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Chapter 1: Problem 73

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

Expert verified
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 \(2.8 \ \text{mL}\) is approximately \(49.14 \ \text{carats}\).

Step by step solution

01

Convert the mass of the diamond to grams.

To find the volume, we should first convert the given mass in carats to grams and use the density formula. To do that, multiply the mass in carats by the conversion factor (1 carat = 0.200 g). Mass in grams = (5.0 carats) × (0.200 g / 1 carat) = 1.0 g
02

Calculate the volume of the diamond.

Now that we have the mass of the diamond in grams, it's time to calculate the volume. We can rearrange the density formula to find the volume: V = m / ρ. Volume = Mass / Density V = 1.0 g / (3.51 g/cm³) V ≈ 0.285 cm³ The volume of the 5.0-carat diamond is approximately 0.285 cm³. b. We want to find the carats of a diamond measuring 2.8 mL. Given information: - The diamond's volume is 2.8 mL. - 1 mL = 1 cm³ (since we are working with cm³ in the rest of the problem, we will use this conversion). - The diamond's density is 3.51 g/cm³. - 1 carat = 0.200 g
03

Convert the volume of the diamond to cm³.

We know that the diamond has a volume of 2.8 mL, which is equal to 2.8 cm³ since 1 mL is equal to 1 cm³. So the volume of the diamond is 2.8 cm³.
04

Calculate the mass of the diamond.

Using the density formula, we can find the mass of the diamond. To do so, rearrange the density formula to m = ρ * V. Mass = Density * Volume m = (3.51 g/cm³) * (2.8 cm³) m ≈ 9.828 g The mass of the diamond is approximately 9.828 g.
05

Convert the mass to carats.

Now, we want to find the mass of the diamond in carats. To do that, divide the mass in grams by the conversion factor (0.200 g = 1 carat). Mass in carats = 9.828 g ÷ (0.200 g/carats) ≈ 49.14 carats The mass of the diamond with a volume of 2.8 mL is approximately 49.14 carats.

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