Problem: Gold is alloyed (mixed) with other metals to increase its hardness in making jewelry.(a) Consider a piece of gold jewelry that weighs 9.85 g and has a volume of 0.675 cm3. The jewelry contains only gold and silver, which have densities of 19.3 g/cm3 and 10.5 g/cm3, respectively. If the total volume of the jewelry is the sum of the volumes of the gold and silver that it contains, calculate the percentage of gold (by mass) in the jewelry. (b) The relative amount of gold in an alloy is commonly expressed in units of karats. Pure gold is 24-karat, and the percentage of gold in an alloy is given as a percentage of this value. For example, an alloy that is 50% gold is 12-karat. State the purity of the gold jewelry in karat.

🤓 Based on our data, we think this question is relevant for Professor Ritz's class at UCI.

FREE Expert Solution

Part A

Step 1

$$\begin{gathered} \mathbf{Mass}\mathbf{ }\mathbf{jewelry}\mathbf{ }\mathbf{=}\mathbf{ }{(\mathrm{Density}\times \mathrm{Volume})}_{\mathbf{gold}}\mathbf{ }\mathbf{+}\mathbf{ }{(\mathrm{Density}\times \mathrm{Volume})}_{\mathbf{silver}} \\ \mathbf{9}\mathbf{.}\mathbf{85}\mathbf{ }\mathbf{g}\mathbf{ }\mathbf{=}\mathbf{ }(19.3 \frac{g}{{\mathrm{cm}}^3})\left(V_{\mathrm{gold}}\right)\mathbf{ }\mathbf{+}\mathbf{\hspace{0.17em}}(10.5 \frac{g}{{\mathrm{cm}}^3})(0.675 {\mathrm{cm}}^3 - V_{\mathrm{gold}}) \\ \mathbf{9}\mathbf{.}\mathbf{85}\mathbf{ }\mathbf{g}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{19}\mathbf{.}\mathbf{3}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{\hspace{0.17em}}\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{5}\mathbf{ }\frac{\mathbf{g}}{\cancel{{\mathbf{cm}}^{\mathbf{3}}}}\mathbf{)}(0.675 \cancel{{\mathrm{cm}}^3})\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{5}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)} \\ \mathbf{9}\mathbf{.}\mathbf{85}\mathbf{ }\mathbf{g}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{19}\mathbf{.}\mathbf{3}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{5}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)}\mathbf{ }\mathbf{+}\mathbf{\hspace{0.17em}}\mathbf{7}\mathbf{.}\mathbf{0875}\mathbf{ }\mathbf{g} \\ \mathbf{9}\mathbf{.}\mathbf{85}\mathbf{ }\mathbf{g}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{7}\mathbf{.}\mathbf{0875}\mathbf{ }\mathbf{g}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{19}\mathbf{.}\mathbf{3}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{5}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)} \\ \mathbf{2}\mathbf{.}\mathbf{7625}\mathbf{ }\mathbf{g}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{8}\mathbf{ }\frac{\mathbf{g}}{{\mathbf{cm}}^{\mathbf{3}}}\mathbf{)}\mathbf{\left(}{\mathbf{V}}_{\mathbf{gold}}\mathbf{\right)} \\ {\mathbf{V}}_{\mathbf{gold}}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{2}\mathbf{.}\mathbf{7625}\mathbf{ }\cancel{\mathbf{g}}}{\mathbf{8}\mathbf{.}\mathbf{8}\mathbf{ }\frac{\cancel{\mathbf{g}}}{{\mathbf{cm}}^{\mathbf{3}}}} \end{gathered}$$

V_gold = 0.314 cm³

Step 2

$$\begin{gathered} \mathbf{mass}\mathbf{ }\mathbf{gold}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{density}\mathbf{ }\mathbf{\times }\mathbf{ }\mathbf{volume} \\ \mathbf{mass}\mathbf{ }\mathbf{gold}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{19}\mathbf{.}\mathbf{3}\mathbf{ }\frac{\mathbf{g}}{\cancel{{\mathbf{cm}}^{\mathbf{3}}}}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{314}\mathbf{ }\cancel{{\mathbf{cm}}^{\mathbf{3}}} \end{gathered}$$

mass gold = 6.059 g