Question

What is the center of mass of the ammonia molecule? This molecule has the chemical formula NH3. It is comprised of one nitrogen atom and three hydrogen atoms. The hydrogens are positioned at the corners of an equilateral triangle (with side length 0.16 nm) which forms the base of a pyramid. The nitrogen atom is at the apex of this pyramid, a vertical distance of 0.037 nm above the plane of the equilateral triangle. The lengths and distances in this question have been given in nanometres or nm, where 1 nm = 1 × 10?9 m.

Answer 1

Answer.1)The center of mass equation we need to use is:

1/M [sum(m_i*r_i)]. This is a summation equation. Basically, we are taking the average of the sum of the particles at their corresponding positions.

m_i = the mass of a molecule
r_i = the position of a molecule
M = Mass of total system

Since there are 4 molecules, i= 4. Thus, the equation becomes:

(1/M)[m₁*r₁ + m₂*r₂ + m₃*r₃ + m₄*r₄]

Where m_i are the masses of the particles m1, m2,m3, etc. and r_i is the distance from the origin to the particle r₁,r₂, r₃, etc. The "Sum" notation means to sum the relation over all particles.

To find the complete center of mass, we generally need to perform the above calculation in all three directions, x, y, and z. The calculation can be greatly simplyfied when there is an axis or plane of symmetry since the center of mass must lie on that axis or plane.

we need to find the center of mass. Because of symmetry, the center of mass lies along the axis from the center of the nitrogen to the center of the hydrogen equilateral triangle. Set the origin at the nitrogen atom. we know that the distance, along the axis, between the nitrogen and the hydrogen triangle is 0.037 nm and the total mass of Hydrogen at that distance is 3 amu (mass of 3 hydrogens). Since the nitrogen is at the origin, its distance is zero. Its mass is 14 amu.

Sum(m_i * y_i) / Sum(m_i) = (14 amu * 0 nm + 3 amu * 0.037 nm) / (14 amu + 3 amu) = 0.00653 amu