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Vectors

Vectors

To understand how the dipole moments of the bonds in a molecule contribute to the net dipole moment of the molecule, it is very helpful to represent the individual bond dipoles as vectors. The net dipole moment of the molecule is then the vector sum of all the individual bond dipoles (plus any contribution from unshared electron pairs).

For our purposes, we define a vector as a mathematical object in three dimensional space that is characterized by two quantities: (1) magnitude (or length), and (2) direction, which can be measured as an angle with respect to a line of reference. The positive x axis is used as the reference line, so that the direction of a vector is the angle it makes with the positive x axis.

We represent a vector as an arrow. The end of the arrow is called its head, while the start of the arrow is called the vector's tail. An important point about vectors is that we can place them anywhere in three dimensional space - i.e., we can pick them up and move them around, but as long as the vector's length and direction are not changed, it is considered the same vector. It is convenient in many cases to place the tail of a vector at the origin of a system of coordinates.

There are two ways we can perform vector addition and subtraction, geometrically or algebraically. These are both illustrated in the figure for the special case of two dimensions. The geometric method for adding two vectors A and B is to place the tail of B at the head of A, and then draw the new vector from the tail of A to the head of B. This is the vector A + B. For subtraction, place the tail of A on the tail of A, and draw the new vector from the head of B to the head of A. This is the vector A – B. Note that B + (A – B) = A. The algebraic method for adding vectors makes use of the coordinates of the heads of the vectors when their tails are at the origin of the coordinate system. If the coordinates of the head of A are (xA, yA), and those of the head of B are (xB, yB),then the coordinates of the head of the vector A + B will be (xA+ xB, yA+ yB). Similarly, the coordinates of the head of the vector A – B will be (xA– xB, yA– yB)
We can use vectors and vector sums to understand why molecules such as CO2 and CCl4 have no net dipole moments. Hence these molecules are considered nonpolar. The reason why CCl4 is nonpolar even though the individual C–Cl bonds are polar is that the vectors corresponding to the individual bond dipoles sum to zero (i.e. a vector with zero magnitude). This is illustrated in the figure below. Here we must use a three dimensional coordinate system since molecules are three dimensional objects. The molecule has a tetrahedral geometry, which means the bonds are directed to the corners of a tetrahedron, and all bond angles are 109.5°. We place the carbon atom of CCl4 at the origin of the coordinate system with one C–Cl bond aligned along the positive z axis. The vectors corresponding to the bond dipole moments (A, B, C, and D) conveniently all have their tails at the origin. The coordinates of the heads of each bond dipole vector will have three components, (x,y,z), and their approximate values are shown in the figure. The length of each of these vectors is 2.5, reflecting the electronegativity difference between C and Cl and the length of the C–Cl bond, although the exact length does not matter in this case as long as all the lengths are equal.
 

The vector A is oriented along the z axis,and the vector B lies in the xz plane, making an angle of 109.5° with A. The relative positions of the vectors C and D are difficult to illustrate in 2D, but C is pointing below the xy plane and out of the plane of the figure (if the plane of the figure is the xz plane), while D is also below the xy plane,and pointing behind the plane of the figure. There is no ambiguity in the coordinate values of the heads of each of the vectors, however, and these can be used to calculate vector sums. The vectors A and B can be summed as shown. The vectors C and D can also be summed, and the net dipole moment is given by the sum (A + B) + (C + D). But the two vectors A + B and C + D are equal in length and opposite in direction. Their sum is therefore zero, which is also demonstrated by the algebraic sum.

Exercises:

(i) Using the formula for the length of a vector A in three dimensions, the tail of which is located at the origin (0, 0, 0), show that the length of the vector C ( ||C|| ) is 2.5. The formula, which is just the Pythagorean theorem in three dimensions, is shown below:
Use the more accurate coordinates
(–0.70698, –1.22496, –0.50071) for the head of C.

(ii) Apply a similar method to show that the net dipole of the carbon dioxide (CO2) molecule is zero.

(iii) Astatine (Element 85) is the heaviest element in Group 7A (the halogens). Hence, it would be expected to be similar in chemical properties to fluorine or chlorine. However, electronegativity generally decreases within a group as atomic mass increases, and astatine (EN = 2.0) is much less electronegative than chlorine (EN = 3.0). Consider the molecule CCl2At2 Using vectors and reasoning based on symmetry, decide whether or not this molecule has a net dipole moment.
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