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B.4 Geometry and trigonometry

Syllabus
2017
Topic
—
Level
A2

Learning objectives

—B.4.1—Angles and regular structuresRecognise angles and shapes in regular two- and three-dimensional structures, including molecular shapes and bond angles.—B.4.2—Representing two- and three-dimensional formsVisualise and represent two- and three-dimensional forms, including drawing isomers. Identifying chiral centres from such representations is an A2 application.—B.4.3—Symmetry of shapesUnderstand the symmetry of two- and three-dimensional shapes. Identifying chiral centres from representations is an A2 application.

Electron regions determine molecular shape and bond angle

Electron-pair repulsion places regions of electron density around a central atom as far apart as possible. Count each single, double or triple bond as one bonding region, then add lone pairs. Name the molecular shape from atom positions, while using lone pairs to explain angle compression.

Regions around centre Lone pairs Molecular shape Ideal/typical angle Example
2 0 linear 180∘180^\circ BeClX2\ce{BeCl2}, COX2\ce{CO2}
3 0 trigonal planar 120∘120^\circ BClX3\ce{BCl3}
4 0 tetrahedral 109.5∘109.5^\circ CHX4\ce{CH4}, NHX4X+\ce{NH4+}
4 1 trigonal pyramidal about 107∘107^\circ NHX3\ce{NH3}
4 2 bent about 104.5∘104.5^\circ HX2O\ce{H2O}
5 0 trigonal bipyramidal 90∘90^\circ, 120∘120^\circ gaseous PClX5\ce{PCl5}
6 0 octahedral 90∘90^\circ SFX6\ce{SF6}

Lone pairs repel more strongly than bonding pairs because their electron density is concentrated near one nucleus. One lone pair changes tetrahedral electron-region geometry into a trigonal-pyramidal molecular shape; two produce a bent shape and compress the bond angle further.

Do not name a shape from the number of bonds alone: include lone pairs on the central atom. A double bond counts as one region for basic shape prediction, and the tabulated angle is not automatically exact when lone pairs or unequal surrounding groups are present.

Preserve connectivity and depth when moving between 2D and 3D

Mark Spatial meaning
ordinary line bond lies in the plane of the page
solid wedge bond points towards the viewer
hashed wedge bond points away from the viewer
crossing lines without a labelled atom not automatically a bond or shared atom

Isomers share a molecular formula but differ in arrangement. First fix connectivity: different carbon skeletons, functional-group positions or substituent positions are structural isomers. With the same connectivity, restricted rotation or a chiral three-dimensional arrangement can produce stereoisomers.

For a substituted ring, fix one substituent as position 1, place the remaining substituent(s) systematically, then remove drawings related only by rotation or reflection of the ring. For two identical substituents on benzene, the distinct relative positions are 1,2-, 1,3- and 1,4-. This symmetry check prevents duplicate counting.

At A2, inspect each tetrahedral carbon and trace its four attached groups. It is a chiral centre only when all four groups are different. A valid pair of enantiomer drawings keeps every bond connection unchanged and reverses the complete three-dimensional arrangement.

Rotating a whole drawing does not create an isomer, while swapping two groups at one chiral centre changes its configuration. Wedges show depth; they must not be added randomly to a flat structure.

Use symmetry to decide whether drawings are equivalent or stereoisomeric

Relationship Structural test Consequence
same object after rotation/reflection in the drawing plane connectivity and full spatial arrangement coincide duplicate representation, not a new isomer
geometric isomers same connectivity; restricted rotation and suitable different substituents distinct E/ZE/Z or cis/trans arrangements
optical isomers same connectivity; non-superimposable mirror images enantiomeric pair

A carbon-carbon double bond prevents free rotation. Geometric isomerism requires each double-bonded carbon to have two different substituents. The E/ZE/Z system compares the higher-priority substituent on each carbon and remains usable where cis/trans labels are ambiguous.

For the single-centre cases required here, a tetrahedral carbon bonded to four different groups is asymmetric and gives two non-superimposable mirror arrangements. A mirror plane through the complete molecule would map one half onto the other and indicates that the structure is not chiral.

Apply the same spatial test to molecular or complex-ion representations: identify which positions are equivalent by symmetry, then decide whether two drawings superimpose or preserve a genuine geometric or mirror-image difference.

A mirror image is not automatically a different optical isomer; it must be non-superimposable. Conversely, two flat drawings that look different may represent the same three-dimensional object after rotation.