Problem 6 - Chemistry

Question

You are beachcombing on summer vacation and find a silver bracelet. You take it to the jeweler and he tells you that it is silver plated and will give you $$\$ 10$$ for it. You do not want to be swindled so you take the bracelet to your chemistry lab and find its mass on a balance \((80.0 \mathrm{~g})\). To measure the volume you place the bracelet in a graduated cylinder (Figure 1.8) containing \(10.0 \mathrm{~mL}\) of water at \(20^{\circ} \mathrm{C}\). The final volume in the graduated cylinder after the bracelet has been added is \(17.61 \mathrm{~mL}\). The density of silver at \(20^{\circ} \mathrm{C}\) is \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\) and \(1 \mathrm{~cm}^{3}=1 \mathrm{~mL}\). What can you conclude about the identity of the metal in the bracelet?

Step-by-Step Solution

Answer

The bracelet's density matches silver, suggesting it may not be just silver-plated.

Step 1: Determine the Volume of the Bracelet

To find the volume of the bracelet, you need to subtract the initial water volume from the final water volume after submerging the bracelet. Initially, the water is at 10.0 mL, and after the bracelet is submerged, the water level rises to 17.61 mL. So, the volume of the bracelet is:\[ \text{Volume of Bracelet} = 17.61 \text{ mL} - 10.00 \text{ mL} = 7.61 \text{ mL} \]

Step 2: Calculate the Density of the Bracelet

Density is defined as mass divided by volume. Using the mass of the bracelet (80.0 g) and the volume we calculated (7.61 mL), we can find the density of the bracelet:\[ \text{Density of Bracelet} = \frac{80.0 \text{ g}}{7.61 \text{ mL}} = 10.51 \text{ g/mL} \]

Step 3: Compare Bracelet Density with Silver Density

Now, compare the calculated density of the bracelet with the known density of silver, which is 10.5 g/cm³ at 20°C. Since the density of the bracelet (10.51 g/mL) is very close to that of silver, it indicates that the bracelet is consistent with being silver.

Key Concepts

Volume calculation

When determining the volume of an object, especially irregular shapes like a bracelet, displacement of water in a graduated cylinder is a useful method. This technique is simple to understand: you fill the cylinder with water, note its volume, immerse the object, and note the new water volume. The difference in water level gives you the object's volume.
This approach leverages Archimedes' principle, which explains that the volume of displaced water equals the volume of the submerged object. By observing that the water rose from 10.0 mL to 17.61 mL, you can conclude that the bracelet occupies 7.61 mL.

Mass measurement

Measuring mass is crucial in calculating density. In this exercise, a balance was used to find that the bracelet's mass is 80.0 g. Accurate mass measurement ensures reliable results in further calculations.
Balances vary in precision, but accurate readings are usually achieved by calibrating the scale prior to use and ensuring no additional objects or substances are accidentally measured along with your target object. Such thoroughness is key to scientific accuracy and calculating density or any other properties.

Density comparison

Density is a ratio of mass to volume and is often used to identify or verify the composition of substances. To find density, simply divide the mass by the volume. For the bracelet, the mass was 80.0 g and the volume was 7.61 mL, resulting in a density of 10.51 g/mL.
This calculated density was then compared to the known density of silver at 20°C, which is 10.5 g/cm³. Since 1 mL is equivalent to 1 cm³, this comparison simplifies. The slightly higher density of the bracelet (10.51 g/mL) closely aligns with pure silver, supporting the conclusion that the metal could indeed be silver.

Silver properties

Silver is a valuable metal, renowned for its conductivity, malleability, and distinct sheen. Known for its density of 10.5 g/cm³ at room temperature, silver is often used in jewelry, electrical components, and as a monetary standard.
Its high density makes it feasible to differentiate it from less dense metal alloys used in plating. Silver's properties not only contribute to its beauty and utility, but also provide a basis for identification, as seen in this bracelet exercise. Understanding these characteristics allows more informed judgments regarding the authenticity or purity of silver products.