Question

Diamonds are measured in carats, and 1 carat \(=0.200 \mathrm{~g}\). The density of diamond is \(3.51 \mathrm{~g} / \mathrm{cm}^{3}\). a. What is the volume of a \(5.0\) -carat diamond? b. What is the mass in carats of a diamond measuring \(2.8 \mathrm{~mL}\) ?

Short answer

a. The volume of a 5.0-carat diamond is approximately \(0.285 \, \text{cm}^3\). b. The mass of a diamond with a volume of \(2.8 \, \text{cm}^3\) is approximately 49.14 carats.

Step-by-step solution

Convert carats to grams

The first step to finding the volume of a 5.0-carat diamond is to convert the diamond's mass from carats to grams. We're given the relationship 1 carat = 0.200 grams. Therefore, we can multiply the mass in carats with the conversion factor. \( 5.0 \, \text{carats} \times \frac{0.200 \, \text{g}}{1 \, \text{carat}} \)

Calculate the volume using the density formula

Now that we have the mass in grams, we can use the density formula to find the volume. The density of diamond is given as 3.51 g/cm³. The formula for density is: \( \text{density} = \frac{\text{mass}}{\text{volume}} \) We want to find the volume, so we can rearrange the formula: \( \text{volume} = \frac{\text{mass}}{\text{density}} \) Now, we plug in the values for the mass and the density: \( \text{volume} = \frac{1 \, \text{g}}{3.51 \frac{\text{g}}{\text{cm}^3}} \)

Calculate the volume

Now we can perform the calculations to find the volume of the 5.0-carat diamond: \( \text{volume} = \frac{1 \, \text{g}}{3.51 \frac{\text{g}}{\text{cm}^3}} = 0.285 \, \text{cm}^3 \) The volume of a 5.0-carat diamond is approximately 0.285 cm³. #b. Mass in carats of a diamond with a volume of \(2.8 \mathrm{~mL}\)#

Convert mL to cm³

Since the density is given in grams per cubic centimeter (g/cm³), we need to first convert the given volume from mL to cm³. The conversion factor is 1 mL = 1 cm³: 2.8 mL = 2.8 cm³

Calculate the mass using the density formula

We want to find the mass of the diamond, so we can rearrange the density formula: \( \text{mass} = \text{density} \times \text{volume} \) Now, we plug in the values for the density and the volume: \( \text{mass} = (3.51 \frac{\text{g}}{\text{cm}^3}) \times 2.8 \, \text{cm}^3 \)

Calculate the mass

Now we can perform the calculations to find the mass of the diamond: \( \text{mass} = (3.51 \frac{\text{g}}{\text{cm}^3}) \times 2.8 \, \text{cm}^3 = 9.828 \, \text{g} \) The mass of the diamond with a volume of 2.8 cm³ is approximately 9.828 grams.

Convert grams to carats

Now we need to convert the mass from grams to carats. We're given the relationship 1 carat = 0.200 grams. Therefore, we can divide the mass in grams by the conversion factor. \( 9.828 \, \text{g} \times \frac{1 \, \text{carat}}{0.200 \, \text{g}} \)

Calculate the mass in carats

Now we can perform the calculations to find the mass of the diamond in carats: \( \text{mass} = 9.828 \, \text{g} \times \frac{1 \, \text{carat}}{0.200 \, \text{g}} = 49.14 \, \text{carats} \) The mass of a diamond with a volume of 2.8 cm³ is approximately 49.14 carats.