What amount (moles) is represented by each of these samples? a. \(20.0 \mathrm{mg}\) caffeine, \(\overline{\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}}\) b. \(2.72 \times 10^{21}\) molecules of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OH}\) c. \(1.50 \mathrm{~g}\) of dry ice, \(\mathrm{CO}_{2}\)

To find the molar mass of each compound, we need to add the molar masses of all the elements present in one molecule of the compound. The molar mass of each element can be found in the periodic table. a. Caffeine, \(\overline{\mathrm{C}_{8}\mathrm{H}_{10}\mathrm{N}_{4}\mathrm{O}_{2}}\): Molar mass = 8 (molar mass of Carbon) + 10 (molar mass of Hydrogen) + 4 (molar mass of Nitrogen) + 2 (molar mass of Oxygen) = \(8\cdot 12.01 + 10\cdot 1.01 + 4\cdot 14.01 + 2\cdot 16.00 = 194.19~\mathrm{g/mol}.\) b. Ethanol, \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{OH}\): Molar mass = 2 (molar mass of Carbon) + 6 (molar mass of Hydrogen) + 1 (molar mass of Oxygen) = \(2\cdot 12.01 + 6\cdot 1.01 + 1\cdot 16.00 = 46.07~\mathrm{g/mol}.\) c. Dry ice (solid CO\(_{2}\)): Molar mass = 1 (molar mass of Carbon) + 2 (molar mass of Oxygen) = \(1\cdot 12.01 + 2\cdot 16.00 = 44.01~\mathrm{g/mol}.\)

To find the amount of substance in moles for samples with given mass, we use the formula: Moles = \( \frac{\text{Mass of sample}}{\text{Molar mass of the compound}} \) a. For the caffeine sample: Moles = \( \frac{20.0~\mathrm{mg}}{194.19~\mathrm{g/mol}} = \frac{0.0200~\mathrm{g}}{194.19~\mathrm{g/mol}} = 1.03\times 10^{-4}~\mathrm{mol}\) c. For the dry ice sample: Moles = \( \frac{1.50~\mathrm{g}}{44.01~\mathrm{g/mol}} \ = 0.0341~\mathrm{mol} \)

To find the amount of substance in moles for samples with given number of molecules, we use the formula: Moles = \( \frac{\text{Number of molecules}}{\text{Avogadro's number}} \) b. For the ethanol sample: Moles = \( \frac{2.72\times 10^{21}~\text{molecules}}{6.022\times 10^{23}~\text{molecules}\cdot\text{mol}^{-1}} = 4.52\times 10^{-3}~\mathrm{mol}\) So, the amount in moles for each sample is: a. Caffeine: \(1.03\times 10^{-4}~\mathrm{mol}\) b. Ethanol: \(4.52\times 10^{-3}~\mathrm{mol}\) c. Dry ice (CO\(_{2}\)): \(0.0341~\mathrm{mol}\)