Problem 66 - Chemistry An Atoms First Approach

Question

What amount (moles) is represented by each of these samples? a. 20.0 mg caffeine, \(C_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\) b. \(2.72 \times 10^{21}\) molecules of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) c. \(1.50 \mathrm{g}\) of dry ice, \(\mathrm{CO}_{2}\)

Step-by-Step Solution

Answer

The moles of each sample are: a. \(1.03 \times 10^{-4} \,-\! \mathrm{mol \, caffeine}\) b. \(4.52 \times 10^{-3} \,-\! \mathrm{mol \, ethanol}\) c. \(0.0341 \,-\! \mathrm{mol \, CO_{2}}\)

Step 1: Calculate the molar mass of caffeine, ethanol, and carbon dioxide.

Use molecular weights for each element and add them to find the molar mass of each compound: Molar mass of caffeine: \(C_{8}H_{10}N_{4}O_{2} = 8 \times 12.01 + 10 \times 1.01 + 4 \times 14.01 + 2 \times 16.00 = 194.19 \,\mathrm{g/mol}\) Molar mass of ethanol: \(C_{2}H_{5}OH = 2 \times 12.01 + 6 \times 1.01 + 1 \times 16.00 = 46.07 \,\mathrm{g/mol}\) Molar mass of carbon dioxide: \(CO_{2} = 1 \times 12.01 + 2 \times 16.00 = 44.01 \,\mathrm{g/mol}\)

Step 2: Calculate moles of each sample.

For each sample, divide the mass (or number of molecules for ethanol) by the corresponding molar mass or Avogadro's number: a. Moles of caffeine: \(20.0 \,\mathrm{mg} \, \times \, \frac{1 \,\mathrm{g}}{1000 \,\mathrm{mg}} \, \times \, \frac{1 \,-\! \mathrm{mol \, caffeine}}{194.19 \,\mathrm{g}} = 1.03 \times 10^{-4} \,\mathrm{mol \, caffeine}\) b. Moles of ethanol: \(\frac{2.72 \times 10^{21} \, \mathrm{molecules}}{6.022 \times 10^{23} \, \mathrm{molecules/mol}} = 4.52 \times 10^{-3} \,\mathrm{mol \, ethanol}\) c. Moles of dry ice: \(\frac{1.50 \,\mathrm{g}}{44.01 \,\mathrm{g/mol}} = 0.0341 \,\mathrm{mol \, CO_{2}}\) So, the amounts of each sample in moles are: a. \(1.03 \times 10^{-4} \,-\! \mathrm{mol \, caffeine}\) b. \(4.52 \times 10^{-3} \,-\! \mathrm{mol \, ethanol}\) c. \(0.0341 \,-\! \mathrm{mol \, CO_{2}}\)

Key Concepts

Caffeine

Caffeine is the bitter substance that we frequently find in coffee, tea, and energy drinks. It is a stimulant that affects the central nervous system, helping us feel more awake and less tired. If you have ever wondered about the chemical composition of caffeine, it is represented by the formula \(C_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\).

To calculate the molar mass of caffeine, you'll need to consider the individual atomic masses from the periodic table. Here's how it works:

• Carbon (C): 12.01 g/mol
• Hydrogen (H): 1.01 g/mol
• Nitrogen (N): 14.01 g/mol
• Oxygen (O): 16.00 g/mol

By adding the masses together for each element in the formula:
8 Carbon atoms \(\times\) 12.01, 10 Hydrogen atoms \(\times\) 1.01, 4 Nitrogen atoms \(\times\) 14.01, and 2 Oxygen atoms \(\times\) 16.00, you get a total of 194.19 g/mol.

Understanding this molar mass is essential for calculating how many moles exist in a given sample size. When we look at 20.0 mg of caffeine, converting that milligram measurement to grams first is crucial. Then using the formula: \[\text{moles} = \frac{\text{mass in grams}}{\text{molar mass}} \]you find it contains \(1.03 \times 10^{-4}\) mol of caffeine. This small amount showcases the precision chemistry requires.

Ethanol

Ethanol, often simply called alcohol, is a colorless, volatile liquid that is a key ingredient in alcoholic beverages. Beyond being a recreational substance, it's also a vital component in the chemical industry, serving as a solvent and a fuel additive.

The chemical formula for ethanol is \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\). To uncover its molar mass, you evaluate the atoms:

• Carbon (C), with two atoms: each weighing 12.01 g/mol
• Hydrogen (H), with six atoms: each weighing 1.01 g/mol
• Oxygen (O): weighing 16.00 g/mol

This gives a final molar mass of 46.07 g/mol when calculated collectively.

However, for ethanol, instead of a mass, you may encounter mole-related problems concerning the number of molecules. If given a number of ethanol molecules, for instance, \(2.72 \times 10^{21}\) molecules, the calculation to find moles utilizes Avogadro's number. Avogadro's number \(6.022 \times 10^{23}\) tells us how many particles are in one mole of a substance.

By dividing the total number of ethanol molecules by Avogadro's number: \[\text{moles} = \frac{2.72 \times 10^{21}}{6.022 \times 10^{23}} \]you determine it equals \(4.52 \times 10^{-3}\) moles of ethanol. This highlights the methodology for switching from a microscopic view of chemicals to their macroscopic measurements.

Carbon Dioxide

Carbon Dioxide, chemically noted as \(\mathrm{CO}_{2}\), is a simple molecule consisting of one carbon atom double-bonded to two oxygen atoms. It's famously known as the gas we exhale and one of the greenhouse gases contributing to global warming.

To find the molar mass of carbon dioxide, you sum the atomic masses of its elements:

• One Carbon atom: 12.01 g/mol
• Two Oxygen atoms: each 16.00 g/mol

This results in a molar mass of 44.01 g/mol.

Carbon dioxide is commonly found as a gas at room temperature, but solid carbon dioxide, known as "dry ice," offers a unique study case. Suppose you have 1.50 grams of dry ice. Then, to find out how many moles this mass corresponds to, apply the following formula: \[\text{moles} = \frac{\text{mass in grams}}{\text{molar mass}} \]Plugging in for \(\frac{1.50}{44.01}\), we discover it equates to 0.0341 moles of \(\mathrm{CO}_{2}\).

This demonstrates how important mole calculations are in converting between the mass of a substance and its quantity on a molecular level. Moles help bridge the microscopic and macroscopic worlds in chemistry.