Question

Detailed calculations show that the value of \(Z_{\text { eff }}\) for the outermost electrons in Na and \(K\) 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 Na and 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 Rb atom based on the calculations for Na and K.

Short answer

The Zeff values calculated using core electron contributions are 1 for both Na and K, while using Slater's rules, the values are 1.3 for Na and 0.3 for K. Slater's rules provide a better estimation in this case. However, neither method perfectly accounts for the gradual increase in Zeff upon moving down a group. Using Slater's rules to predict Zeff for the outermost electron in the Rb atom results in a value of approximately 2.45+.

Step-by-step solution

Find atomic numbers and electronic configurations

For both Na and K, we need to know their atomic numbers and electronic configurations. We can find this information from the periodic table. For Na (Sodium): Atomic number (Z) = 11 Electronic configuration: 1s² 2s² 2p⁶ 3s¹ For K (Potassium): Atomic number (Z) = 19 Electronic configuration: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹

Calculate Zeff with core electron contributions

(a) According to the given condition, core electrons contribute 1.00, while valence electrons contribute 0.00 to the screening constant. For Na: Zeff = Z - core electrons contribution Zeff = 11 - (2+2+6) = 11 - 10 = 1 For K: Zeff = Z - core electrons contribution Zeff = 19 - (2+2+6+2+6) = 19 - 18 = 1

Calculate Zeff using Slater's rules

(b) Using Slater's rules, the calculation of Zeff for outermost electrons in Na and K can be done as follows: For Na: Zeff = Z - σ, where σ is the shielding constant and can be calculated using Slater's rules for s and p electrons. σ = (2 * 0.85) + (8 * 1) = 9.7 Zeff = 11 - 9.7 = 1.3 For K: Zeff = Z - σ σ = (2 * 0.85) + (8 * 1) + (2 * 0.85) + (6 * 1) = 18.7 Zeff = 19 - 18.7 = 0.3

Determine more accurate Zeff

(c) Comparing the values obtained from the given Zeff to the values calculated from both methods: For Na: Given Zeff: 2.51+ Core electron contributions: 1 Slater's rule: 1.3 For K: Given Zeff: 3.49+ Core electron contributions: 1 Slater's rule: 0.3 In this case, Slater's rules provide a better estimation than just the core electron contributions for the Zeff.

Determine if either method accounts for the gradual increase of Zeff

(d) Neither method perfectly accounts for the gradual increase in Zeff that occurs upon moving down a group in the periodic table.

Predict Zeff for outermost electrons in Rb atom

(e) Since Slater's rules provide a more accurate estimation, we can use this method to predict Zeff for Rb outermost electron: Rb (Rubidium): Atomic number (Z) = 37 Electronic configuration: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s¹ Using Slater's rules for Rb: Zeff = Z - σ σ = (2 * 0.85) + (8 * 1) + (2 * 0.85) + (6 * 1) + (2 * 0.85) + (10 * 1) + (6 * 1) = 34.55 Zeff = 37 - 34.55 = 2.45 The predicted Zeff for the outermost electron in Rb atom is approximately 2.45+.

Key concepts

Understanding Slater's Rules

When we discuss the effective nuclear charge (Zeff), we are talking about the net positive charge experienced by an electron in a multi-electron atom. The effective nuclear charge is crucial for determining how strongly an electron is attracted to the nucleus in the presence of other electrons that can shield it. Slater's rules provide a systematic way to calculate this screening effect or 'shielding' by other electrons.

The rules classify electrons into different groups based on their position in the electronic configuration, and each group contributes differently to the shielding constant (σ). The contributions are summed up and subtracted from the actual nuclear charge (Z) to arrive at Zeff. Electrons closer to the nucleus shield more effectively than those further away, which is why their contributions to σ are different.

Key Steps to Calculate Shielding using Slater's Rules

• Identify the electron of interest and its n and l quantum numbers.
• Group surrounding electrons into categories as prescribed by Slater.
• Apply the prescribed shielding values for each electron group.
• Sum these shielding constants to find the total shielding effect (σ).
• Calculate Zeff by subtracting σ from the actual nuclear charge (Z).

To put this into practice, consider a sodium atom with an electronic configuration of 1s² 2s² 2p⁶ 3s¹. According to Slater's rules, the inner-shell electrons effectively shield the outermost electron, resulting in a calculation that reflects a more realistic view of the atom compared to simpler models.

The Role of Atomic Number

The atomic number, often denoted as Z, is a fundamental property of an element that indicates the number of protons in the nucleus of an atom. Aside from identifying the element itself, the atomic number directly establishes the overall positive charge of the nucleus, which influences the strength of the attraction to negatively charged electrons.

This positive charge, however, is not fully experienced by the electrons due to the presence of other electrons that cause a screening effect, as previously explained with Slater's rules. The atomic number, therefore, does not provide a complete picture of the forces acting on an individual electron in a multi-electron atom. It is the starting point, though, for calculating the effective nuclear charge, by providing the absolute positive charge that will then be tempered by the electron-electron interactions.

Deciphering Electronic Configuration

The electronic configuration of an atom describes how electrons are distributed among the different orbitals and energy levels. Each configuration is a map that allows us to understand the structure of an electron's arrangement, which is crucial when applying Slater's rules or when contemplating the properties of an element, like chemical reactivity or magnetic behavior.

For example, when solving the exercise to find the Zeff for potassium, one must identify that potassium has the electronic configuration of 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹. This configuration helps us determine the shielding contribution from each group of electrons, as required by Slater's rules, and highlights the importance of understanding electron configurations fully to be able to calculate the effective nuclear charge accurately.