*Its All about Pj Problem Strings (S _{i}P_{j}A_{jk}) -
7 Spaces Of Interest (S_{i}) and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (A_{jk})*

Expressions Of Pj Problems.

Pj Problems - Overview

Celestial Stars As Expressions Of Pj Problems

The Number Line As Expression Of Pj Problems

Geometries As Expressions Of Pj Problems

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation As Expression Of Pj Problems

The Atom As Expression Of Pj Problems

Survival As Expression Of Pj Problems

Energy As Expression Of Pj Problems

Light As Expression Of Pj Problems

Heat As Expression Of Pj Problems

Sound As Expression Of Pj Problems

Music As Expression Of Pj Problems

Language As Expression of Pj Problems

Stories As Expressions of Pj Problems

Work As Expression Of Pj Problems

States Of Matter As Expressions Of Pj Problems

Buoyancy As Expression Of Pj Problems

Nuclear Reactions As Expressions Of Pj Problems

Molecular Shapes As Expressions Of Pj Problems

Electron Configurations As Expressions Of Pj Problems

Chemical Bonds As Expressions Of Pj Problems

Energy Conversion As Expression Of Pj Problems

Chemical Reactions As Expressions Of Pj Problems

Electromagnetism As Expression Of Pj Problems

Continuity As Expression Of Pj Problems

Growth As Expression Of Pj Problems

Human-cells As Expressions Of Pj Problems

Proteins As Expressions Of Pj Problems

Nucleic Acids As Expressions Of Pj Problems

COHN - Nature's Engineering Of The Human Body

The Human-Body Systems As Expressions Of Pj Problems

Vision As Expression Of Pj Problems

Walking As Expression Of Pj Problems

Behaviors As Expressions Of Pj Problems

Sensors' Sensings As Expressions Of Pj Problems

Beauty As Expression Of Pj Problems

Faith, Love, Charity As Expressions Of Pj Problems

Photosynthesis As Expressions Of Pj Problems

Weather As Expression Of Pj Problems

Systems As Expressions Of Pj Problems

Algorithms As Expressions Of Pj Problems

Tools As Expressions Of Pj Problems

Networks As Expressions Of Pj Problems

Search As Expressions Of Pj Problems

Differential Calculus As Expression Of Pj Problems

Antiderivative As Expression Of Pj Problems

Integral Calculus As Expression Of Pj Problems

Economies As Expresions Of Pj Problems

Inflation As Expression Of Pj Problems

Markets As Expressions Of Pj Problems

Money Supply As Expression Of Pj Problems

Painting As Expressions Of Pj Problems

Molecular Shape - Sigma (σ) Bonds - Pi (π) Bonds

1. Define the following: (a) Resonance Structures (b) Sigma Bonds (c) Pi Bonds

(d) Localized Electrons (e) Delocalized Electrons

2. Identify the bonds (sigma, or pi, or both) that formed each of the following compounds:

(a) H_{2} (Hydrogen) (b) HCl (hydrochloric acid) (c) C_{2}H_{4} (Ethylene)
(d) N_{2} (Nitrogen)

(e) H_{2}CO (Formaldehyde) (f) C_{6}H_{6} (Benzene).

3. What valence atomic orbitals formed the bonds of the compounds listed in problem (2)?

4. What compound listed in problem (2) has Pi (π) bonds that cannot be entirely described as *localized* Pi (π) bonds?

5. Write the general representative strings for Sigma (σ) bonds and Pi (π) bonds.

1(a)
*Resonance structures* (also called resonance forms): the different Lewis structures that are equally acceptable descriptions of a single molecule.

(b) *Sigma (σ) bond*: a covalent bond in which the electron density is concentrated symmetrically about the line joining the nuclei (internuclear axis) of the bonded atoms. In other words, a covalent bond in which the internuclear axis passes through the middle of the region of orbital overlap (figure 9.16).

*Pi (π) bond*: a covalent bond in which the electron density is concentrated above and below the internuclear axis of the bonded atoms. In other words, a covalent bond in which the line passing through the middle of the region of orbital overlap is perpendicular to the internuclear axis of the bonded atoms (figure 9.17).

*Localized electrons*: σ and π electrons totally associated with the atoms that form the bond. Such bonds are *localized bonds*.
*Delocalized electrons*: electrons not totally associated with the atoms that form the bond. Such bonds are *delocalized bonds*. Delocalized bonds are commomly encountered in molecules that have two or more resonance structures involving π bonds. Such bonds are called *delocalzed π bonds*.

2(a) The valence-shell configuration of H = 1s^{1}. Two s orbitals overlap to form H_{2}. The internuclear axis of the atoms passes through the middle of the orbital overlap. So, H_{2} is formed by a sigma (σ) bond (figure 9.19).

(b) The valence-shell configuration of H = 1s^{1}. The valence-shell configuration of Cl = 1s^{2}2s^{2}2p^{6}3s^{2}3p^{5}. The s orbital of H overlaps with the p orbital of Cl to form HCl. The internuclear axis of the atoms passes through the middle of the orbital overlap. So, HCl is formed by a sigma (σ) bond (figure 9.20).

(c) Molecular geometry of C_{2}H_{4} is trigonal planar. This geometry implies the sp^{2} hybrid orbitals on C. The valence-shell configuration of H = 1s^{1}. The valence-shell configuration of C = 1s^{2}2s^{2}2p^{2}. One 2s electron is promoted to the third 2p orbital. This 2s orbital then merges with two 2p orbitals to form 3 sp^{2} hybrid orbitals. Each C atom uses its 3 sp^{2} hybrid orbitals to form σ bonds with the other C atom and two H atoms (a total of 5 σ bonds). The 2 valence electrons in the unhybridized 2p orbitals of the two C atoms form a π bond. Hence the double bond between the two C atoms (one σ bond and one π bond). In total, five σ bonds and one π bond form C_{2}H_{4} (figure 9.21).

(d) The valence-shell configuration of N = 1s^{2}2s^{2}2p^{3}. N_{2} is formed by one σ bond and two π bonds. A total of 3 pairs of 2p orbitals formed the bonds. a pair of 2p orbitals form the sigma bond and 2 pairs of 2p orbitals form the pi bonds (figure 9.22).

(e) Molecular geometry of H_{2}CO is trigonal planar. This geometry implies the sp^{2} hybrid orbitals on C. The valence-shell configuration of H = 1s^{1}. The valence-shell configuration of C = 1s^{2}2s^{2}2p^{2}. One 2s electron is promoted to the third 2p orbital. This 2s orbital then merges with two 2p orbitals to form 3 sp^{2} hybrid orbitals. The C atom uses its 3 sp^{2} hybrid orbitals to form σ bonds with two H atoms and one O atom. The remaining electron is in an unhybridized 2p orbital.
The O atom has three electron domains around it which implies that it has sp^{2} hybrid orbitals. One of the sp^{2} hybrid orbital is used to form the C-O σ bond; the other two hold the two nonbonding elctron pairs of the O atom. The remaining electron is in an unhybridized 2p orbital.
The unhybridized 2p orbital of the C atom and the unhybridized 2p orbital of the O atom formed the π bond between the C atom and the O atom. So, in H_{2}CO, there 3 σ bonds and 1 π bond (figure 9.23).

(f) Molecular geometry of C_{6}H_{6} (Benzene) indicates that each carbon atom is surrounded by three atoms at 120^{o} angles. This geometry implies three sp^{2} hybrid orbitals on each C atom of benzene. Consequently, six C-C σ bonds and C-H σ bonds are formed from the sp^{2} hybrid orbital set. Each C atom has an unhybridized 2p orbital. So, benzene has a total of six unhybridized carbon 2p orbitals in a ring arrangement which form the three π bonds responsible for the *resonance structures* of the benzene molecule (figure 9.24).

3(a) overlap of two s orbitals (b) overlap of an s orbital and a p orbital (c) overlap of four s orbitala and four sp^{2} hybrid orbitals, overlap of two sp^{2} hybrid orbitals, overlap of two unhybridized p orbitals (d) overlap of p orbitals (e) overlap of two s orbitals and two sp^{2} hybrid orbitals, overlap of two sp^{2} hybrid orbitals, overlap of two unhybridized p orbitals (f) overlap of six s orbitals and six sp^{2} hybrid orbitals, overlap of twelve sp^{2} hybrid orbitals, overlap of six unhybridized p orbitals.

4.C_{6}H_{6} (Benzene) is the compound in the list of compounds in problem (2) whose pi (π) bonds cannot be entirely described as localized π bonds. There are three resonance structures presented by the benzene molecule. Two of the resonance structures have localized π bonds while one has delocalized π bonds (figure 9.25).

5. Sigma (σ) and Pi (π) bonds are covalent bonds. The regions where the bondings occur are the valence orbital overlaps. So Sigma (σ) bonds and Pi (π) bonds have: S_{7}P_{6}A_{64} (multi-criteria permutation), S_{7}P_{3}A_{31} (force - pull) and S_{7}P_{6}A_{66} (chemical interaction) as their general representative strings.

Mind Warm Ups

The *point* "**.**" is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

*Problems by Peter O. Sagay*