Question

The carat is the unit of mass used by jewelers. One carat is exactly \(200 \mathrm{mg}\). How many carbon atoms are present in a 2 -carat diamond?

Short answer

A 2-carat diamond contains approximately \( 2.005 \times 10^{22} \) carbon atoms.

Step-by-step solution

Convert Carats to Milligrams

First, we need to convert the mass of the diamond from carats to milligrams. We know that 1 carat is exactly 200 mg. Therefore, a 2-carat diamond would have a mass of:\[ 2 \text{ carats} \times 200 \text{ mg/carats} = 400 \text{ mg} \]

Convert Milligrams to Grams

Next, we convert the mass from milligrams to grams. Since there are 1000 mg in 1 gram, we divide by 1000:\[ 400 \text{ mg} = \frac{400}{1000} \text{ g} = 0.4 \text{ g} \]

Use Molar Mass of Carbon

We use the molar mass of carbon to find the number of moles of carbon present. The molar mass of carbon is approximately 12.01 g/mol. Therefore, the number of moles of carbon in 0.4 grams is:\[ \text{Number of moles} = \frac{0.4 \text{ g}}{12.01 \text{ g/mol}} \approx 0.0333 \text{ moles} \]

Calculate Number of Atoms Using Avogadro's Number

To find the number of atoms, we multiply the number of moles by Avogadro's number, which is approximately \( 6.022 \times 10^{23} \) atoms/mol:\[ \text{Number of atoms} = 0.0333 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 2.005 \times 10^{22} \text{ atoms} \]

Final Answer

After doing all calculations, we find that a 2-carat diamond contains approximately \( 2.005 \times 10^{22} \) carbon atoms.

Key concepts

Molar Mass

Molar mass is a fundamental concept in chemistry related to the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and provides a link between the microscopic atomic scale and the macroscopic scale we can measure. For example, the molar mass of carbon is approximately 12.01 g/mol. This means that one mole of carbon atoms weighs 12.01 grams.

To use molar mass, you measure the mass of your sample and then divide by the molar mass to find the number of moles. This calculation allows chemists to convert between grams and moles, facilitating deeper understanding and manipulation of chemical reactions. In our diamond example, the molar mass of carbon is pivotal in determining how many moles are present in a given mass.

Avogadro's Number

Named after the scientist Amedeo Avogadro, Avogadro's number is a constant that tells you how many particles, typically atoms or molecules, are in one mole of a substance. Its value is approximately \( 6.022 \times 10^{23} \) particles per mole.

This large number is key to bridge the gap between atomic and macroscopic worlds. By using Avogadro's number, you can determine exactly how many atoms are in a known number of moles. For instance, in a diamond weighing 0.0333 moles of carbon, multiplying by Avogadro's number gives us the staggering amount of approximately \( 2.005 \times 10^{22} \) carbon atoms, enabling us to quantify substances at the atomic level.

Unit Conversion

Unit conversion is a crucial skill in chemistry and other sciences, as measurements often need to be converted from one unit to another to solve problems effectively. For example, converting carats to milligrams, and then to grams in our exercise is an essential step.

Let's see a quick guide:

• To convert carats to milligrams, use the conversion factor of 200 mg per carat.
• Once in milligrams, converting to grams involves dividing by 1000, since 1 gram equals 1000 mg.

Proper conversion ensures accuracy in calculations, helping avoid mistakes that could lead to incorrect conclusions.

Carbon Atoms

Carbon atoms serve as a building block in a myriad of structures due to their versatility in forming bonds. In the context of diamonds, pure carbon atoms are arranged in a crystal lattice, which leads to their hardness and brilliance.

Each diamond is essentially a network of carbon atoms bonded in a way that each atom is equidistant from its neighbors, forming a strong but beautiful structure. Counting these atoms is not feasible without chemistry's theoretical tools like moles and Avogadro’s number, as they allow us to estimate the immensely large number of atoms in even small masses.

Jeweler's Measurements

Jeweler's measurements, such as carats, provide a specialized metric for the trade. A carat, equivalent to 200 milligrams, has become a standard unit for weighing gemstones, especially diamonds.

This specific measurement system facilitates uniformity and understanding when dealing with precious stones. However, understanding these weight units in terms of standard scientific measurements, like grams, is vital for those who aim to delve deeper into the science behind jewelry. This overlap between jeweler and scientific measurements highlights the importance of unit conversion in bridging fields and ensuring precise communication.