Chapter 2, Problem 167CP

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, ….

q takes on positive odd integers and q ≤ p

r takes on all even integer values from −q to +q. (Zero is considered an even number.)

$s=+\frac{1}{2}\text{or}-\frac{1}{2}$

a. Sketch what the first four periods of lhe periodic table will look like in this universe.

b. What are the atomic numbers of the first four elements you would expect to be least reactive?

c. Give an example, using elements in the first four rows, of ionic compounds with the formulas XY, XY₂, X₂Y, XY₃, and X₂Y₃.

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?

Interpretation Introduction

Interpretation: Another universe with different physical laws and in which electrons are given to be described by four quantum numbers with similar meaning is to be assumed. The rules for these quantum numbers are to be given. Quantum numbers are to be called as $p,q,r,s$. The values of $p,q,r,s$ are to be given on the basis of certain rules. The questions given in part (a) to (f) based on the given rules are to be stated.

Concept introduction: Quantum numbers are the characteristics of the atomic orbitals. The concept of quantum number is given by the famous physicist Schrodinger. There are total four types of quantum numbers are present in an orbital or each orbital is designated by four set of quantum numbers. The quantum numbers are the characteristic of the electrons.

To determine: The sketch of the first four periods of the periodic table look like in the assumed universe.

Explanation of Solution

Explanation

Given

The given rules for four quantum numbers are as follows,

$\begin{array}{l}p=1,2,3,4,\mathrm{5....}\\ q\le p\\ r=\left(-p\right)\text{to}\left(+p\right)\\ s=\left(+\frac{1}{2}\right)\text{or}\left(-\frac{1}{2}\right)\end{array}$

The sketch of the first four periods of the periodic table based on the given rules is drawn as,

$\begin{array}{cccc}p\text{value}& q\text{value}& r\text{value}& s\text{value}\\ \text{For}p=\text{1}& 0,1& -1,0,+1& \pm \frac{1}{2}\\ \text{For}p=\text{2}& 0,1,2& -2,-1,0,+1,+2& \pm \frac{1}{2}\\ \text{For}p=\text{3}& 0,1,2,3& -3,-2,-1,0,+1,+2,+3& \pm \frac{1}{2}\\ \text{For}p=\text{4}& 0,1,2,3,4& -4,-3,-2,-1,0,+1,+2,+3,+4& \pm \frac{1}{2}\\ \text{For}p=\text{5}& 0,1,2,3,4,5& -5,-4,-3,-2,-1,0,+1,+2,+3,+4,+5& \pm \frac{1}{2}\end{array}$

The principle quantum number is one of the characteristic of the atomic orbitals. Here, it is denoted by the symbol $p$. This quantum number is related to the orbital’s energy and the size. It has the integral values $1,2,\mathrm{3...}$. The smaller value indicates that the electron is present near the nucleus while larger value indicates that the electron is farther from the nucleus. The second quantum number is angular momentum quantum number. Here, it is denoted by the symbol $q$. The third quantum number is magnetic quantum number. Here, it is denoted by the symbol $m$. The fourth quantum number is spin quantum number. Here, it is denoted by the symbol $s$.

The meaning of the given four quantum numbers is similar with the actual quantum numbers. Therefore, the orbital representation is as follows,

$\begin{array}{cc}\text{Period}& \text{Orbitals}\\ {\text{1}}^{\text{st}}& \text{s-orbital}\\ {\text{2}}^{\text{nd}}& \text{s-orbital}\\ {\text{3}}^{\text{rd}}& \text{s,}\text{p-orbitals}\\ {\text{4}}^{\text{th}}& \text{s,}\text{p-orbitals}\end{array}$

Interpretation Introduction

Interpretation: Another universe with different physical laws and in which electrons are given to be described by four quantum numbers with similar meaning is to be assumed. The rules for these quantum numbers are to be given. Quantum numbers are to be called as $p,q,r,s$. The values of $p,q,r,s$ are to be given on the basis of certain rules. The questions given in part (a) to (f) based on the given rules are to be stated.

Concept introduction: Quantum numbers are the characteristics of the atomic orbitals. The concept of quantum number is given by the famous physicist Schrodinger. There are total four types of quantum numbers are present in an orbital or each orbital is designated by four set of quantum numbers. The quantum numbers are the characteristic of the electrons.

To determine: The atomic number of the first four least reactive elements.

Answer to Problem 167CP

Answer

The atomic number of the first four least reactive elements are $2,4,12,20$.

Explanation of Solution

Explanation

The least reactive elements are those elements which contain completely filled subshell. Therefore, the atomic number of the elements which are least reactive is given as,

$\begin{array}{l}2={\text{1s}}^{\text{2}}\\ 4={\text{1s}}^{\text{2}}{\text{2s}}^{\text{2}}\\ 12={\text{1s}}^{\text{2}}{\text{2s}}^{\text{2}}{\text{2p}}^{\text{6}}{\text{3s}}^{\text{2}}\\ 20={\text{1s}}^{\text{2}}{\text{2s}}^{\text{2}}{\text{2p}}^{\text{6}}{\text{3s}}^{\text{2}}3{\text{p}}^{\text{6}}{\text{4s}}^{\text{2}}\end{array}$

Here, all the atomic orbitals have the completely filled subshell.

Interpretation Introduction

Interpretation: Another universe with different physical laws and in which electrons are given to be described by four quantum numbers with similar meaning is to be assumed. The rules for these quantum numbers are to be given. Quantum numbers are to be called as $p,q,r,s$. The values of $p,q,r,s$ are to be given on the basis of certain rules. The questions given in part (a) to (f) based on the given rules are to be stated.

Concept introduction: Quantum numbers are the characteristics of the atomic orbitals. The concept of quantum number is given by the famous physicist Schrodinger. There are total four types of quantum numbers are present in an orbital or each orbital is designated by four set of quantum numbers. The quantum numbers are the characteristic of the electrons.

To determine: The examples of the ionic compounds of the first four rows with the formula $\text{XY,}{\text{XY}}_{\text{2}}\text{,}{\text{X}}_{\text{2}}{\text{Y}}_{\text{2}}\text{,}{\text{XY}}_{\text{3}}\text{and}{\text{X}}_{\text{2}}{\text{Y}}_{\text{3}}$.

Answer to Problem 167CP

Answer

The examples of the ionic compounds of the first four rows with the formula $\text{XY,}{\text{XY}}_{\text{2}}\text{,}{\text{X}}_{\text{2}}{\text{Y}}_{\text{2}}\text{,}{\text{XY}}_{\text{3}}\text{and}{\text{X}}_{\text{2}}{\text{Y}}_{\text{3}}$ are given below.

Explanation of Solution

Explanation

Ionic compounds are those compounds which are combined by an ionic interaction. In these compounds the donation and acceptance of electrons occur. The overall charge on such compounds is to be neutralized. For example the compound $\text{XY}$ is formed by the combination of element $"1"$ and element $"11"$.

This is because both the elements have the charges $+1\text{and}-1$ respectively. In this way the neutralization principle is correctly follows.

Therefore, combinations based on the number of electrons are given as,

$\begin{array}{cc}\text{Type}& \text{combinations}\text{based}\text{on}\text{atomic}\text{numbers}\\ \text{XY}& 1+11,3+19\\ {\text{XY}}_2& 6+11,14+19\\ {\text{X}}_2\text{Y}& 1+10,3+18\\ {\text{XY}}_3& 7+11,15+19\\ {\text{X}}_2{\text{Y}}_3& 7+10,15+18\end{array}$

Interpretation Introduction

Interpretation: Another universe with different physical laws and in which electrons are given to be described by four quantum numbers with similar meaning is to be assumed. The rules for these quantum numbers are to be given. Quantum numbers are to be called as $p,q,r,s$. The values of $p,q,r,s$ are to be given on the basis of certain rules. The questions given in part (a) to (f) based on the given rules are to be stated.

Concept introduction: Quantum numbers are the characteristics of the atomic orbitals. The concept of quantum number is given by the famous physicist Schrodinger. There are total four types of quantum numbers are present in an orbital or each orbital is designated by four set of quantum numbers. The quantum numbers are the characteristic of the electrons.

To determine: The number of electrons for the values $p=4,q=3$.

Answer to Problem 167CP

Answer

The number of electrons have the values $p=4,q=3$ is $\underset{\_}{6e^{-}}$.

Explanation of Solution

Explanation

The given quantum number values, $p=4,q=3$ indicates the presence of $\text{p-orbital}$. It has the $r-\text{values}=-2,0,\text{+2}$. There are three orbitals are present and each orbital contains two electrons. Therefore, the number of electrons have the values $p=4,q=3$ is $\underset{\_}{6e^{-}}$.

Interpretation Introduction

Interpretation: Another universe with different physical laws and in which electrons are given to be described by four quantum numbers with similar meaning is to be assumed. The rules for these quantum numbers are to be given. Quantum numbers are to be called as $p,q,r,s$. The values of $p,q,r,s$ are to be given on the basis of certain rules. The questions given in part (a) to (f) based on the given rules are to be stated.

Concept introduction: Quantum numbers are the characteristics of the atomic orbitals. The concept of quantum number is given by the famous physicist Schrodinger. There are total four types of quantum numbers are present in an orbital or each orbital is designated by four set of quantum numbers. The quantum numbers are the characteristic of the electrons.

To determine: The number of electrons for the values $p=3,q=0,r=0$.

Answer to Problem 167CP

Answer

The number of electrons have the values $p=3,q=0,r=0$ is $\underset{\_}{0e^{-}}$.

Explanation of Solution

Explanation

The given quantum number values, $p=3,q=0,r=0$ indicates the absence of orbital. Because $q\text{value}$ must takes as odd integer. Therefore, there is no orbital is present. Hence, number of electrons have the values $p=3,q=0,r=0$ is $\underset{\_}{0e^{-}}$.

Interpretation Introduction

Interpretation: Another universe with different physical laws and in which electrons are given to be described by four quantum numbers with similar meaning is to be assumed. The rules for these quantum numbers are to be given. Quantum numbers are to be called as $p,q,r,s$. The values of $p,q,r,s$ are to be given on the basis of certain rules. The questions given in part (a) to (f) based on the given rules are to be stated.

Concept introduction: Quantum numbers are the characteristics of the atomic orbitals. The concept of quantum number is given by the famous physicist Schrodinger. There are total four types of quantum numbers are present in an orbital or each orbital is designated by four set of quantum numbers. The quantum numbers are the characteristic of the electrons.

To determine: The number of electrons for the values $p=6$.

Answer to Problem 167CP

Answer

The number of electrons have the values $p=6$ is $\underset{\_}{18e^{-}}$.

Explanation of Solution

Explanation

The given quantum number values, $p=6$ the values of rest quantum numbers are calculated as,

$\begin{array}{ccc}p\text{Values}& q\text{Values}& r\text{Values}\\ p=6& q=5& -4,-2,0,+2,+4\\ p=6& q=3& -2,0,+2\\ p=6& q=1& 0\end{array}$

According to the calculated $r$ values the number of orbitals is nine. Therefore, the number of electrons present are $\underset{\_}{18e^{-}}$.