Question

In the ammonia ${\mathbf{NH}}_{\mathbf{3}}$molecule of Figure 9-40, three hydrogen (H) atoms form an equilateral triangle, with the center of the triangle at distancelocalid="1654497980120" $\mathbf{d}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{40}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\mathit{m}$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 ilocalid="1654497984335" $\mathbf{L}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{14}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}\mathbf{}}\mathbf{m}$.(a) What are the xcoordinates of the molecule’s center of mass and(b) What is the ycoordinates of the molecule’s center of mass?

Short answer

a. x coordinate of center of mass of molecule is 0.

b. coordinate of center of mass of molecule is $3.13\times {10}^{-11}\mathrm{m}$.

Step-by-step solution

Listing the given quantities

Distance of center from each hydrogen atom, $\mathrm{d}=9.40\times {10}^{-11}\mathrm{m}$

Distance of hydrogen atom from nitrogen atom,${\text{L=10.14×10}}^{-11}\mathrm{m}$

Understanding the concept of center of mass

For a system of particles, the whole mass of the system is concentrated at the center of mass of the system.

By the symmetry ofthemolecule, it is clear that the x coordinate of the center of mass will be ontheplane containing hydrogen molecules.Using the given structure of H₃ molecule, we can find the distance between nitrogen molecules totheplane containing all hydrogen atoms. From that distance, we can find the y coordinate of center of mass of nitrogen molecule.

The expression for the coordinates of the center of mass are given as:

${\overrightarrow{r}}_{com}=\frac{1}{M}\sum _{i=1}^n{m}_i{r}_i$ … (i)

Here, $M$is the total mass, $m_i$ is the individual mass of ith particle and $r_i$ is the coordinates of ithparticle.

(a) Determination of the x coordinate of center of mass of particle

From the symmetry of the molecule, we can directly conclude that the center of mass will be on the plane containing the hydrogen atoms. Therefore, the x coordinate of the center of mass will be at the origin.

$x_{com}=0$

Thus, the x coordinate of the center of mass is 0.

(b) Determination of the y coordinate of center of mass of particle

First we found the distance from nitrogen to plane containing hydrogen atoms.

We can use Pythagoras theorem here.

${\mathrm{y}}^2\mathrm{n}+{\mathrm{d}}_2={\left(\mathrm{L}\right)}^2$

Substitute the values in the above equation.

$$\begin{gathered} {\mathrm{y}}_{\mathrm{N}}=\sqrt{{(10.14\times {10}^{-11}\mathrm{m})}^2-{(9.40\times 10-11\mathrm{m})}^2} \\ =3.803\times {10}^{-11}\mathrm{m} \end{gathered}$$

Now, the y coordinate of the center of mass can be calculated as:

$$\begin{gathered} {\mathrm{y}}_{\mathrm{com}}=\frac{{\mathrm{m}}_{\mathrm{N}}{\mathrm{y}}_{\mathrm{N}}}{{\mathrm{m}}_{\mathrm{H}}+3{\mathrm{m}}_{\mathrm{H}}} \\ =\frac{(14.0067\mathrm{u})(3.803\times {10}^{-11})}{(14.0067\mathrm{u})+3(1.0067\mathrm{u})} \\ =3.13\times {10}^{-11}\mathrm{m} \end{gathered}$$

Thus, the $y$coordinate of the center of mass is $3.13\times {10}^{-11}m$.