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 $4f$ orbitals be able to hold?

Short answer

a. An orbital can hold 3 electrons. b. The first period contains 3 elements, and the second period contains 12 elements. c. The first transition metal series contains 15 elements. d. The set of 4f orbitals can hold 21 electrons.

Step-by-step solution

a. Number of electrons an orbital can hold

First, recall that there can be n (principal quantum number) orbitals in each energy level, and that each orbital can accommodate a certain number of electrons depending on the value of m_s. In the usual case, m_s can have two values (+1/2, -1/2), resulting in an orbital holding 2 electrons. However, in this scenario, m_s can have three values. This means that an orbital can now accommodate three times the number of electrons, which is 3 electrons in total.

b. Number of elements in the first and second periods

To determine the number of elements in each period, we need to know how many electrons can be accommodated in each respective energy level. For the first period, there is only one energy level, the 1s level, which consists of one orbital. Thus, since each orbital can now hold three electrons, the first period can accommodate a total of 3*1 = 3 elements. For the second period, there are two energy levels, the 2s and the 2p levels. The 2s level has one orbital, and the 2p level has three orbitals. Consequently, the second period can accommodate a total of 3*(1+3) = 12 elements.

c. Number of elements in the first transition metal series

The first transition metal series contains elements in which the 3d orbitals are being filled. The 3d level has five orbitals, and since each orbital can now accommodate three electrons, the first transition metal series can hold 3*5 = 15 elements.

d. Number of electrons the set of 4f orbitals can hold

The 4f level consists of seven orbitals. With each orbital being able to accommodate three electrons, the set of 4f orbitals can hold a total of 3*7 = 21 electrons.

Key concepts

Orbital Electron Capacity

Understanding the 'orbital electron capacity' provides crucial insight into how atoms are structured and how they bond. Ordinarily, an orbital can hold up to two electrons, one with a spin quantum number ($m_s$) of +1/2 and the other with -1/2. The Pauli Exclusion Principle dictates that no two electrons in an atom can have identical sets of quantum numbers, which normally limits the capacity to these two electrons.

However, when we hypothesize a scenario where the spin quantum number can have three possible values, each orbital's capacity increases. Rather than holding two electrons, each orbital could holds three, since each electron would have a unique set of quantum numbers due to the additional spin state. This change would significantly alter the structure of the periodic table and the chemical properties of elements.

Periodic Table Arrangement

The 'periodic table arrangement' is a systematic organization of chemical elements based on their atomic number, electron configurations, and recurring chemical properties. Elements are arranged in order of increasing atomic number in a tabular display, which is divided into periods (horizontal rows) and groups (vertical columns).

If we apply the hypothetical scenario where an orbital can hold three electrons, the arrangement of the periodic table would drastically shift. The first and second periods would have their lengths modified because each orbital's increased capacity would accommodate more elements. Specifically, the first period would include three elements and the second period would expand to include twelve. This would impact how students learn about electron configuration and the trends seen across periods and groups in the standard table.

Transition Metals

The 'transition metals' are elements found in the center of the periodic table and are characterized by the filling of their d orbitals. These metals are known for their ability to form various oxidation states and coordination compounds, making them extremely valuable in catalysis and material science.

In our theoretical scenario where orbitals can hold more electrons, the first transition metal series would house more elements than it does today. With each 3d orbital holding three electrons, the series would include fifteen elements as opposed to ten. This increase could theoretically introduce new chemical behaviors and a potentially richer chemistry for the elements in these expanded groups.

f-orbitals

Lastly, let's discuss 'f-orbitals,' which are among the more complex orbitals in an atom and play a role primarily in the behavior of heavy elements, including the lanthanides and actinides. An f-orbital is shaped more intricately than s, p, and d orbitals, often described as having a 'flower-like' pattern.

In the current understanding, each f-orbital can hold two electrons. With our hypothetical change of three possible spin states, a set of 4f orbitals would now hold three electrons each, totaling 21 electrons. This indicates a broader capacity for electron accommodation, influencing the chemical and physical properties of the heavy elements that interact through f-orbitals.