Question

The Kohinoor Diamond \(\left(d=3.51 \mathrm{~g} / \mathrm{cm}^{3}\right)\) is 108 carats. If one carat has a mass of \(2.00 \times 10^{2} \mathrm{mg},\) what is the mass of the Kohinoor Diamond in pounds? What is the volume of the diamond in cubic inches?

Short answer

Answer: The mass of the Kohinoor Diamond is approximately 0.98 lb, and its volume is approximately 1.25 in³.

Step-by-step solution

Convert mass in carats to grams

Given, 1 carat = 2.00 x 10^2 mg. The mass of the Kohinoor Diamond is 108 carats. First, we need to convert the mass from carats to grams. To do this, we'll multiply the mass in carats by the conversion factor: Mass_carats = 108 carats Conversion_factor = 2.00 × 10^2 mg/carat 1 g = 1000 mg Mass_grams = Mass_carats × Conversion_factor × (1 g/1000 mg) Mass_grams = 108 carats × 2.00 × 10^2 mg/carat × (1 g / 1000 mg)

Convert mass in grams to pounds

Now, we need to convert the mass from grams to pounds. To do this, we'll use another conversion factor: 1 lb = 453.592 grams Mass_pounds = Mass_grams × (1 lb / 453.592 g) Mass_pounds = (108 carats × 2.00 × 10^2 mg/carat × (1 g / 1000 mg)) × (1 lb / 453.592 g)

Calculate the volume in cubic centimeters

Next, we need to find the volume of the diamond in cubic centimeters. To do this, we'll use the formula: Volume_cm3 = Mass_grams / Density Given, Density (d) = 3.51 g/cm³ Volume_cm3 = Mass_grams / 3.51 g/cm³ Volume_cm3 = (108 carats × 2.00 × 10^2 mg/carat × (1 g / 1000 mg)) / 3.51 g/cm³

Convert volume in cubic centimeters to cubic inches

Finally, we need to convert the volume from cubic centimeters to cubic inches. To do this, we'll use the conversion factor: 1 in³ = 16.3871 cm³ Volume_in3 = Volume_cm3 × (1 in³ / 16.3871 cm³) Volume_in3 = ((108 carats × 2.00 × 10^2 mg/carat × (1 g / 1000 mg)) / 3.51 g/cm³) × (1 in³ / 16.3871 cm³) After calculating the above expressions, we find that the mass of the Kohinoor Diamond in pounds is approximately 0.98 lb, and its volume is approximately 1.25 in³.