Chapter 1, Problem 1.97QE

Gold leaf, which is used for many decorative purposes, is made by hammering pure gold into very thin sheets. Assuming that a sheet of gold leaf is 1.27 × 10⁻⁵ cm thick, how many square feet of gold leaf could be obtained from 28.35 g gold? The density of gold is 19.3 g/cm³.

Interpretation Introduction

Interpretation:

A sheet of gold leaf is $\text{1}\text{.27}\text{×}{\text{10}}^{\text{-5}}\text{cm}$ thick.  The area of gold leaf obtained from $\text{28}\text{.35}\text{g}$ gold in square feet has to be given.  The density of gold is $\text{19}\text{.3}{\text{g/cm}}^{\text{3}}$.

Explanation of Solution

Given, thickness of gold leaf is $\text{1}\text{.27}\text{×}{\text{10}}^{\text{-5}}\text{cm}$, mass of the gold leaf is $\text{28}\text{.35}\text{g}$ and density of gold is $\text{19}\text{.3}{\text{g/cm}}^{\text{3}}$.

From mass and density, the volume of the gold leaf is calculated.

The conversion factor from the density of gold can be derived as

  $\begin{array}{l}\text{1}{\text{cm}}^{\text{3}}\text{gold}\text{=}\text{19}\text{.3}\text{g}\text{gold}\\ \\ \text{conversion}\text{factor}\text{=}\left(\frac{\text{1}{\text{cm}}^{\text{3}}\text{gold}}{\text{19}\text{.3}\text{g}\text{gold}}\right)\\ \\ \text{Applying}\text{this}\text{conversion}\text{factor,}\text{volume}\text{can}\text{be}\text{obtained}\text{as}\\ \text{Volume}\text{=}\frac{\text{Mass}}{\text{Density}}\\ \text{Volume of}\text{gold}\text{=}\text{28}\text{.35}\text{g}\text{×}\left(\frac{\text{1}{\text{cm}}^{\text{3}}\text{gold}}{\text{19}\text{.3}\text{g}\text{gold}}\right)\\ \text{=}\text{1}\text{.47}{\text{cm}}^{\text{3}}\end{array}$

The area can be calculated as

  $\begin{array}{l}\text{volume}\text{=}\text{thichness}\text{×}\text{area}\\ \text{area}\text{=}\frac{\text{volume}}{\text{thickness}}\\ \\ \text{Area}\text{=}\frac{\text{1}\text{.47}{\text{cm}}^{\text{3}}}{\text{1}\text{.27}\text{×}{\text{10}}^{\text{-5}}\text{cm}}\\ \text{=}\text{115748}\text{.03}{\text{cm}}^{\text{2}}\end{array}$

The units of the area are converted into square feet by the conversion factor.

  $\begin{array}{l}\text{1}\text{in}\text{=}\text{2}\text{.54}\text{cm}\\ \text{1}\text{ft}\text{=}\text{12}\text{in}\\ \\ \text{conversion}\text{factor}\text{=}\left(\frac{\text{1}\text{in}}{\text{2}\text{.54}\text{cm}}\right)\text{×}\left(\frac{\text{1}\text{ft}}{\text{12}\text{in}}\right)\\ \\ \text{Area}\text{=}\text{115748}\text{.03}{\text{cm}}^{\text{2}}\text{×}{\left(\frac{\text{1}\text{in}}{\text{2}\text{.54}\text{cm}}\right)}^{\text{2}}\text{×}{\left(\frac{\text{1}\text{ft}}{\text{12}\text{in}}\right)}^{\text{2}}\\ \text{=}\text{115748}\text{.03}{\text{cm}}^{\text{2}}\text{×}\left(\frac{\text{1}{\text{in}}^{\text{2}}}{\text{6}\text{.45}{\text{cm}}^{\text{2}}}\right)\text{×}\left(\frac{\text{1}{\text{ft}}^{\text{2}}}{\text{144}{\text{in}}^{\text{2}}}\right)\\ \text{=}\text{124}{\text{ft}}^{\text{2}}\end{array}$