Question

Detailed calculations show that the value of \(Z_{\text {eff }}\) for the outermost electrons in \(\mathrm{Na}\) and \(\mathrm{K}\) atoms is \(2.51+\) and \(3.49+\), respectively. (a) What value do you estimate for \(Z_{\text {eff }}\) experienced by the outermost electron in both \(\mathrm{Na}\) and \(\mathrm{K}\) 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) Does either method of approximation account for the gradual increase in \(Z_{\text {eff }}\) that occurs upon moving down a group? (e) Predict \(Z_{\text {eff }}\) for the outermost electrons in the \(\mathrm{Rb}\) atom based on the calculations for \(\mathrm{Na}\) and \(\mathrm{K}\).

Short answer

In summary, neither method (a) nor Slater's rules accurately estimate the effective nuclear charge, $Z_{\text {eff }}$, for the outermost electrons in Sodium (Na) and Potassium (K) atoms. Both methods overestimate $Z_{\text {eff }}$ and fail to account for the gradual increase in $Z_{\text {eff }}$ that occurs upon moving down a group. Based on the calculations, it is impossible to make an accurate prediction for $Z_{\text {eff }}$ of the outermost electron in Rubidium (Rb); however, we can roughly estimate that it should be higher than 3.49+. More accurate methods should be employed to estimate $Z_{\text {eff }}$ for the outermost electron in Rb.

Step-by-step solution

Estimating Z_eff using core and valence electron contributions.

First, we need to calculate the effective nuclear charges for the outermost electrons in Na and K, considering that each core electron contributes 1.00 to the screening constant, and valence electrons contribute 0.00. We can estimate the Z_eff using the formula: \[Z_{eff} = Z - \sigma\] Where Z is the atomic number, and σ is the screening constant. In Na, Z = 11, and there are 10 core electrons, so σ = 10. For K, Z = 19, and there are 18 core electrons, so σ = 18. Applying the formula: For Na: \(Z_{eff} = 11 - 10 = 1\) For K: \(Z_{eff} = 19 - 18 = 1\)

Estimating Z_eff using Slater's rules.

Now, we will calculate the effective nuclear charges using Slater's rules. For Na and K, the outermost electrons are in the 3s and 4s orbitals, respectively. According to Slater's rules, the screening constant for the last s electron can be calculated using the following rules: - Electrons in the same group (n-1) and n (if the electron considered is ns) contribute 0.35 each - Electrons in the groups between 1 and n-1 contribute 0.85 each - Electrons in the 1s group contribute 1.00 each For Na: - 2 electrons in the 2s and 2p groups contribute 0.85 each: 2 * 0.85 - 1 electron in the 1s group contributes 1.00: 1 sigma_Na = 2(0.85) + 1 = 2.70 For K: - 2 electrons in the 3s and 3p groups contribute 0.85 each: 2 * 0.85 - 8 electrons in the 2s and 2p groups contribute 1.00 each: 8 - 1 electron in the 1s group contributes 1.00: 1 sigma_K = 2(0.85) + 8 + 1 = 10.70 Now plug these values into the formula for Z_eff: For Na: \(Z_{eff} = 11 - 2.70 = 8.30\) For K: \(Z_{eff} = 19 - 10.70 = 8.30\)

Comparing the estimates with the actual values.

We have inferred that: - Using method (a) (core and valence electrons contributions): Z_eff is 1 for both Na and K. - Using Slater's rules: Z_eff in Na and K is 8.30. The actual values given: - For Na: Z_eff = 2.51+ - For K: Z_eff = 3.49+ Comparing these numbers, we can conclude that neither method estimates Z_eff accurately. Both methods overestimate the effective nuclear charge.

Discussing gradual increase in Z_eff.

Both method (a) and Slater's rules fail to account for the gradual increase in Z_eff that occurs upon moving down a group. In the case of method (a), the Z_eff remains constant, and for Slater's rules, the Z_eff is the same for both Na and K.

Predicting Z_eff for Rb.

Based on the calculations for Na and K, it is impossible to make an accurate prediction for Z_eff of the outermost electron in Rb. However, we can roughly estimate that it should be higher than Z_eff of K (3.49+). It would be advisable to use more accurate methods to estimate the Z_eff for the outermost electron in Rb.

Key concepts

Screening Constant

The concept of the screening constant, often denoted as \(\sigma\), is pivotal in calculating the effective nuclear charge \(Z_{\text{eff}}\). This parameter accounts for the reduction of nuclear charge that an electron experiences due to the presence of other electrons between it and the nucleus.
In essence, electrons in the inner shells "screen" or "shield" the outer electrons from the full effect of the nucleus' positive charge. This screening results in a lower effective nuclear charge experienced by the outermost electrons.

The formula used to calculate the effective nuclear charge is:

• \(Z_{\text{eff}} = Z - \sigma\)

where \(Z\) is the atomic number, which represents the total number of protons in the nucleus. The screening constant is a sum of contributions from the electrons, and its value varies based on which electrons are present and their positions. In basic calculations, core electrons are often considered to contribute a value of 1.00 each to the screening constant, while valence electrons may contribute less or even zero, as seen in simplistic models. This variance is crucial to understanding how shielding impacts the effective nuclear charge across different elements.

Slater's Rules

Slater's rules provide a method to estimate the screening constant more accurately than simple assumptions might. Developed by John C. Slater, these rules offer a systematic approach for calculating \(\sigma\) by considering the type and arrangement of electrons in different shells.
Utilizing Slater's rules involves several guidelines, tailored for different orbitals:

• Electrons in the same principal energy level as the electron being considered contribute 0.35 each, except for the \(1s\) electrons, where the same electron contributes 0.30.
• Electrons in the next lower shell contribute 0.85 each.
• Electrons even deeper, including others across lower shells, contribute 1.00 each to \(\sigma\).

By summing these contributions, you can derive a more nuanced \(\sigma\), leading to a realistic calculation of \(Z_{\text{eff}}\).

When applying Slater's rules, one accounts for the electron distribution in a given atom. For instance, for an electron in the 3s or 3p orbital, the contribution from 2s and 2p electrons and further inner shell electrons will significantly affect the overall screening constant. Slater's rules highlight the differences in how electrons shield one another based on their energy levels and provide a more precise framework for understanding effective nuclear charge.

Valence Electrons

Valence electrons are the electrons found in the outermost shell of an atom. These electrons play a crucial role in chemical reactions, as they are typically the ones involved in bonding and interactions with other atoms. Understanding their behavior is essential for predicting an element's chemical properties.
The effective nuclear charge experienced by valence electrons can be markedly different from the atomic number, due to the screening effects by core electrons. As the outermost electrons, valence electrons feel a diminished attraction to the nucleus due to shielding by inner electrons.
This concept explains why elements in the same group on the periodic table often exhibit similar reactivity, as they have a similar number of valence electrons undergoing similar effective nuclear charge conditions. The strength of these electrons' interactions with other atoms directly influences the element's ability to bond and form compounds.
In calculations, while core electrons might be assumed to fully shield nuclear charge, valence electrons might not contribute to the screening, highlighting their significance in determining the net force felt by an electron in the outer shell. Recognizing the impact of valence electrons in terms of \(Z_{\text{eff}}\) extends one's grasp of both atomic structure and reactivity patterns in chemistry.