Find the center of mass of the ammonia molecule. The chemical ...

Question

Find the center of mass of the ammonia molecule. The chemical formula is NH₃. The hydrogens are at the corners of an equilateral triangle (with sides 0.16 nm) that forms the base of a pyramid, with nitrogen at the apex (0.037 nm vertically above the plane of the triangle).

Accepted Answer

Hey, everyone in this problem, we're asked to calculate the center of mass of a phosphine molecule relative to the center of its base. The molecule consists of a phosphorus atom at the apex of a pyramid and three hydrogen atoms at the corners of an equilateral triangle that forms the base. The side of the triangle is 0.14 nanometers. And the vertical distance from the phosphorus atom to the plane of the hydrogen atoms is 0.1 nanometers. We have four answer choices here all in nanometers. Option A 0.01 option B 0.05 option C 0.091 and option D 0.1. So let's start by kind of drawing out the information we've been given and we're gonna start with the base of this pyramid. OK. So we have three hydrogen atoms that form the base and they form an equilateral triangle. OK. So the side, all of these are the same and we know that that length is 0.14 nanometers and this is gonna be the base of our pyramid. OK. And again, these are all hydrogen atoms at each of the corners. All right. So we're looking for the center of mass relative to the center of the base. And so we can think about finding this center point and that's where we're looking for the center of mass measured from. Right. Now. We think about looking at the pyramid from the side. OK. We have our phosphorus atom that's gonna be at the top. And we know that the vertical distance from the phosphorus atom to the plane where we have the hydrogen atoms is 0.1 nanometers. And so we have this kind of side view of our period. All right. So we want to think about how we can calculate the center of mass. Now because the base is an equilateral triangle, we have some symmetry that is gonna make this a lot easier. So because of this symmetry that we have, we know that the center of mass must lie along the z axis passing through that phosphorus atom. And also the center of the base of the triangle, it must lie along this line that I'm tracing in blue, right? Because again, the center of mass of the three hydrogen atoms is gonna be right at the center of the base because of that symmetry, right. So then all we're doing when we add that phosphorus atom is moving that center of mass upwards along that same line. OK. So it's gonna be along this blue line that we've drawn. And so when we think about calculating the center of mass. All we have to consider now is the Z values and are those height of these atoms. So the center of mass we call it ZCM. It is gonna be equal to what? Well remember we have this weighted average. So we have the mass multiplied by the position for each of the objects that we have. In this case, we're dealing with atoms. So we have four atoms because the hydrogen atoms are all the same because we have symmetry in the distance between them. And the other things that we're measuring is the same, we can consider them all together. So we're gonna have three multiplied by the mass of a hydrogen atom multiplied by the position zh plus the mass of the phosphorus atom multiplied by its position said P all divided by the sum of the masses. OK. So we have three MH plus MP and make sure you don't forget that three in the denominator when we're dealing with hydrogen, that can be easy to do we want the sum of all of the masses. So we have to consider all three hydrogens. Now, we know these values the mass of hydrogen, the mass of phosphorus, those are values that we can look up in a table. In our textbook, we know those values the distance, the vertical distance from the center of the base of the triangle where we wanna measure our center of mass from two the hydrogen atoms we know is just zero. Hey, they're all in that same plane, the vertical distance between them is zero. So Zh is actually gonna be zero. OK. The vertical distance between the plane where the hydrogen atoms are OK. That base of our triangle to where the phosphorus atom is, we know that value as well. We've been told that and the problem is 0.1 nanometers. And so we have everything we need, we just need to substitute and, and solve at this point. So our center of mass is gonna be equal to three multiplied by 1.01 U multiplied by zero plus 30.97 U multiplied by 0.1 nanometers. OK? And this is all divided by three multiplied by 1.01 U plus 30.97. He now our units, in this case, we did not convert OK, we typically convert mass into kilograms. But here we have atomic mass unit U in the numerator and in the denominator, those are gonna divide out. So it doesn't matter what that unit is as long as we're being consistent with it. OK. And then we've left our distance in nanometers because the answer choices are in nanometers. So we don't need to convert to meters and then convert back. We're just gonna leave it as it is. All right. So let's simplify this a little bit. So that first term of the numerator goes to zero. So we're just left with 30.97 U multiplied by 0.1 nanometers divided by 30. Or you, if we simplify the denominator, and again, those us will divide out and we're gonna be left with approximately 0.091 nanometers, right. And so this is the distance from the center of the base of our pyramid to the center of mass. That is the distance that we were looking for in this problem. If we go back to our answer choices and compare what we found, we can see that the correct answer is going to be. Option. C Thanks everyone for watching. I hope this video see you in the next one.

Find the center of mass of the ammonia molecule. The chemical formula is NH₃. The hydrogens are at the corners of an equilateral triangle (with sides 0.16 nm) that forms the base of a pyramid, with nitrogen at the apex (0.037 nm vertically above the plane of the triangle).

Key Concepts

Center of Mass

Molecular Geometry

Equilateral Triangle

Watch next