Question

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

In this scenario, an orbital can hold 3 electrons. The first period would contain 3 elements, and the second period would contain 24 elements. The first transition metal series would contain 15 elements. The set of 4f orbitals can hold 21 electrons.

Step-by-step solution

a. How many electrons would an orbital be able to hold?

An orbital is determined by three quantum numbers: principal quantum number \(n\), angular momentum quantum number \(\ell\), and magnetic quantum number \(m_\ell\). However, it can hold electrons with two different spin magnetic quantum numbers (\(m_s\)) under normal circumstances. In this exercise, we are assuming \(m_s\) can have three values instead of two. So each orbital can now hold 3 electrons instead of 2. Answer: An orbital can hold 3 electrons.

b. How many elements would the first and second periods in the periodic table contain?

In the periodic table, elements are arranged by increasing atomic number. The elements in a given period have the same principal quantum number \(n\). For the first period (n=1), there is only one possible value for \(\ell\), which is \(\ell=0\). Therefore, we have only one orbital (called 1s), and according to the answer in part a, it can hold 3 electrons. This means there would be 3 elements in the revised first period. For the second period (n=2), there are two possible values for \(\ell\), which are \(\ell=0\) and \(\ell=1\). This gives us a total of 2 orbitals for \(\ell=0\) (2s orbitals) and 6 orbitals for \(\ell=1\) (2p orbitals). Each of these orbitals can hold 3 electrons. Therefore, there will be 8 orbitals x 3 electrons per orbital = 24 elements in the second period. Answer: The first period would contain 3 elements, and the second period would contain 24 elements.

c. How many elements would be contained in the first transition metal series?

The first transition metal series contains elements with partially filled d orbitals (3d orbitals, since n=3 and \(\ell=2\)) in their ground state. There are 5 possible values for \(m_\ell\) in the d orbitals, so there are 5 orbitals with 3 electrons per orbital. Thus, there will be 5 orbitals x 3 electrons per orbital = 15 elements contained in the first transition metal series. Answer: The first transition metal series would contain 15 elements.

d. How many electrons would the set of 4f orbitals be able to hold?

For the 4f orbitals, we have n=4 and \(\ell=3\). There are 7 possible values for \(m_\ell\) in f orbitals, so there are 7 orbitals in the 4f subshell. Each of these orbitals can hold 3 electrons. Therefore, the set of 4f orbitals can hold 7 orbitals x 3 electrons per orbital = 21 electrons. Answer: The set of 4f orbitals can hold 21 electrons.

Key concepts

Electron Configuration

Electron configuration is a way of describing the arrangement of electrons in an atom. Think of it as a map that shows where electrons are located around the nucleus. Understanding electron configurations helps explain how atoms interact with each other to form molecules. The configurations follow a specific order based on the energy levels of the orbitals. In a typical scenario, each orbital can hold a certain number of electrons. Based on known principles, each orbital can hold two electrons due to the two possible spin states ( up, down) of electrons. However, our exercise challenges this by proposing that the spin quantum number ( ms) could have three values. With three possible spins, each orbital can now accommodate three electrons instead of the usual two. This change in electron accommodation significantly affects how elements and their periods in the periodic table are structured. Electron configuration provides the foundation for understanding chemical properties and behaviors of the elements. It also offers explanations for the arrangement of the periodic table, periodic trends, and chemical bonds.

Quantum Numbers

Quantum numbers are essential in defining the unique state of electrons in an atom. Each electron in an atom is defined by a unique set of four quantum numbers:

• Principal quantum number ( n): Indicates the main energy level occupied by the electron. It's like a shell in which the electron resides.
• Angular momentum quantum number ( l): Defines the shape of the electron's orbit (e.g., s, p, d, f). For each value of n, l can be a number from 0 up to (n-1).
• Magnetic quantum number ( ml): Describes the orientation of the orbital in space. It takes on values between (- el) and (+ el).
• Spin magnetic quantum number ( ms): Represents the electron's spin direction. In regular conditions, ms can be +1/2 or -1/2. In the exercise scenario, it considers a hypothetical third value, allowing for more electrons per orbital.

Quantum numbers form the basis for constructing the electron configuration of atoms. They also help predict the chemical and physical properties of atoms, allowing chemists to make informed predictions about the behavior of elements and compounds.

Periodic Table

The periodic table is a powerful tool that organizes chemical elements by their atomic number, electron configurations, and recurring chemical properties. It's like a calendar of elements, arranging them into rows called periods and columns known as groups or families. Under normal circumstances:

• The first period of the periodic table consists of hydrogen and helium (2 elements), due to the occupation of the 1s orbital by its two electrons.
• The second period contains 8 elements as the 2s and 2p orbitals are filled by 8 electrons altogether.

However, if the spin quantum number can have three values as suggested in the exercise, the number of elements in each period would change considerably:

• The first period would expand to 3 elements as each 1s orbital can hold 3 electrons.
• The second period would encompass 24 elements due to three electrons each in the 2s and 2p orbitals.

Such adjustments in electron occupancy per orbital also alter the number of elements in other sections, like the transition metals and lanthanides. Understanding these arrangements helps predict chemical behaviors and reactions, illustrating the rich, interconnected tapestry of chemistry that the periodic table presents.