Question

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

a. The volume of a 5.0-carat diamond is approximately \( \dfrac{1}{3.51}\ cm^3 \). b. The mass of a diamond measuring 2.8 mL is approximately 49.14 carats.

Step-by-step solution

a. Finding the Volume of a 5.0-carat Diamond

1. Convert the mass of the diamond from carats to grams. Given the conversion factor 1 carat = 0.200g, we can calculate the mass of the diamond in grams. \( Mass (g) = 5.0\ carats × 0.200\ g/carat = 1.0\ g \) 2. Use the density formula to find the volume. Density = Mass / Volume The density of the diamond is given as 3.51 g/cm^3. Rearrange the formula to solve for volume: \( Volume = \dfrac{Mass}{Density} \) 3. Insert the values and calculate the volume. \( Volume = \dfrac{1.0\ g}{3.51\ g/cm^3} = \dfrac{1}{3.51}\ cm^3 \) 4. Volume of the 5.0-carat diamond: The volume of the 5.0-carat diamond is approximately \( \dfrac{1}{3.51}\ cm^3 \).

b. Finding the Mass of a Diamond with 2.8 mL Volume

1. Convert the volume from mL to cm^3. Since 1 mL = 1 cm^3, the volume of the diamond is 2.8 cm^3. 2. Arrange the density formula to find the mass. Density = Mass / Volume We'll use the provided density of the diamond (3.51 g/cm^3) to find the mass. \( Mass = Density × Volume \) 3. Insert the values and calculate the mass. \( Mass (g) = 3.51\ g/cm^3 × 2.8\ cm^3 = 9.828\ g \) 4. Convert the mass of the diamond from grams to carats. Using the conversion factor, we have 1 carat = 0.200 g. \( Mass (carats) = 9.828\ g × \dfrac{1\ carat}{0.200\ g} = 49.14\ carats \) 5. Mass of the diamond measuring 2.8 mL: The mass of the diamond measuring 2.8 mL is approximately 49.14 carats.

Key concepts

Mass-Volume Relationship

Understanding the mass-volume relationship is essential for solving a wide range of problems in science, specifically when it comes to calculating the density of materials. Density is a measure of how much mass is contained in a unit volume of a material.

Mass and volume are two fundamental properties of matter. Mass is the amount of matter in an object and is usually measured in grams (g) or kilograms (kg), while volume is the amount of space that matter takes up and is typically measured in liters (L), milliliters (mL), or cubic centimeters (cm³).

When we say 'mass-volume relationship', we are often referring to the calculation of density, which is defined by the formula:\[ Density = \frac{Mass}{Volume} \]
In this formula, we can see that if we know the mass and density of an object, we can calculate its volume, and vice versa. This concept is critical and was applied in the exercise to determine the volume of a diamond based on its mass and density.

Unit Conversion

In scientific calculations, unit conversion is a fundamental skill that allows us to convert measurements from one unit to another. This conversion is vital to ensure that all values are expressed in compatible units before performing calculations.

In the provided exercise, we saw two key unit conversions. The first was converting carats, a unit of mass typically used for gemstones, to grams. The conversion factor given was 1 carat equals 0.200 grams. Without converting these units, we would not be able to utilize the density formula effectively, which requires mass in grams.

The second conversion was between milliliters and cubic centimeters. Since 1 mL is exactly equal to 1 cm³, understanding this equivalence is crucial when dealing with volumes in different units. Knowing that these two units are interchangeable without affecting the numerical value is a simple yet powerful insight for solving many types of measurement problems.

Density Formula Application

Applying the density formula can seem daunting at first, but once we understand each component, it becomes a straightforward process. The density formula ties together the concepts of mass, volume, and substance density.

The formula for density is:\[ Density = \frac{Mass}{Volume} \]
When we want to find one of these variables, and we have the other two, we simply rearrange the formula accordingly. If we're looking to find volume, the rearranged formula is:\[ Volume = \frac{Mass}{Density} \]
If we seek to find the mass, we use this version:\[ Mass = Density \times Volume \]
In the exercise, we applied the density formula twice. First, to calculate the volume of a diamond given its mass and density. Then, we calculated the mass of a diamond when given its volume and density. By meticulously rearranging and plugging in the values into the formula, we can solve for the unknown quantity, as demonstrated in the step-by-step solution.