Question

A molecule of methane, CH4, is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the H-C-H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about 109.5° Hint: Take the vertices of the tetrahedron to be the points (1, 0, 0), (0, 1, 0), (0, 0, 1), and (1, 1, 1), as shown in the figure. Then the centroid is : Consider the H-C-H combination consisting of the sole carbon atom and the two hydrogen atoms that are at H1 = (1, 0, 0) and H2 = (0, 1, 0). Then the vectors CH1 and CH2 are given by CH = CH2 = and thus CH, CH2 = ICHT = CH21 = The bond angle, 0 = 2 CH1 CH2, therefore satisfies cos 0 = so that, expressed in degrees and rounded to one decimal place, O

Answer 1

solution CHy = $112,0-72,0-42) = ( 721-72-72) 1/C Hell = 5€)2+(-+)2+(3+)? Eve CH2 = 10–12:1-1210-42) = (-121 YZ-V25 (16112 11 = VE12 122632 ++4)2 = 9 (H1 CH2 = (2) (-1/2) + (-1/2) (12) + (-1/2) (-4/2) =/ CH-CH2 = yle Hill || CH21168€ = 314 Gose 3 logo =-14 Loxe = -43 B= (x-1(+13) = 109.80