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Chapter 7: Problem 158

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 forbitals be able to hold?

Short Answer

Expert verified
a. An orbital could hold up to 3 electrons. b. The first period would have 3 elements, and the second period would have 12 elements. c. The first transition metal series would have 15 elements. d. The set of 4f orbitals could hold 21 electrons.

Step by step solution

01

Determine the number of electrons in an orbital

Using the given magnetic quantum number values \(m_s=-\frac{1}{2}, 0, \frac{1}{2}\), since an orbital is described by a unique set of quantum numbers (\(n, \ell, m_\ell, m_s\)), we observe that for each orbital, the \(m_s\) value can only change. So, each orbital can accommodate three different \(m_s\) values, meaning an orbital can hold up to 3 electrons. Answer to (a): An orbital could hold up to 3 electrons.
02

Calculate the number of elements in the first and second periods of the periodic table

To determine the number of elements in a period, we will use the given rules for quantum numbers: - The number of the first period is when \(n = 1\). In this case, only the 1s orbital is available, and it can hold 3 electrons (as calculated in step 1). Therefore, there are 3 elements in the first period. - The number of the second period is when \(n = 2\). In this case, we have the 2s and 2p orbitals available. The 2s orbital can hold 3 electrons, and the 2p orbitals can hold 3 orbitals * 3 electrons = 9 electrons. Thus, there are a total of 12 elements in the second period. Answer to (b): The first period would have 3 elements, and the second period would have 12 elements.
03

Calculate the number of elements in the first transition metal series

The first transition metal series corresponds to \(n=3\). In this case, we have the 3d orbitals available. There are 5 orbitals in the 3d subshell, and each orbital can hold 3 electrons each. Therefore, there are a total of 5 orbitals * 3 electrons = 15 electrons, which corresponds to 15 elements in the first transition metal series. Answer to (c): The first transition metal series would have 15 elements.
04

Calculate the number of electrons in the set of 4f orbitals

The 4f orbitals correspond to \(n=4\) and \(\ell=3\). There are a total of 7 orbitals in the 4f subshell (since \(m_\ell\) can take values between \(-\ell\) and \(+\ell\), i.e. -3, -2, -1, 0, 1, 2, and 3). Each of the 7 orbitals can hold up to 3 electrons. Hence, the 4f orbitals can hold a total of 7 orbitals * 3 electrons = 21 electrons. Answer to (d): The set of 4f orbitals could hold 21 electrons.

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Key Concepts

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

Electrons
Electrons are subatomic particles that orbit the nucleus of an atom. They have a negative charge and are considered fundamental entities in the study of chemistry and physics. Electrons are organized into levels or shells around the nucleus.
Each shell is composed of one or more subshells characterized by the principal quantum number, represented as \(n\), and each subshell comprises one or more orbitals. Every orbital can hold a certain number of electrons, typically up to two for most configurations due to the Pauli Exclusion Principle.
  • In quantum mechanics, each electron within an atom is described by four quantum numbers: \(n\), \(\ell\), \(m_{\ell}\), and \(m_{s}\).
  • These quantum numbers define the electron's energy levels, angular momentum, magnetic orientation, and spin.
  • In the scenario described in the original exercise, \(m_s\) could have three possible values which is an adjustment from the standard two-bin spin system, allowing orbitals to hold more electrons.
Periodic Table
The periodic table is a systematic arrangement of elements in order of increasing atomic number. This layout not only highlights recurring chemical properties of elements but also supports the prediction of chemical behaviors.
Elements in the periodic table are structured into periods (rows) and groups (columns). Each period signifies a principal energy level filled by the electrons:
  • The first period, usually containing hydrogen and helium, corresponds to \(n = 1\) with only one \(s\)-orbital.
  • The second period incorporates \(s\)- and \(p\)-orbitals for \(n=2\), magnetizing more complexity and room for additional elements.
In the modified question setting, the first period expands to 3 elements, and the second period holds up to 12 due to the 3-electron-per-orbital capacity.
Transition Metals
Transition metals are a category of elements found in the central block of the periodic table, which are distinguished by their ability to form various oxidation states and colorful compounds. These elements have partially filled \(d\)-subshells.
  • The first transition metal series corresponds to the filling of the 3d orbitals.
  • In a typical scenario, these orbitals can host 10 electrons, as seen across elements from Scandium to Zinc.
  • With the exercise's adapted rule for \(m_s\), each 3d orbital now holds 3 electrons, increasing the count of metals to 15 in the first series.
This increased capacity showcases how quantum number variations could drastically change our understanding of elemental properties and arrangements.
Orbitals
Orbitals are regions within an atom where electrons are most likely to be located. Each orbital can be identified by a specific set of quantum numbers and comes in different shapes:\(s\), \(p\), \(d\), and \(f\).
  • \(s\)-orbitals are spherical and can hold up to 2 electrons in most cases.
  • \(p\)-orbitals each have a two-lobed shape and can accommodate up to 6 electrons across three orbitals under standard rules.
  • \(d\)-orbitals feature more complex shapes and normally house 10 electrons.
  • \(f\)-orbitals are even larger, with a standard capacity of 14 electrons across seven orbitals.
In the context of the modified problem, each \(f\) orbital could hold 3 electrons, increasing the potential to 21 electrons total for 4f orbitals.
Understanding these levels and arrangements helps in predicting chemical bonding and properties.

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Most popular questions from this chapter

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