Problem 80 - Chemistry and Chemical Reactivity

Question

Calculate the mass percent of each element in the following compounds: (a) \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{2} \mathrm{O}_{2},\) caffeine (b) \(\mathrm{C}_{10} \mathrm{H}_{20} \mathrm{O},\) menthol (c) \(\mathrm{CoCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}\)

Step-by-Step Solution

Answer

(a) C: 57.82%, H: 6.06%, N: 16.86%, O: 19.26%. (b) C: 76.87%, H: 12.90%, O: 10.23%. (c) Co: 24.76%, Cl: 29.80%, Water: 45.44%.

Step 1: Determine Molar Mass of Caffeine

First, calculate the molar mass of caffeine, \( \mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{2} \mathrm{O}_{2} \).- Carbon (C): 8 atoms × 12.01 g/mol = 96.08 g/mol- Hydrogen (H): 10 atoms × 1.008 g/mol = 10.08 g/mol- Nitrogen (N): 2 atoms × 14.01 g/mol = 28.02 g/mol- Oxygen (O): 2 atoms × 16.00 g/mol = 32.00 g/molAdding these gives the molar mass of caffeine: \(96.08 + 10.08 + 28.02 + 32.00 = 166.18 \text{ g/mol}\).

Step 2: Calculate Mass Percent in Caffeine

Mass percent is calculated as \( \frac{\text{mass of element in 1 mol}}{\text{molar mass of compound}} \times 100\% \). For caffeine:- Carbon: \( \frac{96.08}{166.18} \times 100\% = 57.82\% \)- Hydrogen: \( \frac{10.08}{166.18} \times 100\% = 6.06\% \)- Nitrogen: \( \frac{28.02}{166.18} \times 100\% = 16.86\% \)- Oxygen: \( \frac{32.00}{166.18} \times 100\% = 19.26\% \)

Step 3: Determine Molar Mass of Menthol

Calculate the molar mass of menthol, \( \mathrm{C}_{10} \mathrm{H}_{20} \mathrm{O} \).- Carbon (C): 10 atoms × 12.01 g/mol = 120.10 g/mol- Hydrogen (H): 20 atoms × 1.008 g/mol = 20.16 g/mol- Oxygen (O): 1 atom × 16.00 g/mol = 16.00 g/molTotal molar mass of menthol: \(120.10 + 20.16 + 16.00 = 156.26 \text{ g/mol}\).

Step 4: Calculate Mass Percent in Menthol

For menthol, calculate mass percent as follows:- Carbon: \( \frac{120.10}{156.26} \times 100\% = 76.87\% \)- Hydrogen: \( \frac{20.16}{156.26} \times 100\% = 12.90\% \)- Oxygen: \( \frac{16.00}{156.26} \times 100\% = 10.23\% \)

Step 5: Determine Molar Mass of CoCl2·6H2O

Calculate the molar mass of \( \mathrm{CoCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} \):- Cobalt (Co): 1 atom × 58.93 g/mol = 58.93 g/mol- Chlorine (Cl): 2 atoms × 35.45 g/mol = 70.90 g/mol- Water (H2O): 6 molecules × 18.02 g/mol = 108.12 g/molThe total molar mass is: \(58.93 + 70.90 + 108.12 = 237.95 \text{ g/mol}\).

Step 6: Calculate Mass Percent in CoCl2·6H2O

For \( \mathrm{CoCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} \), compute the mass percent:- Cobalt: \( \frac{58.93}{237.95} \times 100\% = 24.76\% \)- Chlorine: \( \frac{70.90}{237.95} \times 100\% = 29.80\% \)- Water (H2O): \( \frac{108.12}{237.95} \times 100\% = 45.44\% \)

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry used to help determine the mass percent of each element in a compound. It is the total mass of all the atoms in a given molecule, usually expressed in grams per mole (g/mol). To calculate the molar mass, multiply the atomic mass of each element by the number of atoms of that element in the compound, then add all these masses together.
For instance, with caffeine (\( \mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{2} \mathrm{O}_{2}\)), the molar mass is calculated by adding the contributions of each element:

• Carbon (C): \( 8 \times 12.01 \text{ g/mol} = 96.08 \text{ g/mol}\)
• Hydrogen (H): \( 10 \times 1.008 \text{ g/mol} = 10.08 \text{ g/mol}\)
• Nitrogen (N): \( 2 \times 14.01 \text{ g/mol} = 28.02 \text{ g/mol}\)
• Oxygen (O): \( 2 \times 16.00 \text{ g/mol} = 32.00 \text{ g/mol}\)

When you sum these values, you obtain the total molar mass: \( 96.08 + 10.08 + 28.02 + 32.00 = 166.18 \text{ g/mol}\). Understanding molar mass helps in determining how much of each element is present in a compound.

Caffeine Composition

Caffeine, a common stimulant found in coffee and tea, is composed of Carbon, Hydrogen, Nitrogen, and Oxygen with the formula \( \mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{2} \mathrm{O}_{2}\). Knowing how to break down its composition can provide insights into its chemical behavior. Let's look at how we determine the mass percent of each element:
The mass percent is calculated by dividing the total mass of each element by the molar mass of the entire compound and multiplying by 100. For caffeine:

• Percent Carbon: \( \frac{96.08}{166.18} \times 100\% = 57.82\%\)
• Percent Hydrogen: \( \frac{10.08}{166.18} \times 100\% = 6.06\%\)
• Percent Nitrogen: \( \frac{28.02}{166.18} \times 100\% = 16.86\%\)
• Percent Oxygen: \( \frac{32.00}{166.18} \times 100\% = 19.26\%\)

By understanding these percentages, scientists and students can estimate the contribution of each element to caffeine's properties and its role in biological systems.

Menthol Composition

Menthol, with the molecular formula \( \mathrm{C}_{10} \mathrm{H}_{20} \mathrm{O} \), is known for its minty cooling effect in various products. To analyze its composition, starting with the calculation of molar mass is key.
The molar mass of menthol is calculated as follows:

• Carbon (C): \( 10 \times 12.01 \text{ g/mol} = 120.10 \text{ g/mol}\)
• Hydrogen (H): \( 20 \times 1.008 \text{ g/mol} = 20.16 \text{ g/mol}\)
• Oxygen (O): \( 1 \times 16.00 \text{ g/mol} = 16.00 \text{ g/mol}\)

Adding these gives a total molar mass of \( 156.26 \text{ g/mol}\). To find the mass percent, you apply the same method used for caffeine. Thus:

• Percent Carbon: \( \frac{120.10}{156.26} \times 100\% = 76.87\%\)
• Percent Hydrogen: \( \frac{20.16}{156.26} \times 100\% = 12.90\%\)
• Percent Oxygen: \( \frac{16.00}{156.26} \times 100\% = 10.23\%\)

This provides a detailed picture of menthol, explaining why carbon dominates its makeup.

CoCl2·6H2O Composition

Cobalt(II) chloride hexahydrate, \( \mathrm{CoCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}\), is an example of a hydrate—a compound that includes water molecules. Assessing its composition reveals insights into how hydrates work.
First, calculate its molar mass by considering each component:

• Cobalt (Co): \( 1 \times 58.93 \text{ g/mol} = 58.93 \text{ g/mol}\)
• Chlorine (Cl): \( 2 \times 35.45 \text{ g/mol} = 70.90 \text{ g/mol}\)
• Water (\( \mathrm{H}_{2} \mathrm{O}\)): \( 6 \times 18.02 \text{ g/mol} = 108.12 \text{ g/mol}\)

Together, they sum up to \( 237.95 \text{ g/mol}\). Each component's mass percent is calculated similarly:

• Percent Cobalt: \( \frac{58.93}{237.95} \times 100\% = 24.76\%\)
• Percent Chlorine: \( \frac{70.90}{237.95} \times 100\% = 29.80\%\)
• Percent Water: \( \frac{108.12}{237.95} \times 100\% = 45.44\%\)

This breakdown of composition not only highlights the importance of water in the structure but also emphasizes how hydrates retain specific properties through chemically bound water.