Ch.5 - Periodicity & Electronic Structure of Atoms

Problem 135

Chapter 5, Problem 135

Assume that the rules for quantum numbers are different and that the spin quantum number ms can have any of three values, ms = -1/2, 0, +1/2, while all other rules remain the same. (a) Draw an orbital-filling diagram for the element with Z = 25, showing the individual electrons in the outer-most subshell as up arrows, down arrows, or 0. How many partially filled orbitals does the element have?

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1. The atomic number (Z) of the element is 25, which means it has 25 electrons. In the periodic table, this corresponds to the element Manganese (Mn).

2. The electron configuration of Manganese is [Ar] 4s2 3d5. This means that the outermost subshell is the 3d subshell, which has 5 electrons.

3. In the 3d subshell, there are 5 orbitals. Each orbital can hold up to 2 electrons. However, according to the modified rules, each orbital can now hold up to 3 electrons (ms = -1/2, 0, +1/2).

4. To fill the 3d subshell, start by adding one electron to each orbital (with ms = +1/2) until all orbitals have one electron. This is known as Hund's rule. Then, start pairing up the electrons in each orbital (with ms = -1/2). If there are any electrons left, they will have ms = 0.

5. Count the number of orbitals that are not completely filled (i.e., they don't have 3 electrons). This will give you the number of partially filled orbitals.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. The four quantum numbers include the principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s). Each number provides specific information about the electron's energy level, shape, orientation, and spin, which are essential for understanding electron configurations and orbital filling.

Orbital Filling Diagram

An orbital-filling diagram visually represents the distribution of electrons among the various orbitals of an atom. Electrons are filled into orbitals following the Aufbau principle, Hund's rule, and the Pauli exclusion principle. In this scenario, the modified spin quantum number allows for three possible spin states, which will affect how electrons are represented in the diagram, particularly in the outermost subshell.

Electron Configuration

Electron configuration describes the arrangement of electrons in an atom's orbitals. For an element with atomic number Z = 25 (manganese), the electron configuration is typically [Ar] 4s² 3d⁵. Understanding how to write and interpret this configuration is crucial for determining the number of partially filled orbitals, which in this case involves analyzing the 3d subshell where five electrons are present, indicating partial filling.

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Textbook Question

The amount of energy necessary to remove an electron from an atom is a quantity called the ionization energy, Ei. This energy can be measured by a technique called photoelectron spectroscopy, in which light of wavelength l is directed at an atom, causing an electron to be ejected. The kinetic energy of the ejected electron (Ek) is measured by determining its veloc-ity, v (Ek = mv2/2), and Ei is then calculated using the conservation of energy principle. That is, the energy of the incident light equals Ei plus Ek. What is the ionization energy of selenium atoms in kilojoules per mole if light with l = 48.2 nm produces electrons with a velocity of 2.371 * 106 m/s? The mass, m, of an electron is 9.109 * 10-31 kg.

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Textbook Question

X rays with a wavelength of 1.54 * 10-10 m are produced when a copper metal target is bombarded with high-energy electrons that have been accelerated by a voltage difference of 30,000 V. The kinetic energy of the electrons equals the product of the voltage difference and the electronic charge in coulombs, where 1 volt-coulomb = 1 J. (a) What is the kinetic energy in joules and the de Broglie wavelength in meters of an electron that has been accel-erated by a voltage difference of 30,000 V?

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Textbook Question

In the Bohr model of atomic structure, electrons are constrained to orbit a nucleus at specific distances, given by the equation

where r is the radius of the orbit, Z is the charge on the nucleus, a0 is the Bohr radius and has a value of 5.292 * 10-11 m, and n is a positive integer (n = 1, 2, 3...) like a principal quantum number. Furthermore, Bohr concluded that the energy level E of an electron in a given orbit is

where e is the charge on an electron. Derive an equation that will let you calculate the difference ∆E between any two energy levels. What relation does your equation have to the Balmer–Rydberg equation?

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Textbook Question

A minimum energy of 7.21⨉10^-19 J is required to produce the photoelectric effect in chromium metal. (a) What is the minimum frequency of light needed to remove an electron from chromium?

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Textbook Question

A minimum energy of 7.21⨉10^-19 J is required to produce the photoelectric effect in chromium metal. (b) Light with a wavelength of 2.50⨉10^-7 m falls on a piece of chromium in an evacuated glass tube. What is the minimum de Broglie wavelength of the emitted electrons? (Note that the energy of the incident light must be conserved; that is, the photon's energy must equal the sum of the energy needed to eject the electron plus the kinetic energy of the electron.)