Problem 15 - Chemistry

Question

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.)

Step-by-Step Solution

Answer

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

Step 1: 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: \(44.4 \text{ carats} \times 200 \text{ mg/carats} = 8880 \text{ mg}\).

Step 2: 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: \(\frac{8880 \text{ mg}}{1000} = 8.88 \text{ grams}\).

Step 3: 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: \(\frac{8.88 \text{ g}}{28.35 \text{ g/ounce}} \approx 0.313 \text{ ounces}\).

Key Concepts

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’s 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.