Problem 89 - Chemistry and Chemical Reactivity

Question

Diamond has a density of \(3.513 \mathrm{g} / \mathrm{cm}^{3} .\) The mass of diamonds is often measured in "carats," where 1 carat equals \(0.200 \mathrm{g} .\) What is the volume (in cubic centimeters) of a 1.50 -carat diamond?

Step-by-Step Solution

Answer

The volume of a 1.50-carat diamond is approximately 0.0854 cm³.

Step 1: Convert carats to grams

First, convert the mass of the diamond from carats to grams. We know that 1 carat equals 0.200 grams. Therefore, for a 1.50-carat diamond, the mass will be: \ \[ 1.50 \text{ carats} \times 0.200 \text{ g/carats} = 0.300 \text{ g} \]

Step 2: Use the density formula

To find the volume of the diamond, use the formula for density \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Rearrange it to solve for volume: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).

Step 3: Substitute values into the volume formula

Substitute the mass of the diamond and the density into the formula. Given that the mass is 0.300 grams and the density is 3.513 g/cm³, we have: \[ \text{Volume} = \frac{0.300 \text{ g}}{3.513 \text{ g/cm}^3} \approx 0.0854 \text{ cm}^3 \]

Key Concepts

Volume Calculation

To determine the volume of an object, such as our diamond, we use the relationship between mass, volume, and density. The formula that connects these quantities is density

• Density is defined as mass per unit volume.
• We can rewrite the density formula to solve for volume: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]

For our diamond example, we need to find the volume using this principle.
We know the mass (from the mass conversion explained later), and we know the density (as given in the exercise). All you need to do is plug these values into the formula, calculate, and hocus pocus! Volume appears. It’s as simple as that! Using the values 0.300 g for mass and 3.513 g/cm³ for density, the diamond's volume works out to approximately 0.0854 cm³.
This calculation demonstrates the beauty of these formulas and why they are so powerful in understanding physical characteristics.

Mass Conversion

Converting between units is an essential skill, especially when dealing with quantities that are not in standard units like grams or kilograms. In this case, the mass of diamond is given in carats.
To convert carats into grams, use:

• 1 carat = 0.200 grams

For our diamond weighing 1.50 carats, how do we convert this to grams? It’s easy.
Multiply the number of carats by the conversion factor (0.200 grams per carat):\[1.50 \text{ carats} \times 0.200 \text{ grams/carats} = 0.300 \text{ grams} \]
This conversion step is crucial because most density calculations use standard mass units like grams.

Density Formula

Understanding the density formula is key to solving problems involving material properties. The density of a material tells us how much mass is contained within a specific volume.
The basic density equation is:

• \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

In rearranged form, it allows us to find any of the three quantities if the other two are known. This formula is fundamental in not only physics and chemistry but also everyday situations, where knowing one allows you to estimate the others.
For example, knowing the density of diamond as 3.513 g/cm³ helps us calculate the volume of a diamond when we know its mass (as we have calculated in grams).
By inserting our values into \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \), the real power of the density formula is revealed. With such a simple set of calculations, you are on your way to mastering densities!