12.100P - Chemistry: Molecular Nature Of Matter And Change

Question

One of the most important enzymes in the world— nitrogenase, the plant protein that catalyzes nitrogen fixation— contains active clusters of iron, sulfur, and molybdenum atoms. Crystalline molybdenum (Mo) has a body-centered cubic unit cell (d of Mo is $\mathbf{10}\mathbf{.}\mathbf{28}\mathbf{ }\mathbf{g}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$).

(a) Determine the edge length of the unit cell.

(b) Calculate the atomic radius of Mo.

Step-by-Step Solution

Answer

1. The edge length of the unit cell is 3.55 Ǻ
2. The atomic radius of molybdenum is 1.54 Ǻ

Step 1: Unit cell of Molybdenum

Molybdenum (Mo) has a body-centered cubic unit cell with Mo atoms in the corner of the cubic unit cell and one atom in the body center.

The body-centered cubic unit cell consists of 8 atoms in the corner, thus one atom contributing 1/8 to the unit cell and one atom in the body-center

Thus, each unit cell consists of $8\times \left(\frac{1}{8}\right)+1=2$, i.e., two atoms in the unit cell.

Step 2: Calculation of atomic radius for Mo

As in BCC, in total two atoms contribute for the unit cell. The mass of the unit cell would be the twice as the mass of single atom of Mo.

Now, calculating the volume of the atom and thus the radius of the atom,

$$\begin{gathered} \mathrm{Volume}=\frac{\mathrm{Mass}}{\mathrm{Density}} \\ \frac{4}{3}{\mathrm{\pi r}}^3=\frac{95.95}{\left(10.28\times 6.023\times {10}^{23}\right)} \\ {\mathrm{r}}^3=0.37\times {10}^{-23 }{\mathrm{cm}}^3 \\ \mathrm{r}=1.54\times {10}^{-10} \mathrm{m} \end{gathered}$$

The atomic radius of molybdenum is 1.54 Ǻ

Step 3: Calculation of edge length of Mo

Now, considering the body-centered cubic unit cell

Here, the diagonal of the lattice (edge length is denoted by a) will be equal to twice the diameter of the atom, thus

$$\begin{gathered} \sqrt{3}\mathrm{a}=2\times 2\mathrm{r} \\ \mathrm{a}=\frac{\left(4\times 1.54\times {10}^{-10}\right)}{\sqrt{3} \mathrm{m}} \\ \mathrm{a}=3.55\times {10}^{-10} \mathrm{m} \end{gathered}$$