Problem 83 - Chemistry: The Central Science

Question

(a) Diamond is a natural form of pure carbon. How many moles of carbon are in a 1.25-carat diamond (1 carat \(=0.200 \mathrm{~g}\) )? How many atoms are in this diamond? (b) The molecular formula of acetylsalicylic acid (aspirin), one of the most common pain relievers, is \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\). How many moles of \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\) are in a \(0.500-\mathrm{g}\) tablet of aspirin? How many molecules of \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\) are in this tablet?

Step-by-Step Solution

Answer

(a) There are 0.0208 moles of carbon in a 1.25-carat diamond, which contains 1.25 × 10^22 carbon atoms. (b) In a 0.500-g tablet of aspirin, there are 0.00277 moles of C9H8O4, with a total of 1.67 × 10^21 molecules of C9H8O4.

Step 1: Convert carats to grams

First, let's convert the weight of the diamond from carats to grams. We are given that 1 carat = 0.200 g. Thus, 1.25 carats * 0.200 g/carat = 0.250 g

Step 2: Calculate the number of moles

Now, we can calculate the number of moles of carbon. The molar mass of carbon is 12.01 g/mol. So, the number of moles is: Number of moles = (0.250 g) / (12.01 g/mol) = 0.0208 mol

Step 3: Calculate the number of atoms

Finally, we can find the number of carbon atoms in the diamond. To do this, we'll multiply the number of moles by Avogadro's number (6.022 × 10^23): Number of atoms = (0.0208 mol) * (6.022 × 10^23 atoms/mol) = 1.25 × 10^22 atoms (b)

Step 1: Calculate the molar mass of acetylsalicylic acid

First, we need to calculate the molar mass of C9H8O4 based on its molecular formula. The molar masses of carbon (C), hydrogen (H), and oxygen (O) are 12.01 g/mol, 1.01 g/mol, and 16.00 g/mol, respectively. So, the molar mass of C9H8O4 is: (9 × 12.01 + 8 × 1.01 + 4 × 16.00) g/mol = 180.15 g/mol

Step 2: Calculate the number of moles

We are given that the mass of the aspirin tablet is 0.500 g. Then we can find the number of moles of C9H8O4 in the tablet using the molar mass: Number of moles = (0.500 g) / (180.15 g/mol) = 0.00277 mol

Step 3: Calculate the number of molecules

Finally, we can find the number of molecules of C9H8O4 in the aspirin tablet. To do this, we'll multiply the number of moles by Avogadro's number (6.022 × 10^23): Number of molecules = (0.00277 mol) * (6.022 × 10^23 molecules/mol) = 1.67 × 10^21 molecules

Key Concepts

Molar Mass

Molar mass represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). To find the molar mass, you simply add up the atomic masses of all the atoms in a molecule. For example, when calculating the molar mass of acetylsalicylic acid (aspirin) with the molecular formula \( C_9H_8O_4 \), you need to account for the molar masses of carbon (12.01 g/mol), hydrogen (1.01 g/mol), and oxygen (16.00 g/mol).

• Carbon contributes: \(9 \times 12.01 = 108.09\ g/mol\)
• Hydrogen contributes: \(8 \times 1.01 = 8.08\ g/mol\)
• Oxygen contributes: \(4 \times 16.00 = 64.00\ g/mol\)

Adding these values gives the molar mass of aspirin, \(180.15\ g/mol\). Calculating molar mass is crucial for conversions between grams and moles, which is often needed to understand the quantity of a substance in chemical reactions.

Avogadro's Number

Avogadro's number provides a fundamental link between the macroscopic and atomic scales by defining the number of entities (atoms, molecules, ions) in one mole of any substance. This constant value is approximately \(6.022 \times 10^{23}\).
When you have a quantity expressed in moles, multiplying by Avogadro's number will give you the exact count of atoms or molecules. For instance, if you have \(0.0208\ moles\) of carbon, the number of carbon atoms would be:

• \((0.0208\ \text{mol}) \times (6.022 \times 10^{23} \ \text{atoms/mol}) = 1.25 \times 10^{22}\ \text{atoms}\)

Avogadro's number is vital for navigating between mass and the microscopic count of atoms and molecules in chemistry.

Carbon Atoms

Understanding carbon atoms is essential because carbon is a fundamental element in organic chemistry and forms the basis of all known life. In practice, determining the number of carbon atoms in various substances helps in understanding material properties, chemical equations, and more.
Consider a diamond, which is a pure form of carbon. If you weigh a diamond and use molar mass to find the moles, Avogadro's number allows you to calculate the precise number of carbon atoms. This is particularly relevant in both academic exercises and real-world applications, such as gemstone verification and carbon footprint studies.

Acetylsalicylic Acid

Acetylsalicylic acid, commonly known as aspirin, is a widely used medication for pain relief and inflammation reduction. With the molecular formula \(C_9H_8O_4\), aspirin encapsulates nine carbon atoms, eight hydrogen atoms, and four oxygen atoms.
To understand the amount of aspirin in a given sample, you'll need to calculate moles. For example, a tablet weighing \(0.500\ g\) translates to approximately \(0.00277\ mol\) of acetylsalicylic acid using its molar mass. This conversion is key in pharmaceutical purposes, dosing strength determination, and in quality control of medication production.

Molecular Formula

A molecular formula represents the number and type of atoms present in a molecule. This notation is essential for understanding the composition and properties of chemical substances.
For example, the molecular formula \(C_9H_8O_4\) shows that acetylsalicylic acid has a specific arrangement of carbon, hydrogen, and oxygen atoms. The subscripts indicate the quantity of each type of atom, which directly correlates to molar mass calculations and stoichiometric conversions in chemical reactions.
By understanding molecular formulas, one can predict the reactivity, solubility, and other physical and chemical properties of substances, making it indispensable in both theoretical and applied chemistry.