Why is the tetrahedral angle 109.5?

Answer 1

Actually, carbon's bonding angle varies depending on the compound it is in. In methane (CH4), for example, the molecule takes a tetrahedral shape. However, in carbon dioxide (CO2), the molecule takes a linear shape. This is due to VSEPR (Valence Shell Electron Pair Repulsion) theory, which states that lone pair electrons and bonds will spread out as evenly as possible in 3 dimensions The molecular geometry (i.e., shape) of a molecule is determined by the total number of electron pairs and bonds (double and triple bonds count as one bond). The carbon in methane has four bonds and no lone pairs, making its molecular geometry tetrahedral. The angles of a tetrahedron measure 109.6o, so carbon's bonding angle in methane is 109.6o.

Answer 2

That is just how it works. A triangle contains 180 degrees, in an equilateral triangle each angle is 60 degrees. For a tetrahedral it just works out to 109.5.

The following explanation should be intelligible if you draw all the figures.

The easiest way to compute this is to start with a cube with length of side = 1. By drawing lines from the centre of the cube to every other corner, we find we have four lines with equal angles between them; thus, these angles are the tetragonal angles we are interested in.

Consider one face of a cube, and draw a line dividing it in half from corner to corner. This forms two right-angled triangles. The hypotenuse of these, by Pythagoras' theorem, is sqrt(2).

Now construct a rectangle by using two edges of the cube, and two face diagonals. The centre of this rectangle is the centre of the cube, and the sides of the rectangle are 1, 1, sqrt(2), sqrt(2). Draw the diagonals. Compare this shape with the cube. The larger angle formed at the centre between the diagonals is the tetrahedral angle, which we will label A.

Now draw one more line, dividing this tetragonal angle into two. This divides two of the triangles in our figure into pairs of right-angled triangles, each having as one angle A/2. The side opposite to A/2 is now sqrt(2)/2, and the adjacent side is 1/2. Thus tan(A/2) = sqrt(2), meaning that A=2*arctan(sqrt(2)), which is approximately 109.47 degrees.