Chapter 5: Molecular Geometry and Polarity

Summary

This chapter uses VSEPR theory to predict molecular shapes, introduces orbital hybridization and sigma/pi bonding, and explains how molecular geometry determines bond polarity and overall molecular polarity.

Concepts Covered

This chapter covers the following 34 concepts from the learning graph:

VSEPR Theory
Electron Geometry
Molecular Geometry
Linear Geometry
Trigonal Planar Geometry
Tetrahedral Geometry
Trigonal Bipyramidal
Octahedral Geometry
Bent Geometry
Trigonal Pyramidal
Bond Angles
Lone Pair Repulsion
Orbital Hybridization
sp Hybridization
sp2 Hybridization
sp3 Hybridization
Sigma Bonds
Pi Bonds
Bond Polarity
Electronegativity Difference
Polar Covalent Bonds
Nonpolar Covalent Bonds
Dipole Moment
Molecular Polarity
Polar Molecules
Nonpolar Molecules
Naming Covalent Compounds
Network Covalent Solids
Molecular Orbital Theory
Bonding Molecular Orbitals
Antibonding Molecular Orbitals
MO Energy Diagrams
Bond Order from MO Theory
Paramagnetism and Diamagnetism

Prerequisites

This chapter builds on concepts from:

5.1 Why Geometry Matters: From Flat Drawings to Three-Dimensional Reality

A Lewis structure is a powerful tool. It shows which atoms are bonded, how many lone pairs exist, and whether bonds are single, double, or triple. But it is fundamentally a flat, two-dimensional representation of a molecule that exists in three-dimensional space. The actual shape of a molecule — how its atoms are arranged in space — is not captured by a dot structure drawn on paper.

Why should three-dimensional shape matter to a chemist? Because almost everything interesting about a molecule depends on its geometry. The physical properties of a substance — its melting point, boiling point, and solubility — depend on how molecules interact with one another, which depends on their shapes and charge distributions. Biological molecules fold into precise three-dimensional structures that allow enzymes to recognize specific substrates and antibodies to bind particular antigens with extraordinary selectivity. The smell of a spearmint leaf versus a caraway seed comes not from different atoms, but from two mirror-image versions of the same molecule, carvone, whose different geometric arrangements fit different receptors in your nose. The geometry of a water molecule, bent at about 104.5°, is directly responsible for water's remarkable properties — its high surface tension, its ability to dissolve ionic compounds, and its unusual density behavior near the freezing point.

Understanding molecular geometry therefore requires moving beyond the Lewis structure and into three dimensions. The organizing principle that makes this possible is remarkably simple: electron groups repel one another, so they spread as far apart as possible. This idea, formalized as VSEPR theory, provides an accurate and intuitive framework for predicting the shapes of hundreds of molecules from first principles.

5.2 VSEPR Theory: Electrons Spread Out

VSEPR Theory stands for Valence Shell Electron Pair Repulsion. First proposed by Nevil Sidgwick and Herbert Powell in 1940 and later refined by Ronald Gillespie and Ronald Nyholm, VSEPR theory rests on a single central principle: the electron groups around a central atom adopt the arrangement that minimizes their mutual repulsion, placing them as far apart as possible in three-dimensional space.

The word "electron group" is important. An electron group (also called an electron domain) is any collection of electrons occupying a single region around the central atom. This includes:

A single bond (two shared electrons)
A double bond (four shared electrons)
A triple bond (six shared electrons)
A lone pair (two nonbonding electrons)
Occasionally, a single unpaired electron (in radical species)

A critical feature of VSEPR theory is that double and triple bonds count as a single electron group for the purpose of predicting geometry. All six electrons of a triple bond occupy one region of space between two atoms; they repel other electron groups as a unit.

The VSEPR procedure can be summarized in four steps:

Draw the Lewis structure of the molecule or ion
Count the total number of electron groups around the central atom (bonding pairs plus lone pairs)
Identify the electron geometry — the arrangement that minimizes repulsion for that number of groups
Identify the molecular geometry — the arrangement of atoms only, ignoring lone pairs

This distinction between electron geometry and molecular geometry is one of the most important ideas in this chapter and is explored in detail in the sections that follow.

5.3 Electron Geometry: Placing Electron Groups in Space

The electron geometry describes the three-dimensional arrangement of all electron groups — bonding and nonbonding — around the central atom. There are five fundamental electron geometries, each corresponding to a specific number of electron groups:

Electron Groups	Electron Geometry	Bond Angles	Example
2	Linear	180°	CO₂, BeCl₂
3	Trigonal Planar	120°	BF₃, SO₃
4	Tetrahedral	109.5°	CH₄, NH₃, H₂O
5	Trigonal Bipyramidal	90°/120°	PCl₅, SF₄
6	Octahedral	90°	SF₆, XeF₄

Each geometry arises naturally from the mathematics of spreading points on a sphere as far apart as possible. Two points on a sphere are farthest apart at 180°, giving a linear arrangement. Three points spread to the vertices of an equilateral triangle (120°), four to the vertices of a tetrahedron (109.5°), five to a trigonal bipyramid (90° and 120°), and six to an octahedron (90°).

5.4 Molecular Geometry: What You See Without the Lone Pairs

The molecular geometry describes only the positions of atoms, not lone pairs. Because lone pairs are invisible in most experimental measurements of molecular shape, the molecular geometry is often what chemists report and what determines a molecule's physical and chemical behavior.

When all electron groups around the central atom are bonding pairs, the molecular geometry and the electron geometry are identical. When one or more electron groups are lone pairs, the molecular geometry is a subset of the electron geometry, and its name changes accordingly.

5.4.1 Linear Geometry

A molecule has linear geometry when its central atom has exactly two electron groups, both of which are bonding. The three atoms lie on a straight line with bond angles of exactly 180°. Carbon dioxide (CO₂) is the classic example: the carbon atom forms two double bonds to oxygen atoms on opposite sides. Each double bond counts as one electron group, giving two groups total, arranged 180° apart. Beryllium chloride (BeCl₂) and acetylene (HC≡CH) are additional examples of linear geometry.

5.4.2 Trigonal Planar Geometry

When a central atom has three electron groups, all in the same plane, arranged at 120° to one another, the result is trigonal planar geometry. Boron trifluoride (BF₃) is the textbook example: boron forms three single bonds to fluorine atoms with no lone pairs, and all four atoms lie in a flat plane with F–B–F angles of exactly 120°. Formaldehyde (H₂CO) also exhibits trigonal planar geometry around the carbon, which forms one double bond to oxygen and two single bonds to hydrogen atoms, all in the same plane.

5.4.3 Tetrahedral Geometry

Four electron groups arrange themselves at the corners of a tetrahedron, producing tetrahedral geometry with bond angles of 109.5°. Methane (CH₄) is the defining example: carbon's four single bonds to hydrogen atoms point toward the corners of a regular tetrahedron. This geometry is three-dimensional — you cannot draw a true tetrahedron in a flat plane. In structural drawings, bonds in the plane of the paper are shown as regular lines, bonds coming toward you are shown as solid wedges, and bonds pointing away from you are shown as dashed wedges.

5.4.4 Trigonal Bipyramidal Geometry

Five electron groups adopt a trigonal bipyramidal arrangement. This geometry has two distinct types of positions: three equatorial positions in a triangular plane through the center (at 120° to each other) and two axial positions above and below that plane (at 90° to the equatorial positions). Phosphorus pentachloride (PCl₅) illustrates this geometry, with all five positions occupied by bonding pairs. Bond angles are 120° between equatorial bonds and 90° between axial and equatorial bonds.

The distinction between axial and equatorial positions becomes critical when lone pairs are present, because lone pairs preferentially occupy equatorial positions (where they experience less repulsion from neighboring groups, having only two 90° neighbors instead of three).

5.4.5 Octahedral Geometry

Six electron groups adopt an octahedral geometry, with all six positions equivalent and bond angles of exactly 90°. Sulfur hexafluoride (SF₆) is the classic example, with sulfur forming six bonds to fluorine in a perfectly symmetric arrangement. The name "octahedral" refers to the eight-faced geometric solid whose vertices are occupied by the six bonded atoms.

5.5 The Effect of Lone Pairs: Bent and Trigonal Pyramidal Shapes

When lone pairs replace bonding pairs in the tetrahedral and trigonal bipyramidal electron geometries, the molecular geometry changes. The resulting shapes — bent geometry and trigonal pyramidal — are among the most important and most commonly tested in AP Chemistry.

Trigonal Pyramidal

Ammonia (NH₃) has four electron groups around nitrogen: three bonding pairs to hydrogen and one lone pair. The electron geometry is tetrahedral (four groups), but the molecular geometry — based only on atom positions — is trigonal pyramidal. The three hydrogen atoms form the base of a triangular pyramid, and the nitrogen atom sits at the apex. The lone pair occupies the fourth tetrahedral position, invisible in the molecular shape but crucial to the bond angles.

Bent Geometry

Water (H₂O) has four electron groups around oxygen: two bonding pairs to hydrogen and two lone pairs. The electron geometry is again tetrahedral, but the molecular geometry is bent — only three atoms are present, and the two hydrogen atoms form a bent or V-shaped arrangement around the oxygen.

Lone Pair Repulsion and Bond Angle Compression

Lone pair repulsion explains why the actual bond angles in NH₃ and H₂O are smaller than the ideal tetrahedral angle of 109.5°. The ranking of repulsion strength, from strongest to weakest, is:

Lone pair–lone pair repulsion (strongest)
Lone pair–bonding pair repulsion
Bonding pair–bonding pair repulsion (weakest)

Because lone pairs are held closer to the central atom (not shared with another nucleus) and occupy more space, they push bonding pairs closer together. In NH₃, the one lone pair compresses the H–N–H angle to about 107°. In H₂O, two lone pairs compress the H–O–H angle all the way to 104.5°, noticeably smaller than the tetrahedral ideal.

This same principle applies to trigonal bipyramidal electron geometries when lone pairs are present. Because equatorial positions have fewer 90° interactions with neighboring groups, lone pairs always occupy equatorial positions first. This leads to the seesaw shape (1 lone pair, 4 bonded), T-shape (2 lone pairs, 3 bonded), and linear molecular geometry (2 lone pairs, 2 bonded) within the trigonal bipyramidal electron geometry.

Diagram: Interactive VSEPR Molecular Geometry Builder

VSEPR Molecular Geometry Builder (p5.js MicroSim)

Type: MicroSim sim-id: vsepr-geometry-builder
Library: p5.js
Status: Specified

Canvas size: 800×450px, responsive to window resize using windowWidth and windowHeight. Background color: dark navy (#0d1b2a). All text in white or light gray.

Layout: Left control panel (260px wide, full height) with dark gray (#1e2d3d) background. Right panel fills remaining canvas width for the 3D-style molecular visualization.

Control panel contains (top to bottom): - Label "Electron Groups" with a large increment/decrement button pair (– and +), showing current count 2–6 in a large bold number. Default: 4. - Label "Lone Pairs" with a similar increment/decrement control, range 0 to (electron groups – 1). Default: 0. - A read-only text display showing "Electron Geometry:" with the name in yellow (#f4d03f). - A read-only text display showing "Molecular Geometry:" with the name in cyan (#5dade2). - A read-only text display showing "Bond Angles:" in white. - A read-only text display showing "Example:" with a chemical formula in light green (#58d68d). - A color legend: bonding pairs shown as red-orange spheres, lone pairs shown as semi-transparent blue-gray clouds, bonds shown as white cylinders.

Right visualization panel: Draws a pseudo-3D representation of the molecule using p5.js 2D drawing, centered in the panel. The central atom is shown as a medium gray sphere. Bonding pair groups are shown as orange-red spheres connected to the center by thick white lines. Lone pairs are shown as diffuse semi-transparent blue-gray ellipses near the center atom. The arrangement updates smoothly (with a brief animated transition using lerp()) whenever the controls change. Bond angle arcs with degree labels appear between adjacent bonds.

All five electron geometries must be implemented with correct 3D projections: - 2 groups: linear (horizontal) - 3 groups: trigonal planar (flat triangle) - 4 groups: tetrahedral (3D perspective using wedge/dash representation) - 5 groups: trigonal bipyramidal (equatorial triangle + axial positions; lone pairs fill equatorial first) - 6 groups: octahedral (square plane + two axial; lone pairs fill positions such that they are trans to each other)

When lone pairs are added, the corresponding molecular geometry name, bond angles, and example update automatically.

Learning objective: Students will be able to (Apply / Analyze — Bloom's levels 3–4) predict the electron and molecular geometry of a molecule given the number of bonding pairs and lone pairs, and explain how lone pairs alter bond angles relative to ideal values.

5.6 Complete VSEPR Geometry Reference

The table below summarizes all common VSEPR geometries, including how lone pairs transform the electron geometry into specific molecular geometries.

Electron Groups	Lone Pairs	Electron Geometry	Molecular Geometry	Ideal Bond Angle	Example
2	0	Linear	Linear	180°	CO₂, BeCl₂
3	0	Trigonal Planar	Trigonal Planar	120°	BF₃, SO₃
3	1	Trigonal Planar	Bent	<120°	SO₂, O₃
4	0	Tetrahedral	Tetrahedral	109.5°	CH₄, CCl₄
4	1	Tetrahedral	Trigonal Pyramidal	<109.5°	NH₃, PH₃
4	2	Tetrahedral	Bent	<109.5°	H₂O, H₂S
5	0	Trigonal Bipyramidal	Trigonal Bipyramidal	90°/120°	PCl₅
5	1	Trigonal Bipyramidal	Seesaw	<90°/<120°	SF₄
5	2	Trigonal Bipyramidal	T-Shape	<90°	ClF₃
5	3	Trigonal Bipyramidal	Linear	180°	XeF₂
6	0	Octahedral	Octahedral	90°	SF₆
6	1	Octahedral	Square Pyramidal	<90°	BrF₅
6	2	Octahedral	Square Planar	90°	XeF₄

5.7 Orbital Hybridization: Reconciling Atomic Orbitals with Molecular Shapes

VSEPR theory predicts molecular geometry with impressive accuracy. But it raises a deeper question: if carbon's valence electrons occupy one spherical 2s orbital and three dumbbell-shaped 2p orbitals, why does methane (CH₄) have four equivalent bonds arranged at perfect tetrahedral angles? The four atomic orbitals are not equivalent — the 2s and 2p orbitals have different shapes and different energies. Yet methane's four C–H bonds are identical in every measurable way.

The answer lies in orbital hybridization, a mathematical mixing of atomic orbitals that produces new hybrid orbitals with shapes and energies intermediate between the originals. Hybridization is not a physical process that literally happens to electrons; it is a mathematical description that reconciles the observed molecular geometry with the underlying orbital structure. The key rule is straightforward: the number of hybrid orbitals produced always equals the number of atomic orbitals mixed.

5.7.1 sp Hybridization

When one s orbital and one p orbital mix, they produce two equivalent sp hybrid orbitals that point in opposite directions, 180° apart. The remaining two p orbitals on the atom are left unmixed and lie perpendicular to the sp axis. sp Hybridization occurs when the central atom forms two electron groups — giving linear geometry.

In acetylene (C₂H₂), each carbon atom is sp hybridized. The two sp orbitals on each carbon point in opposite directions: one forms a bond to hydrogen, and one forms a bond to the other carbon. The two unhybridized p orbitals on each carbon participate in additional bonding between the two carbons. The result is a carbon-carbon triple bond (one sigma bond through the sp orbitals plus two pi bonds through the unhybridized p orbitals) and a perfectly linear H–C≡C–H geometry.

Beryllium chloride (BeCl₂) also shows sp hybridization: the beryllium atom mixes its 2s and one 2p orbital to produce two sp orbitals pointing in opposite directions, each forming a bond to a chlorine atom.

5.7.2 sp² Hybridization

When one s orbital and two p orbitals mix, they produce three equivalent sp² hybrid orbitals arranged in a flat plane at 120° to one another. One p orbital remains unhybridized and points perpendicular to that plane. sp² Hybridization occurs when the central atom forms three electron groups — giving trigonal planar electron geometry.

Ethylene (C₂H₄) illustrates sp² hybridization beautifully. Each carbon atom mixes its 2s and two 2p orbitals into three sp² hybrids in the same plane, with one unhybridized 2p orbital pointing above and below that plane. Two sp² orbitals on each carbon bond to hydrogen atoms; the third sp² orbital on each carbon bonds to the other carbon (forming one sigma bond). The two leftover p orbitals, one on each carbon, align parallel to each other and overlap side-by-side to form a pi bond. The result is the carbon-carbon double bond characteristic of alkenes.

Boron trifluoride (BF₃) also shows sp² hybridization around the central boron atom, producing three equivalent planar bonds.

5.7.3 sp³ Hybridization

When one s orbital and three p orbitals all mix, they produce four equivalent sp³ hybrid orbitals pointing toward the corners of a tetrahedron at 109.5° angles. sp³ Hybridization occurs when the central atom forms four electron groups — giving tetrahedral electron geometry.

Methane (CH₄) is the canonical sp³ example. Carbon's 2s and all three 2p orbitals mix to form four equivalent sp³ hybrid orbitals, each forming one sigma bond with a hydrogen atom. All four bonds are identical, and the H–C–H angles are all 109.5°.

In ammonia (NH₃), nitrogen is also sp³ hybridized — three of the four sp³ orbitals form bonds to hydrogen, and the fourth sp³ orbital holds the lone pair. In water (H₂O), oxygen is likewise sp³ hybridized — two sp³ orbitals bond to hydrogen and two hold lone pairs. The hybridization is the same in all three cases, but the molecular geometry differs because the number of bonding pairs versus lone pairs differs.

The table below summarizes the three main hybridization types:

Hybridization	Orbitals Mixed	Hybrid Orbitals	Electron Geometry	Bond Angles	Unhybridized p Orbitals	Example
sp	s + p	2	Linear	180°	2	BeCl₂, C₂H₂
sp²	s + p + p	3	Trigonal Planar	120°	1	BF₃, C₂H₄
sp³	s + p + p + p	4	Tetrahedral	109.5°	0	CH₄, NH₃, H₂O

Diagram: Hybridization Orbital Mixing Diagram

sp / sp² / sp³ Orbital Hybridization Visualizer

Type: Infographic / Interactive Diagram sim-id: hybridization-diagram
Library: p5.js
Status: Specified

Canvas size: 800×450px, responsive. Background: white (#ffffff). Three horizontally-arranged panels, one for each hybridization type (sp, sp², sp³), each 260px wide with slight padding and a light gray border.

Each panel contains two rows of orbital diagrams: - Top row labeled "Before Mixing": shows the unmixed atomic orbitals as separate labeled boxes. Each box represents one orbital and is colored by type: s orbitals in soft blue, p orbitals in soft orange. Boxes are labeled (2s, 2px, 2py, 2pz). Unhybridized p orbitals after mixing are shown in a lighter shade. - A downward arrow in the center labeled "Hybridize" with a mathematical plus sign between the contributing orbitals. - Bottom row labeled "After Mixing": shows the resulting hybrid orbitals as larger boxes colored in green (hybrid type: sp = lime green, sp² = teal, sp³ = forest green), labeled with their names (sp, sp², sp³). Any remaining unhybridized p orbitals are shown in orange to the right of the hybrids.

Below each panel: a small 2D geometry icon (line for sp, flat triangle for sp², tetrahedron outline for sp³) and the bond angle value.

Panel headers read "sp Hybridization", "sp² Hybridization", "sp³ Hybridization" in bold dark text.

Clicking any panel highlights it (light yellow background) and displays a text box below the three panels showing: the hybridization name, number of hybrid orbitals, geometry, bond angles, number of unhybridized p orbitals available for pi bonding, and an example molecule.

Learning objective: Students will be able to (Understand / Apply — Bloom's levels 2–3) explain how atomic orbitals combine to form hybrid orbitals, connect hybridization type to molecular geometry, and determine the hybridization of a central atom from the number of electron groups.

5.8 Sigma and Pi Bonds: The Two Faces of Covalent Bonding

Hybridization introduces a distinction that is essential for understanding double and triple bonds: the difference between sigma bonds and pi bonds. These two types of bonds differ in how the orbital lobes overlap and have profoundly different properties.

Sigma bonds (\(\sigma\) bonds) form when two orbitals overlap end-to-end, directly along the internuclear axis — the imaginary line connecting the two nuclei. This head-on overlap produces a bond with electron density concentrated in a cylindrical region around the internuclear axis. Sigma bonds can form between any two hybrid orbitals (sp–sp, sp²–sp², sp³–sp³) or between a hybrid orbital and an s orbital (as in a C–H bond). Every single bond is a sigma bond. Every double bond contains one sigma bond. Every triple bond also contains one sigma bond.

Pi bonds (\(\pi\) bonds) form when two unhybridized p orbitals on adjacent atoms overlap side-by-side, parallel to each other and perpendicular to the internuclear axis. This sideways overlap produces two lobes of electron density — one above and one below the internuclear axis. Pi bonds are always the second bond in a double bond and the second and third bonds in a triple bond. A double bond therefore consists of one sigma bond plus one pi bond; a triple bond consists of one sigma bond plus two pi bonds.

Key differences between sigma and pi bonds:

Sigma bonds allow free rotation around the internuclear axis (rotation does not disrupt the end-on overlap). Pi bonds prevent rotation — rotating one end of a pi bond relative to the other would break the sideways overlap and destroy the bond.
Sigma bonds are generally stronger than pi bonds, because end-on overlap produces a greater concentration of electron density between the nuclei.
Pi bonds can only form after a sigma bond already connects two atoms, because they require the p orbitals to be aligned parallel to each other, which is only guaranteed when the atoms are already bonded.
A carbon-carbon single bond (one sigma) is weaker than a carbon-carbon double bond (one sigma plus one pi), which is weaker than a carbon-carbon triple bond (one sigma plus two pi). More bonds between the same two atoms means shorter, stronger bonds.

The restriction on rotation imposed by pi bonds has enormous consequences in organic chemistry. The carbon-carbon double bond in alkenes creates geometric (cis-trans) isomers, and the rigidity of pi bonds in aromatic rings like benzene gives them their distinctive planar structure and chemical stability.

Diagram: Sigma and Pi Bond Formation Visualization

Sigma and Pi Bond Orbital Overlap Diagram

Type: Static Infographic sim-id: sigma-pi-bond-diagram
Library: p5.js
Status: Specified

Canvas size: 800×400px, responsive. Background: very light gray (#f5f5f5). Two side-by-side panels with titles in bold dark text.

Left panel — "Sigma Bond (σ)": - Shows two atomic nuclei (gray dots) on a horizontal axis labeled "Internuclear Axis". - Two orbital lobes (one from each atom) are drawn as ellipses pointing directly toward each other and overlapping in the middle, colored blue for one atom and red for the other, with the overlap region colored purple. - An arrow below the diagram labeled "End-on overlap" points to the overlap region. - Below the diagram, a cross-section view (looking along the internuclear axis) shows a circular electron density pattern — symmetric around the axis. - Text below: "All single bonds, plus first bond in double/triple bonds. Free rotation allowed."

Right panel — "Pi Bond (π)": - Shows two atomic nuclei (gray dots) on a horizontal axis, closer together than in the sigma diagram to indicate they are already bonded. - Two p orbital lobes (one above the axis, one below) from each atom are drawn as ellipses parallel to each other, perpendicular to the axis. The upper lobes are blue, the lower lobes are red, with the overlap regions purple. - An arrow pointing to the overlap region labeled "Side-on overlap". - A cross-section view from the side shows two lobes of electron density — one above and one below the axis (a figure-eight or dumbbell cross-section), showing the node at the axis. - Text below: "Second bond in double bonds, second and third in triple bonds. Rotation prevented."

A summary line at the bottom of the full canvas reads: "Single bond = 1σ | Double bond = 1σ + 1π | Triple bond = 1σ + 2π"

Color scheme: atomic nuclei gray, atom 1 orbitals blue (#3498db), atom 2 orbitals tomato red (#e74c3c), overlap region purple (#9b59b6). Labels in dark charcoal (#2c3e50).

Learning objective: Students will be able to (Understand / Analyze — Bloom's levels 2–4) distinguish sigma and pi bonds by their orbital overlap patterns, explain why pi bonds restrict molecular rotation, and count sigma and pi bonds in single, double, and triple bonds.

When two identical atoms form a covalent bond — as in H₂, Cl₂, or N₂ — the shared electrons are attracted equally by both nuclei and are distributed symmetrically between the two atoms. This is a nonpolar covalent bond: no net charge separation, no dipole.

When two different atoms bond, however, their different electronegativities mean they attract the shared electrons with different strengths. The more electronegative atom pulls the electron density toward itself, acquiring a partial negative charge (symbolized δ–), while the less electronegative atom is left with a partial positive charge (δ+). This unequal sharing creates a polar covalent bond — a bond with a definite positive end and a negative end.

The degree of bond polarity is determined by the electronegativity difference (\(\Delta\chi\)) between the two bonded atoms:

\[ \Delta\chi = \chi_{\text{more electronegative}} - \chi_{\text{less electronegative}} \]

The conventional scale for classifying bonds by electronegativity difference is:

\(\Delta\chi < 0.4\): Nonpolar covalent bond (essentially equal sharing)
\(0.4 \leq \Delta\chi < 1.7\): Polar covalent bond (unequal sharing, partial charges)
\(\Delta\chi \geq 1.7\): Ionic bond (electron transfer, full charges)

These are guidelines, not sharp boundaries. Many chemists use 0.5 and 2.0 as the cutoffs, and the AP exam typically does not require you to memorize exact values. What matters is the trend: as the electronegativity difference increases, the bond becomes more polar and more ionic in character.

The dipole moment (\(\mu\)) is the quantitative measure of bond polarity. It equals the magnitude of the partial charges multiplied by the distance between them:

\[ \mu = Q \times r \]

where \(Q\) is the magnitude of the partial charges (in units of electron charge, e) and \(r\) is the bond length (distance between the atoms). Dipole moments are reported in units called debyes (D). A larger dipole moment indicates a more polar bond.

Bond polarity and electronegativity differences for common bonds:

Bond	Electronegativity Difference	Bond Type	Polarity
C–H	0.4	Nonpolar covalent	Essentially nonpolar
N–H	0.9	Polar covalent	Moderate
O–H	1.4	Polar covalent	Significant
H–F	1.9	Polar covalent (borderline ionic)	Very large
Na–Cl	2.1	Ionic	Full charge transfer
C–C	0.0	Nonpolar covalent	Nonpolar
C–O	0.9	Polar covalent	Moderate
C–F	1.5	Polar covalent	Large

5.10 Molecular Polarity: Adding Up Dipole Vectors

A single bond dipole, like a tug-of-war between two electronegative players, can be represented as a vector — an arrow pointing from the positive end (\(\delta+\)) toward the negative end (\(\delta-\)) of the bond. Molecular polarity is determined by the vector sum of all individual bond dipoles in the molecule.

This is where molecular geometry becomes decisive. A molecule with polar bonds can still be nonpolar overall if its geometry causes the individual bond dipoles to cancel one another through symmetry. Conversely, even a molecule with only moderately polar bonds can be significantly polar if the geometry is asymmetric and the dipoles add up rather than cancel.

Polar molecules have a net dipole moment — their individual bond dipoles do not cancel. Examples include water (H₂O), ammonia (NH₃), hydrogen chloride (HCl), and sulfur dioxide (SO₂). In water, the two O–H bond dipoles both point roughly toward the oxygen atom, and because the molecule is bent (not linear), they add to produce a net dipole pointing toward the oxygen.

Nonpolar molecules have a zero net dipole moment — either because all bonds are nonpolar, or because the bond dipoles cancel exactly due to molecular symmetry. Examples include carbon dioxide (CO₂), methane (CH₄), boron trifluoride (BF₃), carbon tetrachloride (CCl₄), and nitrogen (N₂). In CO₂, the two C=O bond dipoles are equal in magnitude and point in exactly opposite directions (180° apart), so they cancel completely. In CCl₄, the four C–Cl bond dipoles point symmetrically toward the four tetrahedral corners and cancel exactly, making CCl₄ nonpolar despite having four polar bonds.

The key rules for determining molecular polarity:

If a molecule has no polar bonds, it is nonpolar (e.g., H₂, Cl₂, CH₄)
If a molecule has polar bonds but they are arranged symmetrically so that dipoles cancel, it is nonpolar (e.g., CO₂, BF₃, CCl₄, SF₆)
If a molecule has polar bonds and its geometry is asymmetric — either due to lone pairs or different surrounding atoms — the dipoles do not cancel and the molecule is polar (e.g., H₂O, NH₃, HCl, CH₃Cl)
Lone pairs on the central atom almost always cause asymmetry and therefore polarity

Molecular polarity has direct consequences for physical properties. Polar molecules experience dipole-dipole intermolecular forces and can dissolve in polar solvents like water ("like dissolves like"). Nonpolar molecules experience only London dispersion forces and dissolve preferentially in nonpolar solvents. These ideas are developed further in Chapter 6.

5.11 Connecting Geometry, Hybridization, and Polarity: A Worked Example

To see how VSEPR, hybridization, bond polarity, and molecular polarity work together, consider chloromethane (CH₃Cl):

Step-by-step analysis:

Lewis structure: Carbon forms four bonds — three to hydrogen and one to chlorine. Carbon has no lone pairs.
Electron groups: Four (all bonding). Electron geometry: tetrahedral.
Molecular geometry: Tetrahedral (no lone pairs to alter shape).
Bond angles: Approximately 109.5° (the H–C–Cl angles are slightly compressed from ideal because chlorine is larger than hydrogen, but close to 109.5°).
Hybridization: Carbon has four electron groups → sp³ hybridization.
Bond polarity: C–H bonds have \(\Delta\chi \approx 0.4\) (borderline nonpolar). C–Cl bond has \(\Delta\chi \approx 0.5\) (slightly polar, with Cl being the negative end).
Molecular polarity: The three C–H bond dipoles are small and arranged symmetrically, nearly canceling each other. The C–Cl bond dipole points toward Cl and does not cancel. The molecule has a net dipole moment pointing toward chlorine — CH₃Cl is a polar molecule.

This example illustrates that even a molecule with nearly nonpolar C–H bonds can be polar overall if one substituent is significantly more electronegative than the others.

5.12 Naming Covalent Compounds: The Prefix System

Binary covalent compounds — those consisting of two nonmetal elements — are named using a systematic set of Greek numerical prefixes that indicate how many atoms of each element are present. This naming covalent compounds system avoids the ambiguity of ionic compound names, since nonmetals can form multiple different compounds with each other (carbon and oxygen form both CO and CO₂, for instance).

The prefixes used in naming binary covalent compounds are:

Number	Prefix
1	mono-
2	di-
3	tri-
4	tetra-
5	penta-
6	hexa-
7	hepta-
8	octa-
9	nona-
10	deca-

Rules for naming binary covalent compounds:

The first element in the formula is named first, using the element's full name. The prefix "mono-" is omitted for the first element.
The second element is named with the suffix "-ide" (just as in ionic compounds), and its Greek prefix is always included — even if the prefix is "mono-."
When the prefix ends in "a" or "o" and the element name begins with a vowel, the final vowel of the prefix is dropped to avoid awkward pronunciation (e.g., "monoxide" not "monooxide," "tetroxide" not "tetraoxide").

Examples of naming covalent compounds:

CO: carbon monoxide (the "mono-" applies to the oxygen, not the carbon)
CO₂: carbon dioxide
NO₂: nitrogen dioxide
N₂O₄: dinitrogen tetroxide
N₂O₅: dinitrogen pentoxide
SO₂: sulfur dioxide
SO₃: sulfur trioxide
PCl₃: phosphorus trichloride
PCl₅: phosphorus pentachloride
SF₆: sulfur hexafluoride

Note that many common compounds have traditional names that are used instead of the systematic prefix names: water (H₂O), ammonia (NH₃), and nitric oxide (NO) are always referred to by their common names in both everyday usage and on the AP exam.

5.13 Network Covalent Solids: When Covalent Bonds Extend Throughout a Crystal

Most covalent compounds exist as individual molecules — discrete units of a fixed number of atoms held together by covalent bonds, with the molecules themselves interacting through weaker intermolecular forces. But a special category of covalent substances, called network covalent solids, forms enormous three-dimensional lattices in which every atom is covalently bonded to its neighbors in a continuous, repeating network extending throughout the entire crystal.

In a network covalent solid, there are no individual molecules — the entire crystal is one gigantic covalent "molecule." Because breaking a piece of a network covalent solid requires breaking strong covalent bonds (rather than weak intermolecular forces), these materials have extraordinary hardness and extremely high melting points.

Diamond is the most famous network covalent solid. Each carbon atom in diamond is sp³ hybridized, forming four sigma bonds directed tetrahedrally to four neighboring carbon atoms. This arrangement repeats throughout the entire crystal, creating an incredibly rigid three-dimensional framework. Diamond is the hardest known natural material (10 on the Mohs scale), has a melting point above 3500°C, and does not conduct electricity (because all valence electrons are localized in covalent bonds with no free electrons to carry charge).

Quartz (silicon dioxide, SiO₂) is another classic network covalent solid. Unlike molecular carbon dioxide (CO₂), where each carbon forms two double bonds to two separate oxygen atoms, silicon dioxide consists of a continuous three-dimensional network in which each silicon atom is bonded to four oxygen atoms and each oxygen atom bridges between two silicon atoms. This extended structure — with the empirical formula SiO₂ representing just the ratio of atoms, not a molecular formula — gives quartz its hardness, high melting point (~1700°C), and stability.

Other examples of network covalent solids include:

Silicon carbide (SiC): Used as an industrial abrasive and in high-performance ceramics
Boron nitride (BN): Exists in both a diamond-like cubic form (extremely hard) and a graphite-like hexagonal form (soft lubricant)
Graphite: A layered form of carbon where each carbon is sp² hybridized within a flat hexagonal network; unlike diamond, the delocalized pi electrons between layers allow graphite to conduct electricity

Properties that distinguish network covalent solids from molecular covalent compounds:

Very high melting and boiling points (covalent bonds throughout the entire structure must be broken)
Extreme hardness (diamond, SiC, BN cubic)
Poor electrical conductivity in most cases (electrons localized in bonds), except graphite and related materials
Insoluble in virtually all solvents (no molecular units that can be solvated)
Represented by empirical formulas (such as SiO₂) rather than molecular formulas

The distinction between network covalent solids and molecular covalent substances is one of the key ideas connecting molecular structure to bulk material properties — a theme that runs through AP Chemistry from bonding through thermodynamics.

5.13b Molecular Orbital Theory: Competing Models of Covalent Bonding

The Lewis structure and VSEPR/hybridization approach developed in Chapters 4 and 5 explains a great deal — molecular geometry, bond polarity, and physical properties. But it has a fundamental limitation: it treats electrons as localized between specific pairs of atoms. A more powerful model, Molecular Orbital (MO) theory, treats electrons as delocalized — spread across the entire molecule in orbitals that belong to all the atoms at once.

These two models — Valence Bond (VB) theory, which underlies hybridization, and MO theory — are competing models of bonding. Each has strengths and weaknesses. Understanding both satisfies the AP Chemistry Evaluate-level objective: "Evaluate competing models of bonding."

Forming Molecular Orbitals

When two atomic orbitals on adjacent atoms overlap, they combine mathematically to produce molecular orbitals (MOs). Two AOs always produce two MOs:

Bonding MO (σ or π): constructive interference — the electron density increases between the nuclei, lowering the energy relative to the isolated atoms. Electrons in bonding MOs stabilize the molecule.
Antibonding MO (σ* or π*): destructive interference — electron density decreases between the nuclei, raising the energy above the isolated atoms. Electrons in antibonding MOs destabilize the molecule.

The bonding MO is always lower in energy than the original AOs; the antibonding MO is always higher.

Filling MO Energy Diagrams

Molecular orbitals are filled with electrons following the same three rules as atomic orbitals:

Aufbau principle: fill lowest-energy MOs first
Pauli exclusion principle: each MO holds at most 2 electrons with opposite spins
Hund's rule: when MOs of equal energy are available, place one electron in each before pairing

The MO energy diagram for a homonuclear diatomic molecule (\(\ce{A2}\)) places the bonding MO below, the antibonding MO above, and shows each MO filled from the bottom up with all available valence electrons.

For H₂ (2 total electrons):

MO	Electrons
σ*1s (antibonding)	0
σ1s (bonding)	2

Both electrons go into the bonding MO — the molecule is stable.

Bond Order from MO Theory

Bond order (BO) from MO theory is:

\[\text{Bond order} = \frac{\text{(electrons in bonding MOs)} - \text{(electrons in antibonding MOs)}}{2}\]

A higher bond order means a stronger, shorter bond. A bond order of zero means the molecule does not form.

Molecule	Bonding \(e^-\)	Antibonding \(e^-\)	Bond Order	Bond
\(\ce{H2}\)	2	0	1	Single
\(\ce{He2}\)	2	2	0	Does not exist
\(\ce{N2}\)	8	2	3	Triple
\(\ce{O2}\)	8	4	2	Double
\(\ce{F2}\)	8	6	1	Single

Paramagnetism and Diamagnetism

One of MO theory's greatest triumphs is explaining the magnetic behavior of molecules:

Diamagnetic: all electrons are paired → molecule is weakly repelled by a magnetic field
Paramagnetic: one or more unpaired electrons → molecule is attracted to a magnetic field

Liquid oxygen is visibly attracted to a magnet — it is paramagnetic. The Lewis structure of \(\ce{O2}\) (which shows a double bond and all paired electrons) cannot explain this. MO theory predicts it correctly: the two highest-energy electrons occupy two degenerate (equal-energy) antibonding π* MOs one each, following Hund's rule, leaving them unpaired.

\[\ce{O2}: \quad \text{Bond order} = \frac{8-4}{2} = 2 \quad \text{(paramagnetic — 2 unpaired electrons)}\]

This is a case where MO theory makes a prediction that Lewis structures get wrong, demonstrating the value of the more sophisticated model.

MO Theory vs. Lewis/VB Theory: When to Use Each

Feature	Lewis / VB / VSEPR	MO Theory
Molecular geometry	Excellent	Not directly
Bond polarity	Good	Good
Paramagnetism	Cannot predict	Correctly predicts
Delocalization / resonance	Partial (resonance structures)	Full treatment
Bond order	From bond type	From BO formula
Complexity	Simpler	More complex

For AP Chemistry, use VSEPR and Lewis structures for geometry, polarity, and most bonding questions. Invoke MO theory specifically when asked about: magnetic properties, bond order of species not well-described by Lewis structures, or "competing models of bonding."

Diagram: MO Energy Diagram Explorer

Interactive Molecular Orbital Energy Diagram

Type: microsim sim-id: mo-energy-diagram
Library: p5.js
Status: Specified

Learning objective: Students will be able to (Evaluating) compare MO and Lewis/VB models of bonding by constructing MO energy diagrams for diatomic molecules and predicting bond order and magnetic behavior.

Canvas: 820 × 560 px, responsive to window resize.

Controls (left panel, 180 px wide): - Dropdown: Select molecule: H₂ | He₂ | Li₂ | B₂ | C₂ | N₂ | O₂ | F₂ | Ne₂ | O₂⁻ | O₂⁺ | NO | CO - Display: Total valence electrons count

MO diagram (center, ~500 px wide): - Left column: two identical atomic orbital energy levels labeled "Atom A" (e.g., "1s") - Center column: MO energy levels — bonding MO lower, antibonding MO higher (marked with asterisk *) - Right column: same as left, "Atom B" - Dashed lines connecting AOs to MOs - Electrons shown as pairs of up/down arrows filled into MOs from lowest to highest energy - Color: bonding MOs in blue, antibonding MOs in red, electrons as black arrows

Results panel (right, 140 px wide): - Bonding electrons: N - Antibonding electrons: N - Bond order = (bonding − antibonding) / 2 = X - Bond order shown as a large number with label (Single / Double / Triple / None) - Magnetic character: "Paramagnetic (N unpaired)" or "Diamagnetic (all paired)" - Lewis structure bond notation: "≡" or "=" or "−" or "× (no bond)"

Color coding for bond order: Bond order 3 = green, 2 = blue, 1 = orange, 0 = red.

Implementation: p5.js. Store electron configuration data for each molecule in a lookup object. Draw the diagram procedurally from the configuration data.

5.14 Chapter Summary and Connections

This chapter developed a complete framework for moving from the flat Lewis structure of a molecule to its three-dimensional shape, its bonding character, and its physical properties.

The logical sequence of ideas covered:

VSEPR theory uses electron group repulsion to predict three-dimensional molecular geometry from a Lewis structure.
The electron geometry describes the arrangement of all electron groups; the molecular geometry describes only the atom positions.
The five fundamental electron geometries (linear, trigonal planar, tetrahedral, trigonal bipyramidal, octahedral) give rise to specific bond angles.
Lone pairs exert stronger repulsion than bonding pairs, compressing bond angles and producing bent and trigonal pyramidal molecular geometries.
Orbital hybridization (sp, sp², sp³) reconciles observed geometries with the quantum mechanical orbital model.
Covalent bonds consist of sigma bonds (end-on overlap) and pi bonds (side-on overlap); every bond has one sigma bond, and double and triple bonds also have pi bonds.
Bond polarity is determined by the electronegativity difference between bonded atoms; the dipole moment \(\mu = Q \times r\) quantifies this polarity.
Molecular polarity is the vector sum of bond dipoles; symmetric geometries produce nonpolar molecules even with polar bonds.
Binary covalent compounds are named using Greek prefixes.
Network covalent solids extend covalent bonds throughout a three-dimensional lattice, producing materials with extreme hardness and high melting points.

These concepts provide the foundation for Chapter 6, which examines how molecular shape and polarity determine the intermolecular forces responsible for the physical properties of liquids and solids.

Key Terms

VSEPR Theory: Valence Shell Electron Pair Repulsion — predicts molecular geometry from the repulsion of electron groups around a central atom
Electron Geometry: The three-dimensional arrangement of all electron groups (bonding and lone pairs) around a central atom
Molecular Geometry: The three-dimensional arrangement of atoms only, determined by ignoring lone pairs
Electron Group (Domain): Any region of electron density around the central atom — a single bond, double bond, triple bond, or lone pair all count as one group
Bond Angle: The angle between two adjacent bonds at the central atom, measured between the bond axes
Lone Pair Repulsion: The stronger repulsive effect of nonbonding electron pairs relative to bonding pairs, which compresses bond angles below ideal values
Orbital Hybridization: The mathematical combination of atomic orbitals to produce hybrid orbitals with shapes and energies suited to the observed bonding and geometry
Sigma Bond: A covalent bond formed by end-on (head-on) orbital overlap along the internuclear axis; allows free rotation
Pi Bond: A covalent bond formed by side-on (lateral) overlap of p orbitals perpendicular to the internuclear axis; prevents rotation
Electronegativity Difference: The difference in electronegativity values between two bonded atoms, used to classify bonds as nonpolar covalent, polar covalent, or ionic
Dipole Moment: A quantitative measure of bond or molecular polarity; \(\mu = Q \times r\), reported in debyes (D)
Polar Molecule: A molecule with a net dipole moment — bond dipoles do not cancel due to asymmetric geometry
Nonpolar Molecule: A molecule with zero net dipole moment — either all bonds are nonpolar or bond dipoles cancel due to symmetry
Network Covalent Solid: A solid in which covalent bonds extend throughout the entire crystal, with no discrete molecular units; examples include diamond and quartz

References

See Annotated References