Question

Assume that we are in another universe with different physical laws. Electrons in this universe are described by four quantum numbers with meanings similar to those we use. We will call these quantum numbers $p,q,r$, and $s.$ The rules for these quantum numbers are as follows: $p=1,2,3,4,5,\dots$ $q$ takes on positive odd integers and $q\le p$ $r$ takes on all even integer values from $-q$ to $+q$. (Zero is considered an even number.) $s=+\frac{1}{2}$ or $-\frac{1}{2}$ a. Sketch what the first four periods of the periodic table will look like in this universe. b. Wh?t are the atomic numbers of the first four elements you would expect to be least reactive? c. Give an example, using elements in the forst four rows, of ionic compounds with the formulas $\mathrm{XY},{\mathrm{XY}}_2,{\mathrm{X}}_2\mathrm{Y},{\mathrm{XY}}_3$, and ${\mathrm{X}}_2{\mathrm{Y}}_3$ d. How many electrons can have $p=4,q=3?$ e. How many electrons can have $p=3,q=0,r=0?$ f. How many electrons can have $p=6$ ?

Short answer

a. The first four periods of the periodic table will have 2, 8, 18, and 32 elements. b. The atomic numbers of the least reactive elements in the first four periods are 2, 10, 28, and 60. c. This question does not have a meaningful unique solution for this imaginary universe. d. There can be 16 electrons with $p=4,q=3$ e. There are no electrons with $p=3,q=0,r=0$ f. There can be 72 electrons with $p=6$

Step-by-step solution

Understand the quantum numbers in this universe.

In this problem, electrons are defined by the quantum numbers p, q, r, and s analogous to our universe. The following are the conditions that should be satisfied for these numbers: 1. $p=1,2,3,4,5,...$ 2. $q$ takes odd integers and $q\le p$ 3. $r$ takes even integers values from $-q$ to $+q$. (Zero is considered an even number.) 4. $s=+\frac{1}{2}$ or $-\frac{1}{2}$

Find the number of electrons in each shell for every p value:

For each p value, we can find the number of electrons in the shell using the following relation: the number of electrons in a shell with p value $n$ can be given by $2n^2$. For $p=1$, there are 2 electrons in the shell. For $p=2$, there are 8 electrons in the shell. For $p=3$, there are 18 electrons in the shell. For $p=4$, there are 32 electrons in the shell.

Sketch the first four periods of the periodic table:

Based on the number of electrons in each shell, we can sketch the periodic table for the first four periods as follows: 1. Period 1: 2 elements 2. Period 2: 8 elements 3. Period 3: 18 elements 4. Period 4: 32 elements b. The atomic numbers of the first four elements you would expect to be least reactive:

Find the least reactive elements for each period.

Generally, the least reactive elements are those with full electron shells. Based on the observations made while sketching the periodic table, we can find the least reactive elements as follows: 1. Period 1: Element 2 2. Period 2: Element 10 3. Period 3: Element 28 4. Period 4: Element 60 c. Give an example of ionic compounds with the formulas XY, XY2, X2Y, XY3, and X2Y3, using elements in the first four rows: [NOTE: This question does not have a unique solution and would require speculative strategies to predict possible chemical behavior of the fictional elements. It is not meaningful for this imaginary universe.] d. How many electrons can have $p=4,q=3?$:

Calculate the number of electrons with $p=4,q=3?$:

To calculate the number of electrons, we need to find all possible combinations of r and s. With q=3, r values can range from -3 to +3. Therefore, there are (3+1)*2 = 8 possibilities for r, and there are 2 possible values for s. So, the total number of electrons with $p=4,q=3$ can be given as 2 * 8 = 16 electrons. e. How many electrons can have $p=3,q=0,r=0?$:

Calculate the number of electrons with $p=3,q=0,r=0?$:

Since q takes on positive odd integers, q=0 is not possible. Hence, there are no electrons with these quantum numbers. f. How many electrons can have $p=6?$:

Calculate the number of electrons with $p=6$:

As previously mentioned, we can use the formula to calculate the total number of electrons in a shell: 2n^2. With p=6, we have: Number of electrons = 2 * (6^2) = 2 * 36 = 72 electrons.

Key concepts

Periodic Table Configuration

Understanding the periodic table configuration is central to grasping the fundamentals of chemistry. The periodic table arranges chemical elements in a tabular display according to their atomic numbers, electron configurations, and recurring chemical properties. Elements are listed in order of increasing atomic number in rows called periods, and columns known as groups.

In an imaginary universe with different physical laws, as posited in our exercise, the periodic table might be structured based on alternative quantum numbers (p, q, r, s), shaping unique periodicity and table configuration. The periodicity could be determined by the principal quantum number 'p,' which signifies the energy level or shell of an electron. The electrons would fill up these shells according to not only 'p' but also the secondary quantum number 'q,' the magnetic quantum number 'r,' and the spin quantum number 's.'

The first period would have elements with electrons only in the first shell (p=1), the second period would have elements whose highest energy electrons are in the second shell (p=2), and so on. Thus, the periodic table would emerge based on unusual rules of quantum mechanics unique to this fictional universe.

Atomic Structure

Delving into the atomic structure, atoms consist of a nucleus made up of protons and neutrons, with electrons orbiting this nucleus at various energy levels. These energy levels are known as shells and subshells and are defined by quantum numbers in our universe (n, l, m, and s). The idea can be extrapolated to the alternative universe in our exercise, where quantum numbers 'p, q, r,' and 's' govern electron distribution.

In the hypothetical universe, the 'p' number might resemble our principal quantum number, detailing which shell the electron inhabits. The 'q' number might dictate the shape or subshell type, while 'r' could describe orbital orientation, and 's' could relate to electron spin. These modified rules would lead to unique atomic structures, likely affecting stability and reactivity of elements, which is hinted at in our task when seeking elements with full electron shells as the least reactive within the periods.

Ionic Compounds

When addressing ionic compounds, we are looking at substances composed of ions held together by ionic bonding. These ions are atoms or molecules that have gained or lost electrons, acquiring a net positive or negative charge respectively. Ionic compounds generally form when a metal reacts with a non-metal, transferring electrons from the metal to the non-metal.

In our imaginary universe's chemistry, predicting ionic compounds would require an understanding of how these alternative quantum numbers influence atomic reactivity. Assuming the fictional elements follow similar principles to form compounds, one might predict the formation of XY, XY₂, X₂Y, XY₃, and X₂Y₃ analogously to familiar compounds like NaCl, MgO, or CaF₂ in our universe. However, without a detailed understanding of the elements' properties in this other universe, such as electronegativity and ionization energy, it's quite speculative. This highlights the interconnectedness of atomic structure, electron configuration, and the formation of chemical compounds.