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Chapter 1: Problem 15

Gemstones are weighed in carats, with 1 carat \(=200 \mathrm{mg}\) (exactly). What is the mass in grams of the Hope Diamond, the world's largest blue diamond at \(44.4\) carats? What is this mass in ounces? (See conversion on the inside back cover.)

Short Answer

Expert verified
The mass of the Hope Diamond is 8.88 grams or approximately 0.313 ounces.

Step by step solution

01

Convert Carats to Milligrams

First, you need to convert the size of the Hope Diamond from carats to milligrams. We know that 1 carat equals 200 milligrams. So, multiply the carat weight of the diamond by the milligrams per carat: \(44.4 \text{ carats} \times 200 \text{ mg/carats} = 8880 \text{ mg}\).
02

Convert Milligrams to Grams

Next, convert the mass from milligrams to grams. We know that 1000 mg equals 1 gram. To convert milligrams to grams, divide the number of milligrams by 1000: \(\frac{8880 \text{ mg}}{1000} = 8.88 \text{ grams}\).
03

Convert Grams to Ounces

Now, we need to convert the mass from grams to ounces. Typically, the conversion is 1 ounce equals approximately 28.35 grams. Use this conversion factor to find ounces: \(\frac{8.88 \text{ g}}{28.35 \text{ g/ounce}} \approx 0.313 \text{ ounces}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carats to Milligrams Conversion
When it comes to gemstones, their weight is often expressed in terms of carats. This particular unit is quite fascinating because it links the world of jewelry with that of science.
  • One carat is exactly equal to 200 milligrams.
  • This conversion is crucial when you need to translate the mass into more commonly used metric units.
To convert carats to milligrams, you simply multiply the number of carats by 200. It's that straightforward. If you have a gemstone weighing 44.4 carats like the Hope Diamond, it becomes:\[ 44.4 ext{ carats} \times 200 ext{ mg/carats} = 8880 ext{ mg} \]Understanding this conversion helps in accurate weighing and further conversions.
Milligrams to Grams Conversion
Milligrams and grams are both units within the metric system, which is designed to simplify calculations with its base-10 setup. A milligram is a much smaller unit compared to a gram.
  • 1 gram is equal to 1000 milligrams.
  • This means that to convert milligrams to grams, you will divide by 1000.
This conversion requires dividing the total number of milligrams by 1000. Let鈥檚 apply this to 8880 mg:\[ \frac{8880 ext{ mg}}{1000} = 8.88 ext{ grams} \]So, the mass of the diamond in grams provides a clearer picture when calculating larger weights or when further conversion is required into even larger units.
Grams to Ounces Conversion
Moving from the metric system to the imperial system involves converting grams to ounces, which is often necessary in contexts such as international trade or certain scientific fields.
  • One ounce is roughly equivalent to 28.35 grams.
  • To convert grams to ounces, divide the number of grams by 28.35.
Using this conversion for 8.88 grams provides us:\[ \frac{8.88 ext{ g}}{28.35 ext{ g/ounce}} \approx 0.313 ext{ ounces} \]This final conversion step is particularly useful when dealing with measurements in countries using the imperial system, allowing for easy communication of weight across different regions.

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Most popular questions from this chapter

A sodium chloride solution was prepared in the following manner: \- A \(25.0 \mathrm{~mL}\) volumetric flask (Figure \(1.8)\) was placed on an analytical balance and found to have a mass of \(35.6783 \mathrm{~g}\). \- Sodium chloride was added to flask and the mass of the solid \(+\) flask was \(36.2365 \mathrm{~g}\). \- The flask was filled to the mark with water and mixed well. Calculate the concentration of the sodium chloride solution in units of \(\mathrm{g} / \mathrm{mL}\) and give the answer in scientific notation with the correct number of significant figures.

What is the difference between a hypothesis and theory? (a) A hypothesis provides an explanation for a phenomenon, but a theory does not. (b) A theory provides an explanation for a phenomenon, but hypothesis does not. (c) Both a theory and a hypothesis provide an explanation for a phenomenon, but a theory has been upheld by experimental observations.

Convert the following quantities into SI units with the correct number of significant figures: (a) \(5.4\) in. (b) \(66.31 \mathrm{lb}\) (c) \(0.5521\) gal (d) \(65 \mathrm{mi} / \mathrm{h}\) (e) \(978.3 \mathrm{yd}^{3}\) (f) \(2.380 \mathrm{mi}^{2}\)

How many significant figures are in each of the following measurements? (a) \(35.0445 \mathrm{~g}\) (b) \(59.0001 \mathrm{~cm}\) (c) \(\begin{array}{lll}0.030 & 03 \mathrm{~kg}\end{array}\) (d) \(0.00450 \mathrm{~m}\) (e) \(67,000 \mathrm{~m}^{2}\) (f) \(3.8200 \times 10^{3} \mathrm{~L}\)

Distances over land are measured in statute miles ( \(5280 \mathrm{ft}\) ), but distances over water are measured in nautical miles, where 1 nautical mile was originally defined as 1 minute of arc along an Earth meridian, or \(1 / 21,600\) of the Earth's circumference through the poles. A ship's speed through the water is measured in \(k\) nots, where 1 knot \(=1\) nautical mile per hour. Historically, the unit knot derived from the practice of measuring a ship's speed by throwing a log tied to a knotted line over the side. The line had a knot tied in it at intervals of \(47 \mathrm{ft} .3\) in., and the number of knots run out in 28 seconds was counted to determine speed. (a) How many feet are in a nautical mile? How many meters? (b) The northern bluefin tuna can weigh up to 1500 pounds and can swim at speeds up to 48 miles per hour. How fast is this in knots? (c) A league is defined as 3 nautical miles. The Mariana Trench, with a depth of 35,798 feet, is the deepest point in the ocean. How many leagues deep is this? (d) By international agreement, the nautical mile is now defined as exactly 1852 meters. By what percentage does this current definition differ from the original definition, and by what percentage does it differ from a statute mile?
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