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Chapter 13: Problem 155

If the numbers of orbitals of a particular type were \((3 l+1)\), but spin quantum numbers were only \(+1 / 2\) and \(-1 / 2\), then \(\mathrm{d}\) -type orbitals will contain a maximum of electrons. (a) 10 (b) 14 (c) 7 (d) 5

Short Answer

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Step by step solution

01

Understanding the formula for the number of orbitals

The given formula for the number of orbitals of a particular type is \(3l + 1\). This formula is to be used to find the number of \(d\)-type orbitals, where \(l\) represents the azimuthal quantum number corresponding to the \(d\)-type orbitals.
02

Calculate the number of \(d\)-type orbitals

For \(d\)-type orbitals, the azimuthal quantum number \(l = 2\). Plug this value into the formula to calculate the number of \(d\)-type orbitals: \[\text{Number of } d\text{-type orbitals} = 3(2) + 1 = 7.\]
03

Find the maximum number of electrons in \(d\)-type orbitals

Each orbital can hold a maximum of 2 electrons because there are only two possible spin quantum numbers: \(+1/2\) and \(-1/2\). Therefore, to find the total number of electrons that \(d\)-type orbitals can hold, multiply the number of orbitals by 2: \[\text{Maximum number of electrons in } d\text{-type orbitals} = 7 \times 2 = 14.\]

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Key Concepts

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

Azimuthal Quantum Number
The azimuthal quantum number, also known as the angular momentum quantum number, is represented by the symbol \(l\) and describes the subshell in which an electron resides within a given energy level or shell. It takes integral values ranging from 0 to \(n-1\), where \(n\) is the principal quantum number associated with the energy level of the electron. Each value of \(l\) correlates with a specific subshell: \(s\), \(p\), \(d\), and \(f\), which correspond to values 0, 1, 2, and 3, respectively.

For the \(d\)-type orbitals mentioned in the exercise, the azimuthal quantum number is \(l = 2\). This is important because it defines the number of orbitals within the \(d\) subshell. The given formula \((3l + 1)\) helps calculate the total number of orbitals for a particular subshell, and by knowing the azimuthal quantum number, you can determine the exact number of orbitals where electrons can be found. In chemistry, understanding azimuthal quantum numbers is vital for predicting the electron configuration and chemical bonding properties of an element.
Spin Quantum Numbers
Spin quantum numbers refer to an intrinsic property of electrons known as spin, symbolized as \(m_s\). This quantum number can only have two possible values: \(+1/2\) and \(-1/2\), indicating the two possible directions of an electron's spin, conventionally termed 'up' and 'down'. Each orbital can hold a maximum of two electrons, and these electrons must have opposite spins, a requirement known as the Pauli exclusion principle.

In the context of the original exercise, knowing that each \(d\)-type orbital can contain up to two electrons with opposite spins is crucial. This quantization of electron spin is a fundamental concept in chemistry, particularly in the quantum mechanical model of the atom. It determines how electrons are distributed among orbitals and therefore directly influences an element's electron configuration and overall chemical characteristics.
Electron Configuration
Electron configuration is the arrangement of electrons within the orbitals of atoms or ions. It is denoted using the principal quantum number (\(n\)), the azimuthal quantum number (\(l\)), and the number of electrons in each subshell. The configuration follows a set of rules, including the Aufbau principle (which asserts that electrons fill lower energy orbitals first), the Pauli exclusion principle (stating that no two electrons in an atom can have the same set of four quantum numbers), and Hund's rule (which states that electrons will fill an empty orbital before they pair up).

The textbook exercise deals directly with the electron configuration of \(d\)-type orbitals. By combining the azimuthal quantum number and the spin quantum numbers, we find that \(d\)-type orbitals can hold a maximum of 14 electrons, as determined by the method outlined in the solution steps. This information is paramount to predicting the chemical properties and reactions of elements. Chemists and students use electron configurations to understand the behavior of atoms during chemical bonding and reactions.

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Most popular questions from this chapter

The magnetic moment of \(\mathrm{Ni}^{\mathrm{x}+}\) ion \((Z=28)\) is about \(2.82 \mathrm{~B} . \mathrm{M} .\) The value of \(x\) is (a) 2 (b) 4 (c) 1 (d) 3

If \(\lambda_{0}\) is the threshold wavelength for photoelectric emission from a metal surface, \(\lambda\) is the wavelength of light falling on the surface of metal and \(m\) is the mass of electron, then the maximum speed of ejected electrons is given by (a) \(\left[\frac{2 h}{m}\left(\lambda_{0}-\lambda\right)\right]^{1 / 2}\) (b) \(\left[\frac{2 h c}{m}\left(\lambda_{0}-\lambda\right)\right]^{1 / 2}\) (c) \(\left[\frac{2 h c}{m}\left(\frac{\lambda_{0}-\lambda}{\lambda_{0} \lambda}\right)\right]^{1 / 2}\) (d) \(\left[\frac{2 h}{m}\left(\frac{1}{\lambda_{0}}-\frac{1}{\lambda}\right)\right]^{1 / 2}\)

A proton and an \(\alpha\) -particle are accelerated through the same potential difference. The ratio of their de-Broglie wavelengths is (a) \(1: 1\) (b) \(2: 1\) (c) \(\sqrt{2}: 1\) (d) \(2 \sqrt{2}: 1\)

Which of the following ion have the same number of unpaired electrons as in \(\mathrm{Fe}^{2+}\) \((Z=26) ?\) (a) \(\mathrm{Fe}^{3+}(Z=26)\) (b) \(\mathrm{Ni}^{2+}(Z=28)\) (c) \(\mathrm{Co}^{3+}(Z=27)\) (d) \(\operatorname{Cr}^{+}(Z=24)\)

The electrons, identified by quantum numbers \(n\) and \(l\), (i) \(n=4, l=1\) (ii) \(n=4, l=0\) (iii) \(n=3, l=2\) (iv) \(n=3, l=1\) can be placed in order of increasing energy, from the lowest to highest, as (a) \(\mathrm{iv}<\mathrm{ii}<\mathrm{iii}<\mathrm{i}\) (b) \(\mathrm{ii}<\mathrm{iv}<\mathrm{i}<\mathrm{iii}\) (c) \(i<\) iii \(<\) ii \(<\) iv (d) iii \(<\mathrm{i}<\mathrm{iv}<\mathrm{ii}\)
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