Question

What amount (moles) is represented by each of these samples? a. \(20.0 \mathrm{mg}\) caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\) 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, \(\mathrm{CO}_{2}\)

Short answer

The number of moles for each sample is: a. \(1.03 \times 10^{-4}\, \mathrm{mol}\) of caffeine. b. \(0.00452\,\mathrm{mol}\) of ethanol. c. \(0.0341\,\mathrm{mol}\) of dry ice.

Step-by-step solution

Calculate the molar mass of each compound

First, we need to find the molar mass of caffeine, ethanol, and CO₂. Molar mass of caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4}\mathrm{O}_{2}\): \(= 8 \times 12.01\,\mathrm{(C)} + 10 \times 1.01\,\mathrm{(H)} + 4 \times 14.01\,\mathrm{(N)} + 2 \times 16\,\mathrm{(O)}\) \(= 194.19 \,\,\mathrm{g/mol}\) Molar mass of ethanol, \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}\): \(= 2 \times 12.01\,\mathrm{(C)} + 6 \times 1.01\,\mathrm{(H)} + 1 \times 16.00\,\mathrm{(O)}\) \(= 46.07\,\,\mathrm{g/mol}\) Molar mass of dry ice, \(\mathrm{CO}_{2}\): \(= 1 \times 12.01\,\mathrm{(C)} + 2 \times 16.00\,\mathrm{(O)}\) \(= 44.01\,\,\mathrm{g/mol}\)

Calculate the moles for samples (a) and (c) using mass and molar mass

a. 20.0 mg of caffeine. First, convert mg to g: \(20.0 \, \mathrm{mg} = 0.020 \, \mathrm{g}\) Next, calculate the moles: \(\frac{0.020\,\mathrm{g}}{194.19\,\mathrm{g/mol}} = 1.03 \times 10^{-4}\, \mathrm{mol}\) c. 1.50 g of dry ice. Calculate the moles: \(\frac{1.50\,\mathrm{g}}{44.01\,\mathrm{g/mol}} = 0.0341\, \mathrm{mol}\)

Calculate moles for sample (b) using Avogadro's number

b. \(2.72 \times 10^{21}\) molecules of ethanol. Use Avogadro's number, \(6.022 \times 10^{23}\,\mathrm{mol}^{-1}\): \(\frac{2.72 \times 10^{21} \,\mathrm{molecules}}{6.022 \times 10^{23} \,\mathrm{molecules/mol}} = 0.00452\,\mathrm{mol}\) Now we have the number of moles for each sample: a. \(1.03 \times 10^{-4}\, \mathrm{mol}\) of caffeine. b. \(0.00452\,\mathrm{mol}\) of ethanol. c. \(0.0341\,\mathrm{mol}\) of dry ice.

Key concepts

Molar Mass

Molar mass is an essential concept in chemistry that defines the mass of one mole of a given substance, measured in grams per mole (g/mol). Understanding molar mass allows for conversions between mass and moles.

• For complex molecules, it determines how much a mole of that entire substance weighs by considering the atomic masses of each element within it.
• To find the molar mass, multiply the atomic mass of each element by its number in the formula and sum them all up.

In the calculation step provided, caffeine (\(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4}\mathrm{O}_{2}\)) was calculated as follows:

• Multiply the number of carbon atoms by their atomic mass: \(8 \times 12.01\)
• Follow the same process for hydrogen, nitrogen, and oxygen.
• Add them to get \(194.19 \,\mathrm{g/mol}\)

This principle applies the same way for ethanol and carbon dioxide as well, revealing the mass of each mole of these substances.

Avogadro's Number

Avogadro's number is a fundamental constant, denoted as \(6.022 \times 10^{23}\) \\(\mathrm{molecules/mol}\.\) It represents the number of constituent particles, usually atoms or molecules, in one mole of a substance.

• It serves as a bridge between the atomic scale and macroscopic amounts of material.
• Using Avogadro's number, you can convert from moles to number of particles and vice versa.

In our example of ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5}\mathrm{OH}\)), we converted the provided number of molecules to moles by dividing by Avogadro's number.

• This gives an understanding of the quantity of substance at both the unit (mole) and molecular (particles) level.

This conversion is crucial in determining the mole equivalent of a given sample at a molecular level.

Molecular Weight

Molecular weight is often interchanged with the term molar mass but slightly differs as it usually refers to the mass of a single molecule of a compound, expressed in atomic mass units (amu). However, in chemistry calculations, especially in mole calculations, they can be used interchangeably since they numerically indicate the same magnitude.

• Understanding molecular weight helps in computing molar mass, which is essential for conversions in chemistry.
• Every compound has its unique molecular weight based on its atomic constitution.

For instance, the molecular weight of caffeine is calculated similarly as in the molar mass, which is the sum of weights of all atoms within a single molecule of caffeine.

Chemical Compounds

Chemical compounds are substances made from two or more different elements that are chemically bonded together. Understanding chemical compounds is important for predicting the properties and reactions of substances.

• Each compound has a definite proportion of elements. For instance, water (\(\mathrm{H_{2}O}\)) will always have two hydrogens for every oxygen.
• The composition of a compound affects its properties and how it reacts with other substances.

In the examples given, we have compounds like caffeine, ethanol, and carbon dioxide. Each has a distinct set of properties and uses:

• Caffeine is a stimulant found in coffee and tea.
• Ethanol is an alcohol, often used in drinks and disinfectants.
• Carbon dioxide, known as dry ice in solid form, is used in many cooling applications.

Understanding these compounds also aids in comprehending complex reactions and interactions in various contexts like industrial processes and biological systems.