Problem 42 - Chemistry The Central Science

Question

At least \(25 \mu \mathrm{g}\) of tetrahydrocannabinol \((\mathrm{THC}),\) the active ingredient in marijuana, is required to produce intoxication. The molecular formula of \(\mathrm{THC}\) is \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\). How many moles of THC does this 25 \mug represent? How many molecules? (b) Caffeine, a stimulant found in coffee, contains \(49.5 \% \mathrm{C}\), \(5.15 \% \mathrm{H}, 28.9 \% \mathrm{~N},\) and \(16.5 \% \mathrm{O}\) by mass and has a molar mass of \(195 \mathrm{~g} / \mathrm{mol}\). (c) Monosodium glutamate (MSG), a flavor enhancer in certain foods, contains \(35.51 \% \mathrm{C}, 4.77 \% \mathrm{H}, 37.85 \% \mathrm{O},\) \(8.29 \% \mathrm{~N},\) and \(13.60 \% \mathrm{Na},\) and has a molar mass of \(169 \mathrm{~g} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The number of moles of THC in 25 μg is \(7.95 \times 10^{-8}\, mol\), and the number of molecules is \(4.79 \times 10^{16}\, molecules\). The molecular formula of caffeine is \(C_{8}H_{10}N_{4}O_{2}\), and the molecular formula of MSG is \(C_{5}H_{8}O_{4}N_{1}Na_{1}\).

Step 1: Convert micrograms to grams

To convert 25 μg of THC to grams, we will use the conversion factor: \(1 g = 10^6 μg\) \(25\,μg \times \frac{1\,g}{10^6\,μg} = 25 \times 10^{-6}\,g\)

Step 2: Calculate the molar mass of THC

Molecular formula of THC: \(C_{21}H_{30}O_{2}\) Molar mass of THC: \(21 \times (12.01\,g/mol_C) + 30 \times (1.01\,g/mol_H) + 2 \times (16.00\,g/mol_O) = 314.47\,g/mol\)

Step 3: Calculate the moles of THC

Number of moles: \(\frac{Mass}{Molar\,mass}\) Number of moles = \( \frac{25 \times 10^{-6}\,g}{314.47\,g/mol} = 7.95 \times 10^{-8}\, mol\)

Step 4: Calculate the number of molecules of THC

Number of molecules = moles × Avogadro's number Number of molecules = \(7.95 \times 10^{-8}\,mol \times 6.022 \times 10^{23}\,molecules/mol = 4.79 \times 10^{16}\,molecules\) For problem (b):

Step 1: Calculate moles of each element in 1 mole of caffeine

C: \( \frac{49.5\% \times 195\,g/mol}{12.01\,g/mol_C} = 8\,mol \) H: \( \frac{5.15\% \times 195\,g/mol}{1.01\,g/mol_H} = 10\,mol \) N: \( \frac{28.9\% \times 195\,g/mol}{14.01\,g/mol_N} = 4\,mol \) O: \( \frac{16.5\% \times 195\,g/mol}{16.00\,g/mol_O} = 2\,mol \)

Step 2: Deduce the molecular formula of caffeine

Molecular formula of caffeine: \(C_8H_{10}N_4O_2\) For problem (c):

Step 1: Calculate moles of each element in 1 mole of MSG

C: \( \frac{35.51\% \times 169\,g/mol}{12.01\,g/mol_C} = 5\,mol \) H: \( \frac{4.77\% \times 169\,g/mol}{1.01\,g/mol_H} = 8\,mol \) O: \( \frac{37.85\% \times 169\,g/mol}{16.00\,g/mol_O} = 4\,mol \) N: \( \frac{8.29\% \times 169\,g/mol}{14.01\,g/mol_N} = 1\,mol \) Na: \( \frac{13.60\% \times 169\,g/mol}{22.99\,g/mol_{Na}} = 1\,mol \)

Step 2: Deduce the molecular formula of MSG

Molecular formula of MSG: \(C_5H_8O_4N_1Na_1\)

Key Concepts

Molar Mass Determination

Understanding the molar mass of a substance is fundamental in chemistry, allowing us to convert between the mass of a substance and the number of moles, an essential step for quantitative analysis. The molar mass is defined as the mass of one mole of a substance, typically expressed in grams per mole (g/mol).

To determine the molar mass of a compound, like tetrahydrocannabinol (THC), you sum the molar masses of its constituent atoms according to its molecular formula. For instance, THC's molecular formula is \(C_{21}H_{30}O_{2}\), so its molar mass calculation involves multiplying the number of each atom by its atomic mass (from the periodic table) and summing those values together:

\[Molar\ mass\ of\ THC = 21 \times (12.01\ g/mol_C) + 30 \times (1.01\ g/mol_H) + 2 \times (16.00\ g/mol_O) = 314.47\ g/mol\]
With the molar mass known, it's possible to answer how many moles exist in a given mass of the compound — vital for tasks such as reacting proportions in stoichiometry or calculating concentrations.

Avogadro's Number Application

Avogadro's number, \(6.022 \times 10^{23}\) entities per mole, is a cornerstone of chemistry because it bridges the atomic scale with the macroscopic world. This immense number tells us how many atoms, ions, molecules, or other particles are present in one mole of a substance.

Applying Avogadro's number is crucial when determining the number of molecules in a sample. As seen in the THC example, once we know the number of moles (\(7.95 \times 10^{-8} \ mol\)), we multiply this by Avogadro's number to find the total number of molecules:

\[Number\ of\ molecules = 7.95 \times 10^{-8}\,mol \times 6.022 \times 10^{23}\,molecules/mol = 4.79 \times 10^{16}\,molecules\]
This application of Avogadro's number is especially important when dealing with chemical reactions at the molecular level, as it helps determine how many molecules interact.

Mass Percent Composition

The mass percent composition of an element in a compound represents the fraction of the total mass of the compound that is due to that particular element. It's expressed as a percentage and is vital for deducing the empirical formula and the molecular formula of a substance from experimental data.

In problems like the caffeine and monosodium glutamate (MSG) analysis, knowing the mass percent composition allows you to calculate the moles of each element in one mole of the compound:
\

• For caffeine, starting with a molar mass of \(195\ g/mol\), we use the mass percent and atomic mass to find the moles of each element in a mole of caffeine.
• For MSG, we apply the same method, using its molar mass of \(169\ g/mol\) and the mass percentages for each element.

These calculations are the first step towards finding the molecular formula, an essential piece of information for understanding a compound's chemical properties and behavior during reactions.