Answer
\[1.47\,\,cm\]
Work Step by Step
The molar mass of copper is 63.5 g/mol . Thus, the mass of copper containing\[1.14\,\times \,{{10}^{24}}\,atoms\] is calculated as:
\[\begin{align}
& 1\,\,mol\,\,\,Cu\,\,=\,\,63.5\,\,g \\
& 1\,\,mol\,\,Cu\,\,=\,\,6.022\,\times \,{{10}^{23}}\ \ Cu\,\,atoms \\
& Thus,\,\,\,6.022\,\times \,{{10}^{23}}\ \ Cu\,\,atoms\,\,=\,\,63.5\,\,g \\
& \,\,\Rightarrow 1.14\,\times \,{{10}^{24}}\,\,Cu\,\,atoms\,\,=\,\,63.5\,\,g\,\,\times \,\frac{1.14\,\times \,{{10}^{24}}\,\,Cu\,\,atoms}{\,6.022\,\times \,{{10}^{23}}\ \ Cu\,\,atoms\,\,} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,120.2\,g \\
\end{align}\]
Calculate the volume of copper as:
\[\begin{align}
& \,\,\,\,density\,\,=\,\,\frac{mass}{volume} \\
& \Rightarrow volume\,\,=\,\frac{mass}{density} \\
\end{align}\]
Substitute density of copper \[=8.96\ g/c{{m}^{3}}\] and mass of copper = 120.2 g in the above expression as:
\[volume\,\,=\,\,\frac{120.2\,\,g}{8.96\,\,g/c{{m}^{3}}}\,\,=\,13.41\ c{{m}^{3}}\]
As copper atom is spherical, the volume of a copper atom is given by:
\[\begin{align}
& \left( \frac{4}{3} \right)\prod {{r}^{3}}\,\,\,13.41\,\,c{{m}^{3}} \\
& {{r}^{3}}\,\,\,\,=\,\,\frac{13.41\,\,c{{m}^{3}}\,\,\times \,\,\frac{3}{4}}{\prod } \\
& r\,\,\,\,\,\,\,\,\,\,\,=\,\,\sqrt[3]{\frac{13.41\,\,c{{m}^{3}}\,\,\times \,\,\frac{3}{4}}{\prod }} \\
& r\,\,\,\,\,\,\,\,\,\,=\,\,1.47\,cm \\
\end{align}\]
The radius of a pure copper sphere containing \[1.14\,\times \,{{10}^{24}}\,atoms\]is 1.41 cm.
