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Chapter 40: Problem 59

A recently named element is darmstadtium (Ds), which has 110 electrons. Assume that you can put the 110 electrons into the atomic shells one by one and can neglect any electronelectron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number \(\ell\) for the last electron?

Short Answer

Expert verified
The quantum number \\( \ell \\\) for the last electron is 2.

Step by step solution

01

Understand the Electron Filling Order

Electrons fill atomic orbitals according to the Aufbau principle, which states they occupy the lowest available energy levels first. This is governed by the sequence of energy levels: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The s sublevel can hold 2 electrons, the p sublevel 6, the d sublevel 10, and the f sublevel 14.
02

Fill Electrons in Order by Subshell

Fill the electrons into subshells sequentially: 1s (2 electrons), 2s (2), 2p (6), 3s (2), 3p (6), 4s (2), 3d (10), 4p (6), 5s (2), 4d (10), 5p (6), 6s (2), 4f (14), 5d (10), 6p (6), 7s (2), 5f (14), 6d (10). Add up the total number of electrons after each block to determine when you reach 110.
03

Identify the Filled Subshell

Calculate the number of electrons filled: - Up to 6p adds up to 86 - 7s adds 2 (88 total) - 5f adds 14 (102 total) - 6d requires 8 more electrons to reach 110. Thus, we fill 8 electrons in the 6d subshell.
04

Determine the Quantum Number \\( \ell \\\)

The quantum number \( \ell \) denotes the subshell: 0 for s, 1 for p, 2 for d, and 3 for f. Since the last electron is in the 6d subshell, its \( \ell \) value is 2.

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Key Concepts

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

Electron Configuration
Electron configuration is a way to represent how electrons are distributed among the various atomic orbitals of an element. In the case of our exercise, we are dealing with darmstadtium, which has 110 electrons. Understanding electron configuration helps predict chemical properties and behaviors.
  • The principal energy levels are numbered 1, 2, 3, etc., as the size of the orbital increases with the number.
  • Each energy level contains sublevels labeled s, p, d, and f, which house specific numbers of electrons: s can hold 2, p can hold 6, d can hold 10, and f can hold 14 electrons.
  • The configuration for an element is written as a sequence indicating the number of electrons in each subshell, such as 1s虏 2s虏 2p鈦�.
To determine the configuration, follow the Aufbau principle by filling up from the lowest to the highest energy orbitals. This practice ensures stable and lower energy states for atoms.
Atomic Orbital Filling
Atomic orbitals are filled in a specific order following the principles of quantum mechanics. The process is aided by a set of rules and diagrams, making it systematic. One of the most common is the Aufbau principle.
  • Electrons fill orbitals starting from the lowest energy level upwards.
  • The sequence of orbital filling is determined by energy: first 1s, then 2s, then 2p, and so on.
  • Each element has a unique filling order based on its atomic number and available subshells.
In the case of darmstadtium, the orbitals filled include those up to 7s and continuing through 5f and 6d. The sequential filling ensures compatibility with the laws of quantum mechanics and stability of the atom.
Quantum Numbers
Quantum numbers are crucial in defining the state of electrons within an atom. Each electron in an atom is described by four quantum numbers, which act like an address. These numbers include:
  • Principal quantum number ( \( n \)): Indicates the electron's energy level.
  • Angular momentum quantum number ( \( \,\ell \, \)): Determines the shape of the orbital, where s=0, p=1, d=2, f=3.
  • Magnetic quantum number ( \( m_{\ell} \)): Specifies the orientation of the orbital in space.
  • Spin quantum number ( \( m_s \)): Represents the electron's spin with values of +陆 or -陆.
In darmstadtium, we are asked about the \( \,\ell \, \) quantum number for the last electron in the 6d orbital, which is 2. This indicates the particular type of subshell where the electron resides, influencing its energy and shape.
Electron Subshells
Electron subshells are specific zones within an electron cloud where electrons are likely to be found. Each subshell is associated with a quantum number and has a distinct shape and capacity.
  • Subshells are defined by the angular momentum quantum number ( \( \,\ell \, \)): s, p, d, and f.
  • The number of electrons a subshell can hold is determined by the formula 2(2 \( \,\ell \, \) + 1).
  • For darmstadtium, understanding its electron subshell arrangement helps to comprehend how its 110 electrons are organized.
For instance, the configuration ends in the 6d subshell, where 8 electrons complete the overall filling process. By understanding subshells, scientists can predict reactivity patterns and electronic interactions.

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Most popular questions from this chapter

An electron is in a state with \(\ell=3 .\) (a) What multiple of \(\hbar\) gives the magnitude of \(\vec{L}\) ? (b) What multiple of \(\mu_{\mathrm{B}}\) gives the magnitude of \(\vec{\mu} ?\) (c) What is the largest possible value of \(m_{\ell}\), (d) what multiple of \(\hbar\) gives the corresponding value of \(L_{z}\), and (e) what multiple of \(\mu_{\mathrm{B}}\) gives the corresponding value of \(\mu_{\mathrm{orb}, z}\) ? (f) What is the value of the semiclassical angle \(\theta\) between the directions of \(L_{z}\) and \(\vec{L}\) ? What is the value of angle \(\theta\) for \((\mathrm{g})\) the second largest possible value of \(m_{\ell}\) and \((\mathrm{h})\) the smallest (that is, most negative) possible value of \(m_{\ell}\) ?

The wavelength of the \(K_{\alpha}\) line from iron is \(193 \mathrm{pm}\). What is the energy difference between the two states of the iron atom that give rise to this transition?

A cubical box of widths \(L_{x}=L_{y}=L_{z}=L\) contains eight electrons. What multiple of \(h^{2} / 8 m L^{2}\) gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

A rectangular corral of widths \(L_{x}=L\) and \(L_{y}=2 L\) contains seven electrons. What multiple of \(h^{2} / 8 m L^{2}\) gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

A pulsed laser emits light at a wavelength of \(694.4 \mathrm{~nm}\). The pulse duration is \(12 \mathrm{ps}\), and the energy per pulse is \(0.150 \mathrm{~J}\). (a) What is the length of the pulse? (b) How many photons are emitted in each pulse?
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