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on the Left we have the dot structure for methane and we've seen in an earlier video that this carbon is sp3-hybridized which means that the atoms around that central carbon atom are arranged in a tetrahedral geometry it's very difficult to see tetrahedral geometry on a 2-dimensional Lewis dot structure so it's much easier to see it over here on the right with the three-dimensional representation of the methane molecule so if I were trying to see the four sides of the tetrahedron I can find my first side by connecting these hydrogen atoms like that so there's the first side of my tetrahedron and if I'm going to find the second side I could connect these hydrogen atoms like that and there's my second side and to find my last two sides if I connect this hydrogen atom to this one down here I can now see the four sides of my tetrahedron we're also concerned with the bond angle so what is the what is the bond angle what is the angle between that top hydrogen the central carbon and this hydrogen over here on the Left it turns out that bond angle is 109 point five degrees and it's the same all the way around right so you could say that this angle is 109.5 degrees or this angle back here it's all the same and so an sp3 bond angle is one hundred nine point five and the proof for this was shown to me by two of my students so Anthony greeby and Andrew foster came up with a very nice proof to show that the bond angle of an sp3 hybridized carbon is a hundred and nine point five degrees and what they did was they said let's let's go ahead and take that tetrahedron and let's go ahead and put on XYZ axes and let's put carbon at the center here and we can choose any four points to represent the the four hydrogen atoms of our tetrahedron if we satisfy two conditions each point that we choose for our hydrogen's is equidistant from the other three points and also each point that we choose for our hydrogen's is equidistant from the central carbon atom itself and if you fulfill those two criteria or you guarantee that the points that you choose form a tetrahedron and so here we have the tetrahedron on our axes and let's go ahead and look at the first point so this point right here and they chose this point to to be at square root of 2 1 & 0 meaning positive square root of 2 on the x-axis positive 1 on the y axis and zero on the z axis and then this point over here on the left they were very clever and said this point is going to be in the same plane so this point on the left is in the same plane as the point we just talked about the XY plane and therefore the coordinates for that point would be negative square root of 2 1 & 0 we go to the hydrogen down here so this point of our tetrahedron is located at 0 negative 1 and square root of 2 and then finally this point going away from us right here would be at 0 negative 1 and negative square root of 2 so once again you could choose any points that you want as long as you meet that criteria and orienting the molecule in this way allows us to find this bond angle all right so this is the bond angle that we are going for and we don't know that bond angle yet but we can figure out this angle right here so I'm going to call this theta for this triangle that's formed and I know that this x distance down here is positive square root of 2 and and then we go up 1 on the y axis and then 0 on the z axis so I can find out what theta is because I know that tan of theta is equal to opposite over adjacent so for this triangle I have here the opposite side would be 1 and the adjacent side would be square root of 2 so to find theta all I have to do is take inverse tan so I take inverse tan of 1 over square root of 2 on my calculator and I get 35 point 2 6 degrees so I know that theta this angle right in here is 35 point 2 6 degrees and therefore this angle is also 30 five point two six degrees so this is also going to be theta in here and if I want to find my bond angle and here I know that those three angles have to add up to equal 180 degrees since they're all in the same plane here so to find my bond angle all I have to do is take 180 degrees and from that we're going to subtract two times thirty five point two six degrees and we of course come out with a bond angle of 109 point five degrees so again special thanks to my two students for showing me this proof