Question

In the ammonia $\left({\mathrm{NH}}_3\right)$ molecule, the three hydrogen (H) atoms form an equilateral triangle, the distance between centers of the atoms being $16.28\times {10}^{-11}\mathrm{m}$, so that the center of the triangle is $9.40\times {10}^{-11}\mathrm{m}$ from each hydrogen atom. The nitrogen $(\mathrm{N})$ atom is at the apex of a pyramid, the three hydrogens constituting the base (see Fig. 7-27). The nitrogen/hydrogen distance is $10.14\times {10}^{-11}\mathrm{m}$ and the nitrogen/hydrogen atomic mass ratio is $13.9.$ Locate the center of mass relative to the nitrogen atom.

Short answer

The center of mass, relative to the nitrogen atom, is located $0.56\times {10}^{-11}m$ upward toward the plane of hydrogen atoms.

Step-by-step solution

Identify System Coordinates

The problem has been set so the origin lies at the nitrogen atom, which simplifies matters. The system is symmetric in the plane of the hydrogen atoms, and by symmetry the center of mass must be on the line (z-direction) drawn from the nitrogen atom through the center of the equilateral triangle.

Calculate Ratio of Distances

The center of mass can be described mathematically as follows: $$z_{CM}=\frac{\sum _i{m}_i{z}_i}{\sum _i{m}_i}$$Here $m_i$ and $z_i$ are respective masses and z coordinates. For this problem, $z_{CM}$ will be the desired distance from the nitrogen atom (which is at the origin) to the center of mass. The sum goes over all 4 atoms - one nitrogen and three hydrogens. We assume that the entire mass of each atom is concentrated at it's center, allowing us to ignore size. The z position $z_i$ is negative for nitrogen and positive for each hydrogen. The z coordinate is zero for the center of the base, thus it is $9.4\times {10}^{-11}$ m for each hydrogen atom. By plugging values into the equation, the following is obtained: $$z_{CM}=\frac{(-14.003\times {10}^{-27}\mathrm{kg}(-10.14\times {10}^{-11}\mathrm{m}))+3(1.007\times {10}^{-27}\mathrm{kg}(9.40\times {10}^{-11}\mathrm{m}))}{-14.003\times {10}^{-27}\mathrm{kg}+3(1.007\times {10}^{-27}\mathrm{kg})}$$

Calculate the Center of Mass

After solving the equation from step 2, we find that $z_{CM}$ is equal to $0.56\times {10}^{-11}\mathrm{m}$, which is the distance upward from the nitrogen atom (towards the plane of hydrogen atoms). That is, the center of mass is slightly on the nitrogen side of the geometric center (halfway up). This is expected, because the nitrogen atom is more massive.

Key concepts

Molecular Geometry

Understanding molecular geometry is crucial for various scientific disciplines, including chemistry and molecular biology. It involves the spatial arrangement of atoms in a molecule and influences numerous physical and chemical properties. In the excercise, ammonia (NH₃) is described as having a pyramid shape with a trigonal base, which immediately gives us a clue about its three-dimensional structure.

The geometry of the NH₃ molecule is trigonal pyramidal due to the arrangement of its three hydrogen atoms and one nitrogen atom. The hydrogens form an equilateral triangle at the base, with equal distances between their centers, and the nitrogen atom is at the apex. This shape arises from the sp³ hybridization of the nitrogen atom and the presence of a lone pair of electrons, which gives the molecule a distinct directional character. Understanding this molecular shape is important for predicting the behavior and reactivity of the molecule.

Atomic Mass Ratio

The atomic mass ratio is a fundamental concept in physics and chemistry that helps explain the distribution of mass within a molecule. It's defined as the ratio of the masses of different atoms in a compound. In our example, the atomic mass ratio between nitrogen and hydrogen is given as 13.9, indicating that a nitrogen atom is 13.9 times as massive as a hydrogen atom.

In the computation of center of mass, the atomic mass ratio plays a pivotal role as it's used to weigh the position of each atom. Since the nitrogen atom is significantly more massive compared to hydrogens, its position greatly influences the molecule's center of mass, making it closer to the nitrogen atom than it would be if the atoms were of equal mass. Comprehending the importance of atomic mass ratio is essential for anyone studying molecular dynamics or interpreting the geometric structure of molecules.

Equilateral Triangle Geometry

The equilateral triangle is a geometric figure with three equal sides and angles, which is highly symmetrical. In the context of the exercise, the three hydrogen atoms of the ammonia molecule form an equilateral triangle. This arrangement dictates that each hydrogen atom is the same distance from the center of the triangle, and all angles within the triangle are 60 degrees.

this geometry simplifies calculations related to molecular structure, as symmetry can be used to deduce certain properties. For example, the symmetry in an equilateral triangle allows us to know that the center of mass lies along the axis that runs through one vertex (the nitrogen atom) and the center of the triangle (the midpoint of the hydrogen atoms). Understanding the properties of an equilateral triangle is therefore integral to solving problems that involve symmetrical arrangements of particles or forces.

Coordinate System

A coordinate system is a mathematical construct that allows us to pinpoint the location of a point in space using numbers, known as coordinates. In the provided exercise, the origin of the coordinate system is placed at the nitrogen atom of the ammonia molecule. This choice simplifies the problem as it allows for an easy visualization of the molecule's geometry and determination of the center of mass.

By using a coordinate system, the position of each hydrogen atom is described relative to the nitrogen atom. This assists in analyzing the molecule's mass distribution, as we can assign coordinates to each atom and apply mathematical formulas to find the center of mass. A solid grasp of coordinate systems, therefore, is key to solving physics problems that involve spatial relationships and movement, such as finding the center of mass of molecules or objects.