Problem 59 - Chemistry The Science in Context

Question

The mass of a diamond is usually expressed in carats, where 1 carat \(=0.200 \mathrm{g} .\) The density of diamond is \(3.51 \mathrm{g} / \mathrm{cm}^{3}\) What is the volume of a 5.0 -carat diamond?

Step-by-Step Solution

Answer

Answer: The volume of the 5.0-carat diamond is approximately 0.285 cm³.

Step 1: Convert mass from carats to grams

To convert the mass of the diamond from carats to grams, use the conversion factor given in the exercise: 1 carat = 0.200 g. Multiply the given mass in carats by the conversion factor: Mass (g) = 5.0 carats × 0.200 g/carat

Step 2: Calculate the mass of the diamond in grams

Multiply the given mass in carats by the conversion factor to find the mass in grams: Mass (g) = 5.0 × 0.200 = 1 g

Step 3: Use the formula for density to find the volume

The formula for density is: Density = mass/volume, so Volume = mass/density Use the known values of mass (1 g) and density (3.51 g/cm³) to find the volume: Volume (cm³) = 1 g / 3.51 g/cm³

Step 4: Calculate the volume of the diamond

Divide the mass by the density to find the volume: Volume (cm³) = 1 / 3.51 ≈ 0.285 cm³ The volume of the 5.0-carat diamond is approximately 0.285 cm³.

Key Concepts

Mass Conversion

When working with precious gems like diamonds, it's crucial to convert mass measurements into a standard unit. Often, diamond mass is given in carats. A carat is a specific unit of mass used in the jewelry trade, and it's equivalent to 0.200 grams.
To find the mass in grams, the carat measurement must be multiplied by this conversion factor. This step ensures that we use consistent units for calculations, especially when we eventually apply the density formula. In this exercise, we initially have a diamond weighing 5.0 carats. Applying the conversion factor, we calculate its mass in grams as follows:

• Use the conversion: 1 carat = 0.200 grams.
• Calculate: 5.0 carats × 0.200 g/carat = 1 gram.

This conversion is essential because other measurements, like volume, are often calculated based off grams, making grams the preferable unit for solving density-related problems.

Volume Determination

Once the mass of the diamond is expressed in grams, the next step is to determine its volume. The volume of an object represents the space it occupies. This measurement is critical when applying the density formula.
To find the volume, we will rearrange the commonly used density formula. The density formula is written as:

• Density = Mass / Volume.

By algebraic manipulation, we solve for volume:

• Volume = Mass / Density.

In this exercise, we've already determined the diamond's mass to be 1 gram. Given the density of the diamond is 3.51 g/cm³, the volume calculation will be:

• Volume (cm³) = 1 g / 3.51 g/cm³.
• Resulting in approximately 0.285 cm³.

Understanding how to find volume is vital because it's often the missing link required to harness the power of density equations.

Density Formula

The density formula is a fundamental concept in physics and chemistry, providing insights into the relationships between an object's mass, volume, and how compact matter is within it.
The formula is defined as:

• Density = Mass / Volume.

This equation can be rearranged to solve for mass or volume, depending on which quantities are known. By knowing any two variables, we can solve for the third. In our diamond exercise, we began with known mass and density values to find volume.
For our diamond:

• Known: Mass = 1 gram, Density = 3.51 g/cm³.
• Solve for volume: Volume = Mass / Density.

Thus, by dividing the mass by the density, we found that the volume was approximately 0.285 cm³. Understanding this density relationship aids in a wide array of scientific and practical applications, from geology to materials science.