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Chapter 1: Problem 123

Sterling silver is a solid solution of silver and copper. If a piece of a sterling silver necklace has a mass of 105.0 g and a volume of 10.12 mL, calculate the mass percent of copper in the piece of necklace. Assume that the volume of silver present plus the volume of copper present equals the total volume. Refer to Table 1.5. Mass percent of copper $=\frac{\text { mass of copper }}{\text { total mass }} \times 100$

Short Answer

Expert verified
The mass percent of copper in the sterling silver necklace piece is calculated by first finding the mass relation between silver and copper: \(\frac{m_{silver}}{m_{copper}} = \frac{10.5}{8.96}\). Then, we express the mass of silver in terms of the mass of copper and plug this into the total mass equation: \(m_{copper} = 105 - \frac{10.5}{8.96} \times m_{copper}\). Solve for the mass of copper and then calculate the mass percent of copper: Mass percent of copper = \(\frac{m_{copper}}{total_\ mass} \times 100\).

Step by step solution

01

Convert the volume of the necklace (10.12 mL) into masses of silver (m_silver) and copper (m_copper) using their densities. m_silver = density_silver * volume_silver m_copper = density_copper * volume_copper The volumes of silver and copper are unknown, but their sum equals the total volume of the necklace: volume_silver + volume_copper = 10.12 mL #Step 2: Calculate mass relation between silver and copper

Divide the two mass equations we obtained in step 1 to eliminate the volume terms: \(\frac{m_{silver}}{m_{copper}} = \frac{density_{silver}}{density_{copper}}\) #Step 3: Plug in the known values
02

Replace the known densities of silver (10.5 g/mL) and copper (8.96 g/mL) in the equation from step 2 and solve for the mass relation between silver and copper: \(\frac{m_{silver}}{m_{copper}} = \frac{10.5}{8.96}\) #Step 4: Use mass relation to find the mass of copper

Since we have the mass relation between silver and copper from step 3, we can express the mass of silver in terms of the mass of copper. From the total mass of necklace (105 g), we can express the mass of copper as follows: m_copper = 105 - m_silver Using the mass relation found in step 3: m_silver = \(\frac{10.5}{8.96}\) * m_copper #Step 5: Solve for the mass of copper
03

Now we have two equations with one unknown (m_copper). Let's substitute the equation for m_silver in terms of m_copper found in step 4 into the equation for m_copper from step 4: m_copper = 105 - \(\frac{10.5}{8.96}\) * m_copper #Step 6: Calculate m_copper and mass percent

Solve the equation obtained in step 5 to find the mass of copper (m_copper). Once you find the mass of copper, you can calculate the mass percent using the given formula: Mass percent of copper = \(\frac{m_{copper}}{total_\ mass} * 100\) Make sure to plug the values of mass of copper and total mass (105.0 g) into the formula to find the mass percent of copper in the piece of necklace.

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