Problem 63 - Chemistry: Principles and Reactions

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?

Step-by-Step Solution

Answer

Answer: To find the mass in pounds and the volume in cubic inches of the Kohinoor Diamond, follow these steps: 1. Convert the mass of the diamond from carats to grams using the conversion factor 1 carat = 2.00 × 10² mg and then to grams: mass in grams = mass in carats × (2.00 × 10² mg / 1 carat) × (1 g / 1000 mg). 2. Calculate the mass of the diamond in pounds using the conversion factor 1 pound = 453.59237 grams: mass in pounds = mass in grams × (1 pound / 453.59237 grams). 3. Calculate the volume of the diamond in cubic centimeters using the mass in grams and the density formula: volume in cubic centimeters = mass in grams / density in g / cm³. 4. Convert the volume of the diamond to cubic inches using the conversion factor 1 cubic inch = 16.3871 cubic centimeters: volume in cubic inches = volume in cubic centimeters × (1 cubic inch / 16.3871 cm³).

Step 1: Convert mass from carats to grams

The mass of the Kohinoor Diamond is given in carats (unit of mass), we need to convert it to grams. Given that 1 carat has a mass of 2.00 × 10² mg, we can convert the mass of Kohinoor Diamond (108 carats) to grams using the conversion factor: mass in grams = mass in carats × (2.00 × 10² mg / 1 carat) × (1 g / 1000 mg)

Step 2: Calculate mass in pounds:

Now that we have the mass in grams, we can convert it to pounds using the conversion factor (1 pound = 453.59237 grams): mass in pounds = mass in grams × (1 pound / 453.59237 grams)

Step 3: Calculate volume in cubic centimeters:

Now we will use the density formula to calculate the volume of the diamond: density = mass/volume Rearrange to solve for volume: volume = mass/density We know the density (d) and mass (m) of the diamond in grams, so we can find the volume in cubic centimeters: volume in cubic centimeters = mass in grams / density in g / cm³

Step 4: Convert volume to cubic inches:

Now that we have the volume in cubic centimeters, we can convert it to cubic inches using the conversion factor (1 cubic inch = 16.3871 cubic centimeters): volume in cubic inches = volume in cubic centimeters × (1 cubic inch / 16.3871 cm³)

Key Concepts

Density

Density is a measure of how much mass is contained within a given volume. It is commonly expressed in units such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). Understanding density helps us determine how tightly packed the particles in a substance are. For the Kohinoor Diamond, given a density of 3.51 g/cm³, it means that each cubic centimeter of the diamond weighs 3.51 grams.
Calculations involving density often use the formula:

• Density = Mass / Volume

This formula can be rearranged to find the mass or volume of a substance if the other two quantities are known. In practical applications, understanding density allows predictions about whether objects will float or sink in various mediums. It also helps in identifying substances based on their mass-to-volume ratio.

Mass Conversion

Mass conversion involves changing mass units from one form to another to simplify calculations or match the required units of a problem. The original mass of the Kohinoor Diamond was given in carats, a unit commonly used for gemstones, but needed conversion into grams and eventually pounds for further calculations.

• 1 carat = 200 milligrams (mg)
• 1 gram (g) = 1000 milligrams (mg)

When converting the mass from carats to grams, the formula used was:

• Mass in grams = Mass in carats × (200 mg/1 carat) × (1 g/1000 mg)

Understanding these conversions is crucial in various scientific and engineering disciplines to ensure accurate and dimensionally consistent calculations.

Volume Conversion

Volume conversion deals with changing the measurement of volume from one unit to another. This is particularly useful in applications where space or capacity is involved. In the case of the Kohinoor Diamond, converting the volume from cubic centimeters to cubic inches was a necessary step to provide a familiar unit of measurement for those using the Imperial system.
The basic conversion formula for cubic centimeters to cubic inches is:

• Volume in cubic inches = Volume in cubic centimeters × (1 cubic inch / 16.3871 cubic centimeters)

Conversions such as these are important in international contexts where different units of measure are standard. Understanding and being able to perform these conversions ensure precision and clarity when communicating measurements globally.

Unit Conversion

Unit conversion is the process of changing the units of a measured quantity into different forms without affecting its intrinsic value. This can involve converting between metric and imperial units, such as grams to pounds or cubic centimeters to cubic inches. Mastery of unit conversion is essential for scientists, engineers, and students alike to solve problems accurately.
In our problem, the steps included converting carats to grams, grams to pounds, and cubic centimeters to cubic inches. It is critical to understand the conversion factors:

• 1 carat = 200 mg
• 1 g = 1000 mg
• 1 pound = 453.59237 grams
• 1 cubic inch = 16.3871 cubic centimeters

Correctly applying these conversion factors ensures that calculations remain consistent and reliable.

Cubic Inches

Cubic inches are units of volume commonly used in the United States, specifically within the imperial system of measurements. One cubic inch is defined as the volume of a cube with sides of exactly one inch in length. This unit is often used for smaller volumes as it provides a manageable scale.
For the Kohinoor Diamond, converting its volume to cubic inches provides a tangible sense of size for those familiar with imperial units. Understanding cubic inches can be important for designing spaces, packing, and determining storage capacity, among other applications. It is equivalent to approximately 16.387 cubic centimeters, providing an easy way to convert from metric volumes.

Cubic Centimeters

Cubic centimeters (cc or cm³) are metric units of volume used predominantly in scientific contexts and throughout countries using the metric system. A cubic centimeter is defined as the volume of a cube with edges that are each one centimeter long. It is a convenient unit for small volumes, making it ideal for precise measurements in laboratory settings.
In the problem, the volume of the diamond was initially calculated in cubic centimeters using its mass and density. This is because density is often expressed in g/cm³, allowing direct calculation of volume when mass is known through the formula:

• Volume = Mass / Density

Understanding cubic centimeters and their conversion to and from other units is essential in fields such as chemistry, physics, and medicine.

Carats

Carats are a unit of mass used primarily for measuring gemstones and pearls. One carat is defined as 200 milligrams, which is 0.2 grams. This unit is integral in the diamond industry and often used in jewelry to express the size and weight of stones.
In our example, the Kohinoor Diamond's mass was given in carats, necessitating a conversion to grams to accommodate further calculations involving density and volume in the metric system. The value in carats often signifies not only the mass but also the perceived value and pricing of gemstones, making it a crucial concept in gemology.

Pounds

Pounds are a unit of mass and weight used extensively in the imperial system, especially in the United States. The pound is widely used for measuring body weight, groceries, and various large objects. In scientific calculations, however, pounds often need to be converted into metric units for consistency.
For the Kohinoor Diamond problem, the conversion from grams to pounds was a crucial step to present the mass in a commonly accepted unit in regions using the imperial system. The conversion factor for this is:

• 1 pound = 453.59237 grams

Understanding how to convert between pounds and metric units allows for seamless transitions between measurement systems, ensuring accuracy in commerce, science, and engineering.