Problem 40 - Chemistry: Atoms First

Question

A large piece of jewelry has a mass of 132.6 g. A graduated cylinder initially contains 48.6 mL water. When the jewelry is submerged in the graduated cylinder, the total volume increases to \(61.2 \mathrm{mL}\). (a) Determine the density of this piece of jewelry. (b) Assuming that the jewelry is made from only one substance, what substance is it likely to be? Explain.

Step-by-Step Solution

Answer

The density of the jewelry is approximately 21.6 g/cm^3. It's likely made of platinum, which has a density of around 21.45 g/cm^3.

Step 1: Calculate the volume of the jewelry

Determine the volume of the jewelry by subtracting the initial volume of water in the graduated cylinder from the final volume after the jewelry has been submerged. Use the formula: Volume of jewelry = Final volume with jewelry - Initial volume without jewelry.

Step 2: Convert mass from grams to kilograms

Convert the mass of the jewelry from grams (g) to kilograms (kg) because density is commonly expressed in kg/m^3. Note that 1 g = 0.001 kg.

Step 3: Calculate the density of the jewelry

Use the formula: Density = Mass / Volume to calculate the density of the jewelry, using the mass in kilograms and the volume in cubic meters (m^3). Remember that 1 mL = 1 cm^3 and 1,000,000 cm^3 = 1 m^3.

Step 4: Determine the likely substance

Compare the calculated density with known densities of materials to identify the likely substance the jewelry is made of. If the density matches closely with that of a particular material, it is likely the jewelry is made from that substance.

Key Concepts

Mass and Volume

Understanding the relationship between mass and volume is crucial for calculating density, which is a measure of how much mass is contained in a given volume. Mass is the amount of matter in an object and is typically measured in grams (g) or kilograms (kg), whereas volume is the amount of space the object occupies and is often measured in cubic centimeters (cm³) or milliliters (mL), which are equivalent.

In the context of the exercise about finding the density of a piece of jewelry, we can determine the volume of the jewelry by using a graduated cylinder. When the jewelry is submerged, the water level rises from the initial 48.6 mL to 61.2 mL. This means the volume of the jewelry alone is the difference between these two measurements, which is a straightforward concept, yet essential for moving on to more complex steps such as density calculation.

Units Conversion

When solving science and math problems, it's often necessary to convert units to ensure that we're comparing apples to apples, so to speak. Units conversion can be particularly important when dealing with measurements involving mass and volume. In our exercise, for example, although mass starts in grams, we need to convert it into kilograms for standard density calculations. Similarly, volume measured in milliliters should be converted into cubic meters when needed.

For mass, remember the quick conversion that 1 gram equals 0.001 kilograms (kg). For volume, it's useful to recall that 1 milliliter (mL) is equal to 1 cubic centimeter (cm³), and 1,000,000 cubic centimeters make up 1 cubic meter (m³). By understanding these conversions, students can adeptly navigate through exercises involving different units and apply their knowledge across various contexts.

Material Identification

Once we have determined the density of an object, we can often use that information for material identification. Density is a physical property that is unique to each material, which is why it can be used for such identification purposes. In the exercise provided, by calculating the piece of jewelry's density, we can compare it to the known densities of various materials.

If the calculated density of our jewelry aligns closely with that of gold, silver, or another material, we can suggest the likely composition of the item. This process requires keen observation, accurate calculations, and sometimes further testing if materials have similar densities. Thus, understanding density not only helps us solve homework problems but also lies at the heart of material identification in disciplines such as chemistry and engineering.