Calculating Center of Mass in NH3 Molecule

In summary, the x coordinate of the molecule's center of mass is 9.40 ✕ 10^−11 m and the y coordinate of the molecule's center of mass is 10.14 ✕ 10^−11 m.
  • #1
JessicaHelena
188
3

Homework Statement


In the ammonia (NH3) molecule of the figure, the three hydrogen (H) atoms form an equilateral triangle, with the center of the triangle at distance d = 9.40 ✕ 10^−11 m from each hydrogen atom. The nitrogen (N) atom is at the apex of a pyramid, with the three hydrogen atoms forming the base. The nitrogen-to-hydrogen atomic mass ratio is 13.9, and the nitrogen-to-hydrogen distance is L = 10.14 ✕ 10^−11 m.

(a) What is the x coordinate of the molecule's center of mass?

(b) What is the y coordinate of the molecule's center of mass?

Homework Equations



x_cm = sum of (m*x)/M
(same for y)

The Attempt at a Solution


(a) I'm not quite sure how to deal with the two hydrogens towards the negative x-axis — they're not exactly on the axis at all.

(b) None of the H's have any y values, so I did 13.9*sqrt(L^2-d^2)/(13.9+3*1) where 3*1 accounted for the masses of the three H's, but apparently that answer is wrong, and I've no idea why it is. (I got 1.137*10^-(10).)
 

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  • #2
You need to work out the geometry better for part (a). This is really just a geometry problem, with an equilateral triangle, and the origin (x=0) is at the centroid of that equilateral triangle. This can be readily shown to be the x-center of mass, and we can show that once we have the coordinates. ## \\ ## (Edit: Go to the bottom, and read that part first. Looking over what you did, you have already done most of what the problem asks for). ## \\ ## (They really should call their y-direction z, and the plane of the 3 hydrogen atoms should be the x-y plane, but I think they were trying to make it overly simple). I would recommend you make this change, and compute the location of the coordinates of the vertices of an equilateral triangle in the x-y plane, when the centroid is at ##(x,y) ## of ## (d,0 ) ##, and compute ## (x,y) ## for the other two vertices. Then we can show with a little arithmetic that the center of mass in the x-y plane is at (0,0). Otherwise, we can simply assume that the x-center of mass is ## x=0 ## as defined in this problem, but that makes it almost too easy.## \\ ## ## \\ ## The location of the center of mass in the ## y ## direction is very readily computed. They give you the hypotenuse distance, but you need to compute the distance ## d ## from the centroid of the equilateral triangle to the hydrogen atom(s). (Edit: They give you that distance ## d ##. That makes it very easy). Once you have that, it is a simple matter using the Pythagorean theorem to compute the y-height of the nitrogen atom. Computing the y-center of mass then follows with a very simple equation. (Edit: And I see you did that. Very good ).## \\ ## Edit: Looking over your work for part (b), it looks correct. Check your arithmetic=I get a very different answer.
 
Last edited:
  • #3
@Charles Link
for the x_cm, I too thought that it would be 0, but the answer is 9.40 ✕ 10^−11 m... Do you have any idea why this could be?
 
  • #4
JessicaHelena said:
@Charles Link
for the x_cm, I too thought that it would be 0, but the answer is 9.40 ✕ 10^−11 m... Do you have any idea why this could be?
That is incorrect. The way they drew their coordinate system, they have the (x-coordinate of the) center of mass at ## x=0 ##. (The center of mass (x-coordinate) for the 3 hydrogen atoms is at ## x=0 ## with the nitrogen atom directly above it ). They did not put the origin of x at the position of the hydrogen atom that is located on the x-axis. The textbook you have needs to be more accurate. ## \\ ## In any case, please try your arithmetic again for part (b). You should be able to get the same answer that I did.
 
  • #5
Oh yes, I did — (b)'s alright now. Thank you.
 
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Likes Charles Link

Related to Calculating Center of Mass in NH3 Molecule

1. What is the center of mass in NH3?

The center of mass in NH3, or ammonia, refers to the point around which the mass of the molecule is evenly distributed. It is also known as the center of gravity.

2. How is the center of mass calculated in NH3?

The center of mass in NH3 can be calculated by taking the sum of the products of each particle's mass and its distance from a chosen reference point, divided by the total mass of the molecule.

3. Why is the center of mass important in NH3?

The center of mass is important in NH3 because it helps us understand how the molecule will behave in certain physical situations, such as when it is in motion or under the influence of external forces.

4. How does the center of mass change in NH3 when a proton is added?

When a proton is added to NH3, the center of mass will shift slightly towards the added proton due to its increased mass. However, the exact location of the center of mass will depend on the position of the added proton in relation to the other particles in the molecule.

5. Can the center of mass in NH3 be outside of the molecule?

No, the center of mass in NH3 will always be located within the molecule itself. This is because the center of mass is a point that represents the average position of all the particles in the molecule, and cannot be outside of its boundaries.

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