Problem 111 - Chemistry The Molecular Science

Question

It's final exam time and a student drinks a 1.93 -oz bottle of 5-Hour Energy \(\mathbb{B}\) to stay awake. The drink contains, among other substances, \(212 \mathrm{mg}\) caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\) (a) Calculate the mass percent nitrogen in caffeine. (b) Calculate the number of caffeine molecules that the student ingested. (c) Calculate the number of carbon atoms in this mass of caffeine. (d) An 8-oz cup of regular coffee contains approximately \(100 \mathrm{mg}\) caffeine. Calculate how many times greater the caffeine concentration (mg/oz) is in the 5 -Hour Energy \(^{\oplus}\) drink than in the regular coffee.

Step-by-Step Solution

Answer

(a) 28.86%, (b) \(6.58 \times 10^{20}\) molecules, (c) \(5.26 \times 10^{21}\) carbon atoms, (d) 8.79 times greater.

Step 1: Determine the molecular weight of caffeine

Caffeine (\(\mathrm{C}_{8}\mathrm{H}_{10}\mathrm{N}_{4}\mathrm{O}_{2}\)) consists of 8 carbon, 10 hydrogen, 4 nitrogen, and 2 oxygen atoms. Calculate the molecular weight by adding the respective weights: - Carbon: \(8 \times 12.01\,\mathrm{g/mol} = 96.08\,\mathrm{g/mol}\) - Hydrogen: \(10 \times 1.008\,\mathrm{g/mol} = 10.08\,\mathrm{g/mol}\)- Nitrogen: \(4 \times 14.01\,\mathrm{g/mol} = 56.04\,\mathrm{g/mol}\)- Oxygen: \(2 \times 16.00\,\mathrm{g/mol} = 32.00\,\mathrm{g/mol}\)Therefore, the molecular weight of caffeine is: \[96.08 + 10.08 + 56.04 + 32.00 = 194.20\,\mathrm{g/mol}\]

Step 2: Calculate the mass percent nitrogen in caffeine

The total mass of nitrogen in caffeine is \(56.04 \,\mathrm{g/mol}\). To find the mass percent of nitrogen:\[\text{%N} = \left(\frac{56.04}{194.20}\right) \times 100\% = 28.86\%\]

Step 3: Calculate the moles of caffeine ingested

The student ingests \(212 \,\mathrm{mg}\) of caffeine which is equivalent to \(0.212 \,\mathrm{g}\). Calculate the moles of caffeine:\[\text{moles of caffeine} = \frac{0.212}{194.20} = 0.001092 \,\mathrm{mol}\]

Step 4: Calculate the number of caffeine molecules ingested

Use Avogadro's number \(6.022 \times 10^{23}\) to find the molecules:\[0.001092 \,\mathrm{mol} \times 6.022 \times 10^{23} = 6.58 \times 10^{20} \text{ molecules of caffeine}\]

Step 5: Calculate the number of carbon atoms ingested

Each molecule of caffeine contains 8 carbon atoms. Therefore, the number of carbon atoms:\[6.58 \times 10^{20} \text{ molecules} \times 8 = 5.26 \times 10^{21} \text{ carbon atoms}\]

Step 6: Compare caffeine concentration (mg/oz)

For the 5-Hour Energy drink: \[\frac{212 \,\mathrm{mg}}{1.93 \,\mathrm{oz}} = 109.84 \,\mathrm{mg/oz}\]For coffee: \[\frac{100 \,\mathrm{mg}}{8 \,\mathrm{oz}} = 12.5 \,\mathrm{mg/oz}\]Therefore, the concentration ratio is:\[\frac{109.84}{12.5} = 8.79\]

Step 7: Final Results

(a) The mass percent of nitrogen is 28.86%. (b) The number of caffeine molecules ingested is \(6.58 \times 10^{20}\). (c) The number of carbon atoms ingested is \(5.26 \times 10^{21}\). (d) The caffeine concentration in the 5-Hour Energy drink is 8.79 times greater than in coffee.

Key Concepts

Molecular Weight Calculation

Molecular weight, also known as molecular mass, is the sum of the atomic masses of all the atoms in a molecule. To calculate the molecular weight of caffeine (\(\mathrm{C}_{8}\mathrm{H}_{10}\mathrm{N}_{4}\mathrm{O}_{2}\)), we add the atomic masses of carbon, hydrogen, nitrogen, and oxygen according to their respective numbers in the molecule:

• Carbon (C): 8 atoms, \(8 \times 12.01 \,\mathrm{g/mol} = 96.08 \,\mathrm{g/mol}\)
• Hydrogen (H): 10 atoms, \(10 \times 1.008 \,\mathrm{g/mol} = 10.08 \,\mathrm{g/mol}\)
• Nitrogen (N): 4 atoms, \(4 \times 14.01 \,\mathrm{g/mol} = 56.04 \,\mathrm{g/mol}\)
• Oxygen (O): 2 atoms, \(2 \times 16.00 \,\mathrm{g/mol} = 32.00 \,\mathrm{g/mol}\)

By adding up all these values, the molecular weight of caffeine becomes \(194.20 \,\mathrm{g/mol}\). Understanding molecular weight is crucial because it serves as a foundation for further calculations, such as determining moles and mass percentages.

Mass Percent Composition

Mass percent composition tells us the proportion by mass of each element in a compound. It's useful for understanding how much of a particular element is present in a compound. For caffeine, calculating the mass percent of nitrogen involves comparing the mass of nitrogen in one mole of caffeine to the total molecular weight of caffeine. Caffeine contains 4 nitrogen atoms, which contribute \(56.04 \,\mathrm{g/mol}\) to its total molecular weight of \(194.20 \,\mathrm{g/mol}\).To find the mass percent of nitrogen, you use the formula:\[\text{%N} = \left( \frac{\text{mass of nitrogen}}{\text{molecular weight of caffeine}} \right) \times 100\%\]Substituting in the values:\[\text{%N} = \left( \frac{56.04}{194.20} \right) \times 100\% = 28.86\%\]This means that 28.86 percent of the mass of caffeine is made up of nitrogen. Understanding mass percent composition helps when analyzing chemicals and determining the presence of specific elements within compounds.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry representing the number of atoms, molecules, or ions in one mole of a substance. This value is approximately \(6.022 \times 10^{23}\). It's important because it allows us to convert between the number of particles and the amount of substance in moles.To calculate how many caffeine molecules were ingested, first determine the moles of caffeine. Given 212 mg of caffeine is consumed, which is 0.212 g, the amount in moles is calculated as:\[\frac{0.212 \,\mathrm{g}}{194.20 \,\mathrm{g/mol}} = 0.001092 \,\mathrm{mol}\]Using Avogadro’s number, the number of molecules is:\[0.001092 \,\mathrm{mol} \times 6.022 \times 10^{23} = 6.58 \times 10^{20} \text{ molecules}\]This calculation enables chemists to understand the scale of reactions and interactions in terms of individual molecules.

Concentration Comparison

Concentration comparison helps us determine the intensity or strength of a solution or mixture. Specifically, in this scenario, we compare the concentration of caffeine between two beverages: 5-Hour Energy drink and regular coffee. Concentration is expressed in milligrams of caffeine per ounce (mg/oz).For the 5-Hour Energy drink with 212 mg of caffeine in 1.93 ounces, the calculation is:\[\frac{212 \,\mathrm{mg}}{1.93 \,\mathrm{oz}} = 109.84 \,\mathrm{mg/oz}\]For regular coffee, with 100 mg of caffeine in 8 ounces, the concentration is:\[\frac{100 \,\mathrm{mg}}{8 \,\mathrm{oz}} = 12.5 \,\mathrm{mg/oz}\]The comparison of caffeine concentration between these two beverages is then:\[\frac{109.84}{12.5} = 8.79\]This means the 5-Hour Energy drink has a caffeine concentration 8.79 times that of regular coffee, making it significantly more caffeinated per ounce.