# Question c698b

Dec 5, 2015

$\text{0.35 nm}$

#### Explanation:

Your strategy here will be to use the definition of the unified atomic mass unit, $\text{u}$, to find the mass of a single nickel atom, then use the characteristics of a face-centered cubic structure to find the total mass of a unit cell.

Once you know the mass of a cell, use nickel's density to determine the cell's volume.

So, the unified atomic mass unit is defined as $\frac{1}{12} \text{th}$ of the mass of a single neutral carbon-12 atom in its ground state and is equal to

$\text{1 u" = 1.660539 * 10^(-24)"g}$

The mass of a single nickel atom expressed in grams will thus be

58.7 color(red)(cancel(color(black)("u"))) * (1.660539 * 10^(-24)"g")/(1color(red)(cancel(color(black)("u")))) = 9.747 * 10^(-23)"g"

Now, a face-centered cubic cell structure is characterized by a total of $14$ lattice points distributed as follows

• one lattice point in every corner of the cell $\to$ a total of eight corner lattice points
• one lattice point in every face of the cell $\to$ a total of six face lattice points

Each corner lattice point will hold $\frac{1}{8} \text{th}$ of an atom and each face lattice point will hold $\frac{1}{2}$ of an atom. The total number of atoms of nickel that can be found in a face-centered cubic cell will thus be

$\text{no. of atoms" = overbrace(1/8 * 8)^(color(blue)("corners")) + overbrace(1/2 * 6)^(color(red)("faces")) = "4 atoms}$

So, if each cell contains a total of $4$ nickel atoms ,and each atom has a mass of $9.474 \cdot {10}^{- 23} \text{g}$, it follows that the total mass of a cell will be

4 color(red)(cancel(color(black)("atoms"))) * (9.747 * 10^(-23)"g")/(1color(red)(cancel(color(black)("atom")))) = 3.899 * 10^(-22)"g"

Use the given density to find the volume of a unit cell

3.899 * 19^(-22) color(red)(cancel(color(black)("g"))) * "1 cm"^3/(8.9 color(red)(cancel(color(black)("g")))) = 4.381 * 10^(-23)"cm"^3

As you know, the volume of a cube is given by the formula

$\textcolor{b l u e}{V = l \times l \times l = {l}^{3}} \text{ }$, where

$l$ - the length of the cube

This means that you have

$l = \sqrt[3]{V}$

In your case, this is equivalent to

l = root(3)(4.381 * 10^(-23)"cm"^3) = 3.525 * 10^(-7)"cm"

The length of the unit cell is usually given in nanometers, so convert the result to get

3.525 * 10^(-8) color(red)(cancel(color(black)("cm"))) * (10^7"nm")/(1color(red)(cancel(color(black)("cm")))) = "0.3525 nm"#

Rounded to two sig figs, the number of sig figs you have for nickel's density, the answer will be

$l = \textcolor{g r e e n}{\text{0.35 nm}}$