Problem 14 - Chemistry Central Science

Question

Detailed calculations show that the value of \(Z_{\text {eff }}\) for \(\mathrm{Si}\) 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 \(\mathrm{Si}\) 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

In summary, the calculated Z_eff values using given assumptions in part (a) are 4 for Si and 7 for Cl, while using Slater's rules in part (b), the values are 4.4 for Si and 5.65 for Cl. Comparing these values with the given detailed calculations (4.29+ for Si and 6.12+ for Cl), Slater's rules provide a more accurate estimate of Z_eff and also more accurately account for the steady increase in Z_eff across a period.

Step 1: Review the concepts of effective nuclear charge and screening constant

Effective nuclear charge (Z_eff) is the net positive charge experienced by an electron in an atom. The screening constant is a value that accounts for the shielding effect of inner (core) electrons on outer (valence) electrons.

Step 2: Calculate Z_eff using given assumptions in part (a)

For Silicon (Si) with atomic number 14: \(Z = 14\) For Chlorine (Cl) with atomic number 17: \(Z = 17\) Assuming core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant, we can find Z_eff for Si and Cl as follows: For Si: Z_eff = Z - Screening_constant = 14 - (10 * 1) = 4 For Cl: Z_eff = Z - Screening_constant = 17 - (10 * 1) = 7

Step 3: Calculate Z_eff using Slater's rules in part (b)

Slater's rules for calculating the screening constant (S) are as follows: 1. For each electron in the same group, count 0.35. 2. For each electron in the next lower group, count 0.85. 3. For each electron in the "s" and "p" orbitals of the two lower groups, count 1.00. Using Slater's rules, we can calculate the screening constant, S, for Si and Cl atoms: For Si (with electron configuration \(1s^2 2s^2 2p^6 3s^2 3p^2\)): S = (2 * 0.35) + (8 * 0.85) + (2 * 1.00) = 9.6 For Cl (with electron configuration \(1s^2 2s^2 2p^6 3s^2 3p^5\)): S = (5 * 0.35) + (8 * 0.85) + (2 * 1.00) = 11.35 Now, we can calculate Z_eff in both Si and Cl according to Slater's rules: For Si: Z_eff = Z - S = 14 - 9.6 = 4.4 For Cl: Z_eff = Z - S = 17 - 11.35 = 5.65

Step 4: Compare the results for Z_eff from both methods with the given detailed calculations

Given detailed calculations show that Z_eff for Si and Cl are 4.29+ and 6.12+, respectively. Calculated Z_eff with given assumptions in part (a): Si - 4 Cl - 7 Calculated Z_eff with Slater's rules in part (b): Si - 4.4 Cl - 5.65

Step 5: Determine which approach provides a more accurate estimate of Z_eff

Comparing the Z_eff values from both methods with the given detailed calculations, we observe that Slater's rules yield a closer approximation to the detailed results for both Si and Cl than the method used in part (a). Therefore, Slater's rules provide a more accurate estimate of Z_eff.

Step 6: Determine which method more accurately accounts for the steady increase in Z_eff across a period

As we move from left to right across a period, Z_eff increases because the nuclear charge is increasing without significant shielding from additional core electrons. Comparing the estimates from the methods in part (a) and (b), it can be observed that Slater's rules account for the steady increase in Z_eff more accurately than the method in part (a), as values of Z_eff using Slater's rules provide a better approximation of the actual values.

Key Concepts

Screening Constant

The concept of the screening constant is key to understanding how electrons within an atom interact with each other. In simple terms, while electrons are attracted to the nucleus due to its positive charge, they are also repelled by other electrons. This is where the screening constant comes into play.
The screening constant, often symbolized as \( S \), quantifies the amount of this electron repulsion. It helps in determining the "effective nuclear charge" or \( Z_{eff} \), which is the net positive charge felt by an electron in an atom. The principle is fairly straightforward: core electrons, very close to the nucleus, do most of the screening, reducing the positive charge felt by outer electrons.

• Core electrons typically contribute significantly, usually a value close to one, for each electron.
• Valence electrons, being part of the outermost shell, contribute very little as they experience a larger fraction of the nuclear charge.

When calculating \( Z_{eff} \) for atoms like Silicon and Chlorine, it becomes crucial to account for these interactions properly to predict chemical properties accurately.

Slater's Rules

Slater's Rules offer a systematic way to estimate the effective nuclear charge by considering shielding effects in a more structured manner. These rules provide a method to calculate the screening constant \( S \) more accurately than simpler estimations.
Here's how Slater’s Rules work:

• Electrons in the same group shield each other partially, and each contributes 0.35 to \( S \).
• Electrons in the next lower group provide more substantial shielding, each contributing 0.85.
• Electrons in even lower "s" and "p" orbitals contribute a full 1.00 to the screening constant. This acknowledges the significant repulsion they cause.

For example, using the electron configurations of Silicon (\(1s^2 2s^2 2p^6 3s^2 3p^2\) ) and Chlorine (\(1s^2 2s^2 2p^6 3s^2 3p^5\) ), we apply Slater's Rules to compute \( S \). For Silicon, \( S \) comes out to be 9.6, and for Chlorine, it is 11.35. These calculated \( S \) values refine our estimation of \( Z_{eff} \), allowing us to better understand the interaction within the atom's electron cloud.

Electron Configuration

Understanding electron configuration is vital for predicting an atom's chemical behavior, as it describes how electrons are distributed in an atom's orbitals.
Each atom has a unique electron configuration that can be depicted in notations showing the distribution of electrons across various orbital levels (1s, 2s, 2p, etc.). These configurations are crucial in determining the screening effect and effective nuclear charge. Here’s why:

• The order and filling of electron orbitals inform us about how electrons interact within an atom.
• Core and valence electrons can be identified readily from configurations, which is important for calculations involving screening constants and \( Z_{eff} \).

Take Silicon (\(1s^2 2s^2 2p^6 3s^2 3p^2\) ) and Chlorine (\(1s^2 2s^2 2p^6 3s^2 3p^5\) ) as examples. These configurations tell us:

• How electrons are layered, influencing shielding and attractions.
• The state of core versus valence electrons in determining effective nuclear charge.

By understanding electron configurations, one grasps how atomic structure affects an atom’s chemical properties and reactions across the periodic table. Each electron's role becomes clear, leading to accurate predictions of atomic behavior.