Problem 14 - Chemistry: The Central Science

Question

Detailed calculations show that the value of \(Z_{\text {eff }}\) for \(S i\) 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 \(\mathrm{Cl}\) 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?

Step-by-Step Solution

Answer

The estimated values of \(Z_{eff}\) for Silicon (Si) and Chlorine (Cl) using the core and valence electron approximation are 4 and 7, respectively, while using Slater's rules give an estimate of 5.6 for Si and 8.6 for Cl. Slater's rules provide more accurate estimates of \(Z_{eff}\) when compared to the given values (4.29 for Si and 6.12 for Cl) and also more accurately account for the steady increase in \(Z_{eff}\) that occurs upon moving left to right across a period.

Step 1: Electronic Configuration of Si and Cl

Before estimating the \(Z_{eff}\), we need to know the electronic configuration of Silicon (Si) and Chlorine (Cl) atoms. Silicon (Si) has an atomic number (Z) of 14, and its electronic configuration will be: \(1s^2, 2s^2, 2p^6, 3s^2, 3p^2\) Chlorine (Cl) has an atomic number (Z) of 17, and its electronic configuration will be: \(1s^2, 2s^2, 2p^6, 3s^2, 3p^5\)

Step 2: Estimating \(Z_{eff}\) using the core and valence electron approximation

For this approximation, we assume core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant. For Silicon (Si): Total core electrons = 10 (from 1s, 2s, and 2p shells) Total valence electrons = 4 (from 3s and 3p orbitals) \(Z_{eff} (Si) = Z - S\) where \(S\) is the screening constant. \(Z_{eff} (Si) = 14 - (10*1 + 4*0) = 4\) For Chlorine (Cl): Total Core Electrons = 10 (from 1s, 2s, and 2p shells) Total Valence Electrons = 7 (from 3s and 3p orbitals) \(Z_{eff} (Cl) = Z - S\) \(Z_{eff} (Cl) = 17 - (10*1 + 7*0) = 7\) We estimated \(Z_{eff}\) as 4 for Si and 7 for Cl using this approximation.

Step 3: Estimating \(Z_{eff}\) using Slater's rules

Slater's rules for calculating \(Z_{eff}\) can be remembered by the following rules for the screening constant \(S\): 1. \(1s\) electrons in a group are shielded by other \(1s\) electrons with a factor of 0.3. 2. Electrons in the same group (other than 1s electrons) are shielded by other electrons in that group with a factor of 0.35. 3. Electrons in the next group (higher principle quantum number) are shielded by electrons with a factor of 0.85. 4. Electrons in groups of lower principle quantum numbers from the electron in question are shielded with a factor of 1.00. Using the rules mentioned above for Slater's rules, we can calculate the screening constant (S) for Si and Cl: For Silicon (Si): \(3s^2\) electrons: 2 \((2 - 1) \times 1\) + 6 \((2 - 1) \times 0.85\) + 2(0.35) = 9.1 \(3p^2\) electrons: 8 \((2 - 1) \times 1\) + 2 \((2 - 1) \times 0.85\) + 2(0.35) = 7.7 We took the average of \(S_{3s}\) and \(S_{3p}\) for total screening constant: \(S_{total}\) = \(0.5(S_{3s} + S_{3p})\) = 8.4 \(Z_{eff} (Si) = Z - S = 14 - 8.4 = 5.6\) For Chlorine (Cl): \(3s^2\) electrons: 2 \((2 - 1) \times 1\) + 6 \((2 - 1) \times 0.85\) + 5 \((2 - 1) \times 0.35\) = 9.1 \(3p^5\) electrons: 8 \((2 - 1) \times 1\) + 2 \((2 - 1) \times 0.85\) + 4 \((2 - 1) \times 0.35\) = 7.7 We took the average of \(S_{3s}\) and \(S_{3p}\) for total screening constant: \(S_{total}\) = \(0.5(S_{3s} + S_{3p})\)= 8.4 \(Z_{eff} (Cl) = Z - S = 17 - 8.4 = 8.6\) We estimated \(Z_{eff}\) as 5.6 for Si and 8.6 for Cl using Slater's rules.

Step 4: Comparing the results

Now, we are comparing the results of two approximation methods with given values of \(Z_{eff}\). Given values of \(Z_{eff}\): \(Z_{eff}\) (Si) = 4.29 \(Z_{eff}\) (Cl) = 6.12 Core and Valence electron approximation: \(Z_{eff}\) (Si) = 4 \(Z_{eff}\) (Cl) = 7 Slater's rules: \(Z_{eff}\) (Si) = 5.6 \(Z_{eff}\) (Cl) = 8.6 (c) Considering the given values of \(Z_{eff}\) and comparing the results from both methods, Slater's rules give a more accurate estimate for \(Z_{eff}\). (d) Slater's rule more accurately accounts for the steady increase in \(Z_{eff}\) that occurs upon moving left to right across a period since it is a more complex and accurate method, considering a more specific configuration of electrons and their shielding effects.

Key Concepts

Electronic Configuration

The electronic configuration of an atom provides a detailed account of the arrangement of electrons across various shells and subshells. This configuration plays a crucial role in determining the chemical properties and reactivity of an element. In the context of Silicon (Si) and Chlorine (Cl), understanding their electronic configuration is the first step towards estimating the effective nuclear charge, or \(Z_{eff}\). For Si, with an atomic number of 14, the electrons are distributed as follows: \(1s^2, 2s^2, 2p^6, 3s^2, 3p^2\). For Cl, which has an atomic number of 17, the configuration is \(1s^2, 2s^2, 2p^6, 3s^2, 3p^5\).
These configurations highlight the core and valence electrons for each element. Core electrons occupy the inner shells (\(1s, 2s, 2p\) for both Si and Cl), while valence electrons (\(3s, 3p\)) are in the outer shell, significantly influencing each element's chemical behavior.
Knowing this configuration helps us apply various rules and calculations to determine how much of the nuclear charge an outer electron 'feels', known as the effective nuclear charge.

Slater's Rules

Slater's rules offer a systematic way to calculate the effective nuclear charge \(Z_{eff}\) of an electron in an atom by accounting for electron shielding. Shielding occurs when inner-shell electrons repel outer electrons, reducing the full charge of the nucleus experienced by the outer electrons.
Slater's rules involve certain criteria:

• Electrons within the same group (except \(1s\)) shield each other with a factor of 0.35.
• Electrons in \(1s\) shield each other with a factor of 0.3.
• Electrons one shell lower shield with a factor of 0.85.
• Electrons two or more shells lower shield completely with a factor of 1.00.

To calculate \(Z_{eff}\) for Silicon and Chlorine using Slater's rules, calculate the screening constant \(S\) first, then subtract it from the atomic number \(Z\). For example, because Silicon's outer electrons are in the third shell, electrons in the first and second shells provide a shielding effect, along with electrons within the same New Zealand. This rule set makes Slater's method more precise than simply considering core and valence electrons, improving accuracy for \(Z_{eff}\) calculations as seen in the exercise results.

Screening Constant

The screening constant \(S\) is vital in determining the effective nuclear charge \(Z_{eff}\), which helps us understand the "pull" felt by outer electrons in an atom from its nucleus. \(Z_{eff}\) is calculated using the formula: \(Z_{eff} = Z - S\), where \(Z\) is the atomic number. Understanding the screening effect helps explain trends in the periodic table and chemical properties of elements.
The role of the screening constant is to quantify how much of the nuclear charge gets 'screened' by other electrons. Core electrons close to the nucleus effectively reduce the nuclear charge that the valence, more distant, electrons feel. In the simplest method, core electrons typically count as 1 each towards \(S\), while valence electrons contribute little or nothing. However, in reality, the situation is more complex, as Slater's rules describe, with varying shielding effects.
Through the screening constant, we gain insight into the variance of\(Z_{eff}\) across different elements. Elements with higher \(Z_{eff}\) have outer electrons that feel a stronger pull toward the nucleus, influencing their atomic behavior and how these elements interact with others.