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Chapter 7: Problem 170

Answer the following questions assuming that \(m_{s}\) could have three values rather than two and that the rules for \(n, \ell\), and \(m_{\ell}\) are the normal ones. a. How many electrons would an orbital be able to hold? b. How many elements would the first and second periods in the periodic table contain? c. How many elements would be contained in the first transition metal series? d. How many electrons would the set of \(4 f\) orbitals be able to hold?

Short Answer

Expert verified
a. An orbital can hold up to 3 electrons. b. The first period contains 3 elements, and the second period contains 12 elements. c. The first transition metal series would contain 15 elements. d. The set of \(4f\) orbitals can hold a total of 21 electrons.

Step by step solution

01

a. Electrons in an orbital

To determine how many electrons an orbital can hold, we need to consider the spin quantum number, \(m_{s}\). Normally, \(m_{s}\) can have two values, +1/2 and -1/2, which means that an orbital can have two electrons. In this exercise, we assume that \(m_{s}\) can have three values rather than two. Let's call the three values +1, 0, and -1. An orbital can hold one electron with each of these values without violating the Pauli Exclusion Principle. Therefore, in our scenario, an orbital can hold up to 3 electrons instead of 2.
02

b. Elements in the first and second periods

The number of elements in a period of the periodic table is determined by the number of available orbitals in each energy level and the number of electrons each orbital can hold. To find the total number of elements in the first two periods, we need to break down the energy levels and orbitals in each period. 1. The first period contains only the 1s orbital (\(n=1, \ell=0\)). With our modified \(m_{s}\) values, the 1s orbital can hold up to 3 electrons. Thus, the first period contains 3 elements. 2. The second period contains 2s and 2p orbitals (\(n=2, \ell=0,1\)). The 2s orbital can hold 3 electrons, and since there are three 2p orbitals (\(m_{\ell}=-1,0,1\)), each 2p orbital can hold 3 electrons as well. Therefore, the 2p subshell holds 9 electrons (3 electrons x 3 2p orbitals). In total, the second period contains 3 + 9 = 12 elements.
03

c. Elements in the first transition metal series

The first transition metal series refers to the first series of elements where the \(3d\) orbitals are being filled. This series spans from scandium (Sc, atomic number 21) to zinc (Zn, atomic number 30). Normally, the \(3d\) orbitals can hold 10 electrons. However, with our modified \(m_{s}\) values, each of the 5 \(3d\) orbitals (\(m_{\ell}=-2,-1,0,1,2\)) can hold 3 electrons. Thus, the first transition metal series can hold 15 electrons, instead of the usual 10. This means the first transition metal series would contain 15 elements.
04

d. Electrons in \(4f\) orbitals

The set of \(4f\) orbitals has 7 orbitals (\(m_{\ell}=-3,-2,-1,0,1,2,3\)). Normally, each \(4f\) orbital can hold 2 electrons, making a total of 14 electrons. However, with our modified \(m_{s}\) values, each \(4f\) orbital can hold up to 3 electrons instead of 2. Therefore, the set of \(4f\) orbitals can hold a total of 21 electrons (3 electrons x 7 orbitals).

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Key Concepts

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

Spin Quantum Number
In quantum mechanics, each electron in an atom is described by a set of four quantum numbers. The spin quantum number, denoted by \( m_{s} \), is one of these four numbers. Normally, \( m_{s} \) can adopt two values: +1/2 and -1/2. This represents the two possible orientations of an electron's intrinsic spin, often visualized as 'spin up' and 'spin down'.

It鈥檚 a critical concept because it influences how electrons arrange themselves in the orbitals of an atom. According to the exercise, if \( m_{s} \) could have three values (like +1, 0, and -1), it alters how many electrons an orbital can accommodate. This theoretical scenario allows each orbital to hold three electrons instead of the usual two, potentially transforming the structure of the periodic table.
Pauli Exclusion Principle
The Pauli Exclusion Principle is a fundamental principle in quantum chemistry and physics. It states that no two electrons in an atom can have the same set of all four quantum numbers. This principle explains why electrons in the same orbital must have different spins; with two electrons per orbital, each has an opposite spin, resulting in distinct quantum states.

In the modulated version from the problem, with \( m_{s} \) values of +1, 0, and -1, the principle still holds. An orbital can host up to three electrons, each with a different spin quantum number. This hypothetical scenario stretches the traditional limits, providing a unique way to think about electron distribution and chemical complexity.
Electronic Configuration
Electronic configuration describes how electrons are distributed among the various orbitals around an atom鈥檚 nucleus. It's a crucial concept because it dictates how elements react chemically and their position on the periodic table. Normally, electron configurations follow the typical rules with a possible maximum of 2 electrons in an orbital.

With a change in the spin quantum number as suggested in the problem, electron configurations become more complex. For instance, the first and second periods of the periodic table would have 3 and 12 elements, respectively, compared to the standard 2 and 8. This reflects the increased electron capacity per orbital resulting from the added \( m_{s} \) value. It showcases the potential for more variety in elemental behavior and properties.
Periodic Table Periods
The periodic table is organized into distinct rows known as periods. Each period corresponds to the filling of a particular set of atomic orbitals. Regularly, the number of elements in a period is determined by the number of electrons that can fit in the available orbitals at that energy level.

For the first period utilizing only the 1s orbital and the second period filling the 2s and 2p orbitals, the number of elements significantly increases when we assume the possibility of a third value for \( m_{s} \). This hypothetical adjustment leads to 3 elements in the first period and 12 in the second. Such changes underscore how quantum numbers shape the overall layout and understanding of the periodic table.

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Most popular questions from this chapter

For hydrogen atoms, the wave function for the state \(n=3, \ell\) \(=0, m_{\ell}=0\) is $$ \psi_{300}=\frac{1}{81 \sqrt{3 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(27-18 \sigma+2 \sigma^{2}\right) e^{-\sigma \beta} $$ where \(\sigma=r / a_{0}\) and \(a_{0}\) is the Bohr radius \(\left(5.29 \times 10^{-11} \mathrm{~m}\right)\). Calculate the position of the nodes for this wave function.

It takes \(208.4 \mathrm{~kJ}\) of energy to remove \(1 \mathrm{~mole}\) of electrons from an atom on the surface of rubidium metal. How much energy does it take to remove a single electron from an atom on the surface of solid rubidium? What is the maximum wavelength of light capable of doing this?

An electron is excited from the \(n=1\) ground state to the \(n=3\) state in a hydrogen atom. Which of the following statements are true? Correct the false statements to make them true. a. It takes more energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be shorter than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1\). d. The wavelength of light emitted when the electron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\). e. For \(n=3\), the electron is in the first excited state.

The ionization energy for a \(1 s\) electron in a silver atom is \(2.462 \times 10^{6} \mathrm{~kJ} / \mathrm{mol}\) a. Determine an approximate value for \(Z_{\text {eff }}\) for the Ag \(1 s\) electron. Assume the Bohr model applies to the \(1 s\) electron. \(Z_{\text {eff }}\) is the apparent nuclear charge experienced by the electrons. b. How does \(Z_{\text {eff }}\) from part a compare to \(Z\) for Ag? Rationalize the relative numbers.

An excited hydrogen atom emits light with a wavelength of \(397.2 \mathrm{~nm}\) to reach the energy level for which \(n=2\). In which principal quantum level did the electron begin?
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