Answer:
%cu = 13.95%
%au=86.05%
Explanation:
Since the total volume is the sum of both volumes (gold, copper) we can say:
[tex]V_{T} = V_{AU} + V_{CU}[/tex]
We know that [tex]V_{AU} = \frac{m_{AU}}{\rho_{AU}}[/tex] and [tex]V_{CU} = \frac{m_{CU}}{\rho_{CU}}[/tex]. Replacing these values into our volume equation:
[tex]V_{T} = \frac{m_{AU}}{\rho_{AU}} + \frac{m_{CU}}{\rho_{CU}}[/tex]
We also know that: [tex]m_{T}=m_{AU}+m_{CU}[/tex]
Solving this equation system:
[tex]\left \{ {{V_{T} = \frac{m_{AU}}{\rho_{AU}} + \frac{m_{CU}}{\rho_{CU}}} \atop {m_{T}=m_{AU}+m_{CU}}} \right.[/tex]
[tex]m_{CU}=1.01g = 13.95%[/tex]
[tex]m_{AU}=6.23g = 85.94%[/tex]
Percentage by mass of Gold is about 86%
Percentage by mass of Copper is about 14%
[tex]\texttt{ }[/tex]
Let's recall Density formula as follows:
[tex]\boxed {\rho = m \div V}[/tex]
where:
ρ = density of object ( kg/m³ )
m = mass of object ( kg )
V = volume of object ( m³ )
Let us now tackle the problem!
[tex]\texttt{ }[/tex]
Given:
total mass of jewelry = m = 7.24 g
total volume of jewelry = V = 0.436 cm³
density of gold = ρ_g = 19.3 g/cm³
density of copper = ρ_c = 8.96 g/cm³
Asked:
percentage by mass of gold = m_g = ?
percentage by mass of copper = m_c = ?
Solution:
[tex]\texttt{Total Volume = Volume of Gold + Volume of Copper}[/tex]
[tex]V = V_g + V_c[/tex]
[tex]\frac{m}{ \rho} = \frac{m_g}{ \rho_g} + \frac{m_c}{\rho_c}[/tex]
[tex]\frac{m_g + m_c}{ \rho} = \frac{m_g}{ \rho_g} + \frac{m_c}{\rho_c}[/tex]
[tex]\frac{m_g}{\rho} + \frac{m_c}{\rho} = \frac{m_g}{ \rho_g} + \frac{m_c}{\rho_c}[/tex]
[tex]\frac{m_g}{\rho} - \frac{m_g}{ \rho_g} = \frac{m_c}{\rho_c} - \frac{m_c}{\rho}[/tex]
[tex]m_g ( \frac{1}{\rho} - \frac{1}{\rho_g}) = m_c ( \frac{1}{\rho_c} - \frac{1}{\rho})[/tex]
[tex]m_g ( \frac{ \rho_g - \rho}{ \rho\ \rho_g} ) = m_c ( \frac{ \rho - \rho_c} { \rho\ \rho_c} )[/tex]
[tex]\frac{m_g}{m_c} = \frac{\rho_g}{\rho_c}\ \frac{\rho - \rho_c}{ \rho_g - \rho}[/tex]
[tex]\frac{m_g}{m_c} = \frac{19.3}{8.96}\ \frac{ \frac{7.24}{0.436} - 8.96}{ 19.3 - \frac{ 7.24}{0.436} }[/tex]
[tex]\frac{m_g}{m_c} = 6.112[/tex]
[tex]\boxed {m_g = 6.112 m_c}[/tex]
[tex]\texttt{ }[/tex]
[tex]\texttt{Percentage by mass of Gold} = \frac{m_g}{m_g + m_c} \times 100\%[/tex]
[tex]\texttt{Percentage by mass of Gold} = \frac{6.112m_c}{6.112m_c + m_c} \times 100\%[/tex]
[tex]\texttt{Percentage by mass of Gold} = \frac{6.112}{6.112 + 1} \times 100\%[/tex]
[tex]\texttt{Percentage by mass of Gold} = \frac{6.112}{7.112} \times 100\%[/tex]
[tex]\texttt{Percentage by mass of Gold} \approx 86 \%[/tex]
[tex]\texttt{Percentage by mass of Copper} \approx 100\% - 86 \% \approx 14 \%[/tex]
[tex]\texttt{ }[/tex]
[tex]\texttt{ }[/tex]
Grade: High School
Subject: Physics
Chapter: Physical Quantities
