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University of North Texas

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Problem 57 Hard Difficulty

A molecule of methane, $ CH_4 $, 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 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^\circ $. [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 $ (\frac{1}{2}, \frac{1}{2},\frac{1}{2}) $.]

Answer

$\theta=\arccos \left(-\frac{1}{3}\right) \approx 109.5^{\circ}$

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Video Transcript

so they give us a hint on how to start this, and I'll kind of explain a little bit as to why they gave us that hint. So we're just kind of using it blindly. So if you were to look at methane methane looks like this as a carbon in the center as four hydrogen surrounding it. And, um, there's this thing called Vesper theory, which essentially says that every time we have four atoms surrounding something, it gives us a tetrahedron for the shape. Or I guess we would actually say it's Tetra. He'd roll. So this is where they're getting that idea from, because essentially we have one of these facing away from us, one facing towards us and then another one in the plane so you can see kind of how, like these two hydrogen is on the bottom. Here are in the plane of like, the X Y playing here, and then this hydrogen is facing away from us, and this other one is facing towards us. Since it would be, I guess it would be more technically over here facing out. But that's where they came up with this hint for us. But now We just need to kind of use the hint. So we need to figure out Well, what is our center, where our carbon is. So this is going to be what we just add all these up and divide by four. So it be 1/4 times one plus zero. Actually, I'll just add all the components together. I'll be a little lazy. So it be one plus zero plus one plus zero, and then we add up all the second one. So that would be 00 zero plus zero plus one plus one. And then lastly, um well, the same thing you get to zero plus zero plus one plus one. And now, if we go ahead and actually expand the screen, um, add everything up. That is going to give us 1/4. 222 Which gives this one half one half. What happened? Let me draw those two little bit better. One half. One half. One half. Now, all we need to do is figure out what are the vectors between our center and two of our hydrogen or carbon to hydrogen. So I'll call this over here a hydrogen A and this one hydrogen B So if we want to go from C to A, this is going to be just c R C A minus c. And then if we want to go from sea to be this is going to be B minus c. So this is going to be me down a little. So a is 100 minus and then the one half one half one half over here. B is supposed to be 010 and then see again is one half one half one half. And now we can go ahead and just subtract. So we'd get one half or negative, I should know. Yet that's just what happened in the next two are negative one half over here. It would be negative one half, one half and then negative one half. Now, if we want to find the angle between these two vectors, we can take the dark product because remember the dark product. One way to do this is going to be the magnitude of the first times, the magnitude of the second times, the coast sine of the angle between them. And then we can just go ahead, rearrange doing a little bit of algebra, and we should end up with this, which I think they also just give this team in the book. All right, so we first need to find our magnitudes for each of these, and then we find the dot product, and then we'll just plug it all in over here. The magnitude of this. So it's going to be one half squared, plus negative one half squared, plus negative one half squared, all square rooted. So one half square is 1/4 and then we just add that three times. So that would be route 3/4. And you might notice that this if I square all of them is going to give me the same thing again. So this is just going to be route three over for as well, for the magnitude. Okay. So we can go ahead and plug those into our denominators down here. Would be co sign universe of so route three times or route three or four times through three or four days is going to be the fourth. Now, we need to find the dark product between these actually me script this down. So? So if we find the dot product. The other equation we can use is just where we multiply the components and add them up. So this is going to be one half times negative. One half, one half times negative, one half and then we're going to add this too negative, one half times, one half negative, one half, one half and then plus, um, what color did I not use? Great. So negative. One half times negative, one half so negative. One half times negative, one half. So the negatives are canceled. That would just be one fourth, but this and that would cancel, so we would just be left with negative 1/4. So now we can come up here and plug that into we have negative 1/4 Um, the fours are going to cancel out with each other, so we're going to end up with co sign universe of negative one third. And if we plug that into our calculators, making sure they're in radiant are not radiant degrees. This is going to give something around 109.47 degrees, which, if we were to around to one decimal place, is exactly what they were showing us to ship Yeah, we try that again. And if we rounded to one decimal place, that is exactly what they were telling us to show for the actual angle.