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(a) The energy of an electron in the 4\(s\) state of sodium is \(-1.947 \mathrm{eV} .\) What is the effective net charge of the nucleus "seen" by this electron? On the average, how many electrons screen the nucleus? (b) For an outer electron in the 4\(p\) state of potassium, on the average 17.2 inner electrons screen the nucleus. (i) What is the effective net charge of the nucleus "seen" by this outer electron? (ii) What is the energy of this outer electron?

Short Answer

Expert verified
(a) Sodium 4s electron: \(Z_{eff} = 1.95\), 9 electrons screen. (b) Potassium 4p electron: \(Z_{eff} = 1.8\), energy is \(-0.765\) eV.

Step by step solution

01

Understanding the Problem

We are given the energy of an electron in the 4\(s\) state of sodium and need to determine the effective nuclear charge and the number of screening electrons. We are also given the number of screening electrons for a 4\(p\) state electron in potassium and need to calculate the effective nuclear charge and energy.
02

Calculate the Effective Nuclear Charge for Sodium 4s

The formula to calculate the effective nuclear charge \(Z_{eff}\) is \(Z_{eff} = Z - ext{S}\), where \(Z\) is the atomic number and \(S\) is the screening constant or the number of electrons that screen the nucleus. For sodium, \(Z = 11\). First, calculate \(Z_{eff}\) using the provided 4\(s\) state energy \(-1.947\) eV and appropriate equations or assumptions.
03

Determine Screening for Sodium 4s

Subtract the calculated \(Z_{eff}\) from the atomic number of sodium, \(Z\). This will give you the average number of screening electrons for the electron in the 4\(s\) state.
04

Calculate the Effective Nuclear Charge for Potassium 4p

For potassium, we know \(Z = 19\) and 17.2 electrons screen the nucleus. Use the formula \(Z_{eff} = Z - ext{S}\) to calculate \(Z_{eff}\) for the 4\(p\) state electron.
05

Calculate the Energy for Potassium 4p

Use the calculated effective nuclear charge \(Z_{eff}\) for the potassium 4\(p\) electron to determine the energy. The energy can be calculated using the relation \(E = -13.6 \cdot \frac{Z_{eff}^2}{n^2}\), where \(n = 4\) for the 4\(p\) state.

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Key Concepts

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

Electron Screening
Electron screening, also known as electron shielding, refers to the phenomenon where inner electrons in an atom partially shield outer electrons from the full effect of the positive charge of the nucleus. This can be understood as the "effective nuclear charge" that each electron feels, which is less than the actual nuclear charge due to the presence of intervening electrons. As a result, electrons further away from the nucleus experience a reduced electrostatic pull.
The core purpose of electron screening is to explain why electrons in outer shells, or energy levels, experience a different attraction to the nucleus compared to those in inner shells. For instance:
  • Inner electrons are closer to the nucleus and are not shielded by any other electrons, so they experience a greater nuclear charge.
  • Outer electrons are shielded by inner electrons, reducing the effective nuclear charge they experience.
Understanding electron screening is crucial for predicting the chemical properties of atoms, as it affects atomic size, ionization energy, and overall atomic reactivity.
Atomic Structure
Atomic structure is the framework that describes how electrons are arranged around the nucleus of an atom. At the core of every atom is the nucleus, which is positively charged due to the presence of protons and neutral neutrons. Surrounding the nucleus are electrons, which are negatively charged particles that occupy various energy levels or shells.
The atomic structure can be broken down into:
  • Protons and neutrons within the nucleus — they define the element and its isotopes.
  • Electrons in orbitals or shells around the nucleus — they determine chemical behaviors and play a critical role in bonding.
Each shell or energy level is further divided into subshells (such as s, p, d, and f), which are characterized by specific numbers of electron orbitals, affecting the distribution and energy of electrons in an atom. Understanding the atomic structure and arrangement of electrons helps in predicting how an atom will interact with other atoms and form bonds.
Energy Levels
Energy levels in an atom, also known as electron shells, are regions around the nucleus where electrons are most likely to be found. Each level is associated with a specific energy, and the electrons in higher energy levels have more energy than those closer to the nucleus. The energy level of an electron is an important factor in determining its chemical behavior and the types of bonds it can form.
Key characteristics of energy levels include:
  • The principal quantum number, denoted as n, which defines the size and energy of the shell. For example, the 4s state of sodium refers to an electron in the fourth energy level.
  • The number of subshells within each energy level, which determines the possible electron configurations.
In scenarios involving effective nuclear charge, understanding these energy levels is vital, as electrons in higher energy levels are more shielded from the nucleus and experience a different effective nuclear charge. This concept helps explain variations in ionization energies and atomic radii across different elements.

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Most popular questions from this chapter

(a) If the value of \(L_{z}\) is known, we cannot know either \(L_{x}\) or \(L_{y}\) precisely. But we can know the value of the quantity \(\sqrt{L_{x}^{2}+L_{y}^{2}} .\) Write an expression for this quantity in terms of \(l, m_{1}\) and \(\hbar .\) (b) What is the meaning of \(\sqrt{L_{x}^{2}+L_{y}^{2}} ?\) (c) For a state of nonzero orbital angular momentum, find the maximum and minimum values of \(\sqrt{L_{x}^{2}+L_{y}^{2}} .\) Explain your results.

For germanium (Ge, \(Z=32 )\) , make a list of the number of electrons in each subshell \((1 s, 2 s, 2 p, \ldots) .\) Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3 .

A hydrogen atom in an \(n=2, l=1, m_{l}=-1\) state emits a photon when it decays to an \(n=1, l=0, m_{l}=0\) ground state. (a) In the absence of an external magnetic field, what is the wave-length of this photon? (b) If the atom is in a magnetic field in the \(+z\) -direction and with a magnitude of 2.20 \(\mathrm{T}\) , what is the shift in the wavelength of the photon from the zero-field value? Docs the magnetic field increase or decrease the wavelength? Disregard the effect of electron spin. [Hint: Use the result of Problem \(39.56(\mathrm{c}) . ]\)

Effective Magnetic Field. An electron in a hydrogen atom is in the 2\(p\) state. In a simple model of the atom, assume that the electron circles the proton in an orbit with radius \(r\) equal to the Bohr-model radius for \(n=2 .\) Assume that the speed \(v\) of the orbiting electron can be calculated by setting \(L=m v r\) and taking \(L\) to have the quantum-mechanical value for a 2\(p\) state. In the frame of the electron, the proton orbits with radius \(r\) and speed \(v\) . Model the orbiting proton as a circular current loop, and calculate the magnetic field it produces at the location of the electron.

An electron in a hydrogen atom is in an \(s\) level, and the atom is in a magnetic field \(\vec{B}=B \hat{k} .\) Explain why the "spin up" state \(\left(m_{s}=+\frac{1}{2}\right)\) has a higher energy than the "spin down" state \(\left(m_{s}=-\frac{1}{2}\right)\)

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