Question

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

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

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

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

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.

Key concepts

Sterling Silver

Sterling silver is a popular alloy, particularly known for its use in high-quality jewelry. This is a mixture predominantly composed of silver, with a small percentage of copper. The typical composition is 92.5% silver and 7.5% copper. This combination not only preserves the brilliant luster associated with silver but also enhances the strength and durability, thanks to the copper content.
The inclusion of copper makes sterling silver more resistant to scratching and bending. This property is essential for items such as necklaces or rings that undergo daily wear and tear. Understanding the composition of sterling silver helps in calculating the mass percent of the metals present, which is necessary for tasks such as ensuring the authenticity of jewelry or determining corrosion resistance.

Density

Density is a fundamental property used to characterize materials. It is defined as mass per unit volume, typically expressed in grams per milliliter (g/mL) for liquids and solids. For sterling silver calculations, knowing the densities of the individual metals is crucial. The density of silver is approximately 10.5 g/mL, while copper's density is about 8.96 g/mL.
When calculating mass from a given volume, the density values are indispensable. In the case of our sterling silver necklace, converting the total volume into masses of silver and copper involves using these density values directly. Doing so allows us to set up equations representing the relationship between the mass and volume of each metal, which is the first step in calculating the mass percent.

Mass Relation

In the context of the problem, mass relation refers to the ratio between the masses of silver and copper within the necklace. This relationship is derived from their respective densities. By establishing this ratio, we can describe how much of each metal is present without necessarily knowing the individual volumes.
For the necklace, we set up a proportion:

• \(\frac{m_{silver}}{m_{copper}} = \frac{density_{silver}}{density_{copper}}\)

This equation implies that, given the densities of the two metals, we know how their masses compare to one another. It provides a pathway to eliminate volume variables by relating these masses directly. Consequently, we can express one mass in terms of the other, significantly simplifying our calculations.

Copper

Copper is an essential element in the composition of sterling silver, providing crucial properties to the alloy. Though comprising only about 7.5% of the alloy, its presence is vital for enhancing strength and ductility. It also lends a slight reddish hue to sterling silver, distinguishing it from pure silver.
From a chemical standpoint, the appropriate proportion of copper ensures that sterling silver remains easy to work with while maintaining its aesthetic appeal. When calculating the mass percent of copper, it's essential to isolate the mass of just the copper from the mixture. This involves solving equations to determine how much of the total mass of 105 g can be attributed to copper.

Silver

Silver is the primary component in sterling silver, recognized for its illustrious shine and high value. Being more abundant than copper in this alloy, it dictates the overall properties of the mixture. Silver's density, along with copper's, is used to set up the mass relation, a vital calculation in finding the mass percent of each metal.
In the mass calculation for sterling silver, silver takes the larger portion, with its mass and volume contributing significantly to the overall measurements. Solving the problem involves finding how the mass of silver relates to that of copper, through their established density ratio. This relationship is key to identifying the precise composition of the necklace in terms of tailors' mass percents.