A typical sugar cube has an edge length of 1 cm. If you had a cubical box that contained a mole of sugar cubes, what would its edge length be? (${}^{\mathbf{One}\mathbf{}\mathbf{mole}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}\mathbf{}\mathbf{units}}}$.)

The edge length of the sugar cube is${}^{\mathbf{1}\mathbf{}\mathbf{cm}}$.

$\mathbf{1}\mathbf{}\mathbf{mole}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}\mathbf{}}\mathbf{}\mathbf{units}$

In this problem, the volume of one mole of sugar cubes will be equal to the cubical box that contains one mole of sugar cubes.

The volume of a cube is given as:

$\mathrm{V}={\mathrm{a}}^3$ … (i)

Here, ${}^a$ is the edge length of cube.

Using equation (i), the volume of one sugar cube is calculated as:

$$\begin{gathered} {\mathbf{V}}_{\mathbf{sc}}\mathbf{}\mathbf{=}{\left(1.0\mathrm{cm}\right)}^{\mathbf{3}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\left(1.0\mathrm{cm}\times \frac{{10}^{-2}m}{1\mathrm{cm}}\right)}^{\mathbf{3}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}{\mathbf{m}}^{\mathbf{3}} \end{gathered}$$

As we know,

$\mathbf{1}\mathbf{}\mathbf{mole}\mathbf{}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}\mathbf{}}\mathbf{units}$.

The volume of the cubical box is equal to the volume of one mole of sugar cubes. Therefore, volume of cubical box is,

$$\begin{gathered} \mathbf{V}\mathbf{}\mathbf{=}\mathbf{}\left(1.0\times {10}^{-6}{m}^3\right)\mathbf{\times }\left(6.02\times {10}^{23}\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{17}\mathbf{}}{\mathbf{m}}^{\mathbf{3}} \end{gathered}$$

Thus, the volume of cubical box is${}^{\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{17}\mathbf{}}{\mathbf{m}}^{\mathbf{3}}}$ .

Using equation (i), the edge length of cubical box is calculated as:

$$\begin{gathered} \mathbf{a}\mathbf{\text{'}}\mathbf{}\mathbf{=}\mathbf{}\sqrt[\mathbf{3}]{\mathbf{V}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{}\mathbf{\times }{\mathbf{10}}^{\mathbf{17}\mathbf{}}{\mathbf{m}}^{\mathbf{3}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}\mathbf{.}\mathbf{4}\mathbf{}\mathbf{\times }\mathbf{}{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{m} \end{gathered}$$

Thus, the edge length of cubical box is${}^{\mathbf{}\mathbf{8}\mathbf{.}\mathbf{4}\mathbf{}\mathbf{\times }\mathbf{}{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{m}}$ .