}\) for Cl: \( Z_{\text{eff}} = 17 - 11.25 = 5.75\)

03

Compare the results with the given values

Comparing the given values and the estimated values using the different methods, we find that: a) Core and valence electron contributions: Si (4) and Cl (6) b) Slater's rules: Si (9.90) and Cl (5.75) The given values are Si (4.29+) and Cl (6.12+). It seems that the Slater's rules provide a more accurate estimate for Si, and the core and valence electron contributions provide a closer estimate for Cl.

04

Determine the method accounting for a steady increase in \(Z_{\text {eff}}\)

The Slater's rules method accounts for the steady increase in \(Z_{\text {eff}}\) that occurs upon moving left to right across a period as it considers both core and valence electrons with appropriate contributions.

05

Predict \(Z_{\text {eff}}\) for a valence electron in P (Phosphorus)

Based on the Slater's rules, we can predict the \(Z_{\text {eff}}\) for a valence electron in P (Phosphorus). Phosphorus (P) has an atomic number (Z) of 15 and follows the same pattern as Si and Cl. We will calculate the shielding constant (S) for P: S = 0.85(2) + 0.85(8) + 0.35(0) = 1.70 + 6.80 = 8.50 Now, we will calculate the \(Z_{\text {eff}}\) for P: \( Z_{\text{eff}} = 15 - 8.50 = 6.50\) So, the predicted \(Z_{\text {eff}}\) for a valence electron in P (Phosphorus) using Slater's rules is 6.50.

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