Making a connection between mathematics and chemistry (determining the optimal angle between the atoms of covalent bonds) should help answer the trigonometry student's question, "Why do we need to learn these identities and when will we ever use them?" Several trigonometric identities are necessary when doing the proof of the optimal angle for a molecule with four identical atoms bonded to a central atom that has a complete valence shell. This lesson could be taught collaboratively by a chemistry and a mathematics teacher. Manipulatives should be used to aid students' understanding of the lesson.

BASIC INFORMATION

Atoms form covalent bonds with other atoms to create molecules. A covalent bond is formed when two atoms share a pair of electrons. The number of covalent bonds that an atom can form depends on the number of available electrons found in its outermost (valence) shell. In a single covalent bond, the sharing of a pair of electrons forms the bond that holds two atoms together. However, when considering a polyatomic molecule (a molecule in which there are two or more atoms bonded to a central atom) it is important to realize that there are interactions that occur between the covalent bonds that determine the three-dimensional shape of the molecule.

What are these interactions that occur between covalent bonds? An electron is by definition a negatively charged atomic particle. In a polyatomic molecule, there are two or more covalent bonds. Because each bond is composed of negatively charged electrons. the negative charges found on the electrons that compose the bonds repel each other. Ultimately, the molecule will be arranged in three dimensions such that the repulsion between the electron pairs of different bonds is at a minimum. The repulsive forces between the electron pairs of different covalent bonds causes the bonds to remain as far apart as possible. The valence-shell electron-pair repulsion (VSEPR) model is used by scientists to account for the geometric arrangements of covalent bonds around a central atom that minimize the repulsion between the electron pairs of the covalent bonds.

The simplest molecular shape that can be explained
by the VSEPR model is that of a molecule in which two
atoms are bonded covalently to a central atom to complete
its valence shell. Carbon dioxide (with the molecular formula
CO_{2})
is an example of a molecule in which two atoms are bonded
covalently to the central atom (C), leaving no nonbonding
pairs of electrons.

A Lewis structure is a two-dimensional
representation of a molecule's structure. The Lewis structure
for CO_{2} appears in Figure 1. The electron pairs that
create the covalent bonds between the carbon atom and the oxygen
atoms repel each other. In order to minimize the repulsion
between the covalent bonds, the bonds must be separated from
each other by 180°. In this case, the Lewis structure
accurately describes both the two-dimensional and three-dimensional
shape of the molecule. A polyatomic molecule that is composed of
two atoms covalently bonded to a central atom (leaving no
non-bonding pairs of electrons) takes on a linear conformation
and a characteristic bond angle of 180°.

Using the VSEPR model, students can examine the geometry of a
molecule that is composed of three identical atoms covalently
bonded to a central atom, leaving no nonbonding electrons.
Boron trifluoride (BF_{3} ) is a molecule that fits this
description. The Lewis structure of BF_{3} that appears
in Figure 1 accounts for molecular shape in only two dimensions.
In reality, the molecule exists in three dimensions. The
two-dimensional molecular model in this figure suggests that the
optimal bond angle for BF_{3} is 120°, and that all
four atoms of the molecule are in the same plane.

Is there a
three-dimensional conformation that would result in a greater bond
angle and thus a greater distance between the bonds? Intuitively,
the answer is no. If boron were moved out of the plane, the angle
in question (FBF) would become smaller, less than 120°. When this
angle is reduced, the distance between the two fluorine atoms
of the angle is reduced as well.
If the two fluorine atoms move closer to each other, the
electrons that form the bonds are also brought closer
together. If these electrons are brought closer together,
they will experience more repulsion. The three-dimensional
conformation that BF_{3} must take on in order to
minimize the repulsion between the covalent bonds is
a trigonal planar conformation with an optimal bond
angle of 120°.

POLYATOMIC MODELING

A more interesting problem of molecular geometry is
encountered when dealing with a molecule comprised
of four atoms covalently bonded to a central atom leaving no
nonbonding electron pairs. A common example of such a molecule
is methane (CH_{4}). The Lewis structure for
CH_{4} also appears in Figure 1. The Lewis structure
suggests that the optimal bond angle for methane is 90°.
Does a three-dimensional conformation exist for methane that
would allow bond angles greater than 90°? If such a
conformation exists, the hydrogen atoms would be farther apart
from each other. How does one go about finding the optimal bond
angle that places these four hydrogen atoms at points in
space that are the greatest distance from each other?

THREE-DIMENSIONAL MODELS

Students should be encouraged to experiment with manipulatives, such as gum drops and toothpicks or straws and marshmallows, to build the three-dimensional models of carbon dioxide, boron trifluoride, and methane (Figure 2). The three-dimensional model of methane is a tetrahedron, with the carbon atom at the center of the tetrahedron and the four hydrogen atoms at the vertices (Figure 3). Students should be encouraged to construct cardboard or paper triangles for Figures 4, 5, and 6 and to use a protractor to measure the bond angles of each of their models. This will help students to visualize the methane model and understand the following discussion.

To determine the optimal bond angle, draw a
perpendicular line from the carbon atom (C) to the plane
containing three of the hydrogen atoms. Let *Q* represent
the foot of this perpendicular line (Figures 3 and 4), and
let *y* represent the distance between the carbon atom and
any of the hydrogen atoms. Let *a* represent the distance
from *Q* to one of the hydrogen atoms, and let *x* represent
the measure of the required bond angle.

In Figure 3, note that *Q* is the circumcenter of the
equilateral triangle formed by the three hydrogen atoms
that lie in the bottom plane. Because the triangle HHH is
equilateral, each of the angles HQH measures 120°. In
Figure 4, the measure of angle HCQ = (180-x)° and the measure
of angle CHQ = (x-90)°. It follows that

or a = y cos(x- 90)°. Using the
trigonometric identities

cos(-A) = cos(A) and cos(90-A)° = sin(A), it follows
that a = y cos(90 -x)° and that a = y sin(x).

Now, examine the triangle formed by two hydrogen atoms and point Q (Figure 5). The altitude from point Q divides the triangle HQH into two congruent triangles HQT and HQT (hypotenuse - leg theorem). So, the vertex angle HQH is divided into two angles whose measures are each 60°.

Figure 6 represents the triangle formed by the carbon atom and two hydrogen atoms.
Using the definition of sine, it can be shown that
,
where x is the required bond angle. Now substituting the
half angle identity, and the
result from above, a = y sin(x), the following equation is obtained:

.

Squaring both
sides, one obtains the following result: .

Using the identity ,
it can be shown that 3cos^{2}(x) - 2cos(x) - 1 = 0 from which cos(x) = -.33 or
cos(x) = 1. Finally, the value of x is 109.5°, which is the measure of the
required bond angle.

CONNECTING MATH TO SCIENCE

It is important for students to make connections between mathematics and other disciplines. Knowledge of mathematics means much more than just memorizing information or facts; it requires the ability to use information to reason, think, and solve problems. By themselves, trigonometric identities are just facts, but applying them to a real-world problem will give students a deeper appreciation of those identities and of mathematics. This manipulation of several trigonometric identities allows students to discover for themselves that the optimal bond angle for methane is 109.5°, not 90° as suggested by the 2-dimensional representation. Hopefully, students will begin to value and use the connections between mathematics and other disciplines.

*This article was published in the February 1998 issue of*

**The Science Teacher**. At the time, Dr. Pleacher was a medical student at the Medical College of Virginia, Richmond, Virginia.