/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Detailed calculations show that ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 15

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

Expert verified
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

01

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: 1s虏 2s虏 2p鈦� 3s鹿 For K (Potassium): Atomic number (Z) = 19 Electronic configuration: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 4s鹿
02

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
03

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 - 蟽, where 蟽 is the shielding constant and can be calculated using Slater's rules for s and p electrons. 蟽 = (2 * 0.85) + (8 * 1) = 9.7 Zeff = 11 - 9.7 = 1.3 For K: Zeff = Z - 蟽 蟽 = (2 * 0.85) + (8 * 1) + (2 * 0.85) + (6 * 1) = 18.7 Zeff = 19 - 18.7 = 0.3
04

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.
05

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.
06

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: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 4s虏 3d鹿鈦� 4p鈦� 5s鹿 Using Slater's rules for Rb: Zeff = Z - 蟽 蟽 = (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+.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 (蟽). 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 蟽 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 (蟽).
  • Calculate Zeff by subtracting 蟽 from the actual nuclear charge (Z).
To put this into practice, consider a sodium atom with an electronic configuration of 1s虏 2s虏 2p鈦� 3s鹿. 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 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 4s鹿. 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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

One way to measure ionization energies is ultraviolet photoelectron spectroscopy (PES), a technique based on the photoelectric efect. (Section 6.2) In PES, monochromatic light is directed onto a sample, causing electrons to be emitted. The kinetic energy of the emitted electrons is measured. The diference between the energy of the photons and the kinetic energy of the electrons corresponds to the energy needed to remove the electrons (that is, the ionization energy). Suppose that a PES experiment is performed in which mercury vapor is irradiated with ultraviolet light of wavelength 58.4 nm. (a) What is the energy of a photon of this light, in joules? (b) Write an equation that shows the process corresponding to the first ionization energy of Hg. (c) The kinetic energy of the emitted electrons is measured to be \(1.72 \times 10^{-18} \mathrm{J} .\) What is the first ionization energy of \(\mathrm{Hg},\) in \(\mathrm{kJ} / \mathrm{mol} ?(\mathbf{d})\) Using Figure \(7.10,\) determine which of the halogen elements has a first ionization energy closest to that of mercury.

Consider the first ionization energy of neon and the electron affinity of fluorine. (a) Write equations, including electron configurations, for each process. (b) These two quantities have opposite signs. Which will be positive, and which will be negative? (c) Would you expect the magnitudes of these two quantities to be equal? If not, which one would you expect to be larger?

Consider the isoelectronic ions \(\mathrm{Cl}^{-}\) and \(\mathrm{K}^{+}\) . (a) Which ion is smaller? (b) Using Equation 7.1 and assuming that core electrons contribute 1.00 and valence electrons contribute nothing to the screening constant, \(S\) , calculate \(Z_{\text { eff}}\) for these two ions. (c) Repeat this calculation using Slater's rules to estimate the screening constant, \(S.\) (d) For isoelectronic ions, how are effective nuclear charge and ionic radius related?

(a) Why is calcium generally more reactive than magnesium? (b) Why is calcium generally less reactive than potassium?

Predict whether each of the following oxides is ionic or molecular: \(\operatorname{Sn} \mathrm{O}_{2}, \mathrm{Al}_{2} \mathrm{O}_{3}, \mathrm{CO}_{2}, \mathrm{Li}_{2} \mathrm{O}, \mathrm{Fe}_{2} \mathrm{O}_{3}, \mathrm{H}_{2} \mathrm{O} .\)
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Analysis

Read Explanation

Physical Chemistry

Read Explanation

Nuclear Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Organic Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.