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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

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

Step by step solution

01

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:\[ 2 \text{ carats} \times 200 \text{ mg/carats} = 400 \text{ mg} \]
02

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:\[ 400 \text{ mg} = \frac{400}{1000} \text{ g} = 0.4 \text{ g} \]
03

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:\[ \text{Number of moles} = \frac{0.4 \text{ g}}{12.01 \text{ g/mol}} \approx 0.0333 \text{ moles} \]
04

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:\[ \text{Number of atoms} = 0.0333 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 2.005 \times 10^{22} \text{ atoms} \]
05

Final Answer

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

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Key Concepts

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

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.

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Most popular questions from this chapter

Propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) is a minor component of natural gas and is used in domestic cooking and heating. (a) Balance the following equation representing the combustion of propane in air: $$ \mathrm{C}_{3} \mathrm{H}_{8}+\mathrm{O}_{2} \longrightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} $$ (b) How many grams of carbon dioxide can be produced by burning 3.65 mol of propane? Assume that oxygen is the excess reactant in this reaction.

Mustard gas \(\left(\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{Cl}_{2} \mathrm{~S}\right)\) is a poisonous gas that was used in World War I and banned afterward. It causes general destruction of body tissues, resulting in the formation of large water blisters. There is no effective antidote. Calculate the percent composition by mass of the elements in mustard gas.

The molar mass of caffeine is \(194.19 \mathrm{~g}\). Is the molecular formula of caffeine \(\mathrm{C}_{4} \mathrm{H}_{5} \mathrm{~N}_{2} \mathrm{O}\) or \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2} ?\)

When \(0.273 \mathrm{~g}\) of \(\mathrm{Mg}\) is heated strongly in a nitrogen \(\left(\mathrm{N}_{2}\right)\) atmosphere, a chemical reaction occurs. The product of the reaction weighs \(0.378 \mathrm{~g}\). Calculate the empirical formula of the compound containing \(\mathrm{Mg}\) and \(\mathrm{N}\). Name the compound.

Hydrogen fluoride is used in the manufacture of Freons (which destroy ozone in the stratosphere) and in the production of aluminum metal. It is prepared by the reaction $$ \mathrm{CaF}_{2}+\mathrm{H}_{2} \mathrm{SO}_{4} \longrightarrow \mathrm{CaSO}_{4}+2 \mathrm{HF} $$ In one process, \(6.00 \mathrm{~kg}\) of \(\mathrm{CaF}_{2}\) is treated with an excess of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) and yields \(2.86 \mathrm{~kg}\) of \(\mathrm{HF}\). Calculate the percent yield of HF.

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