Problem 122 - Chemistry and Chemical Reactivity

Question

Capsaicin, the compound that gives the hot taste to chili peppers, has the formula \(\mathrm{C}_{18} \mathrm{H}_{27} \mathrm{NO}_{3}\) (a) Calculate its molar mass. (b) If you eat 55 mg of capsaicin, what amount (moles) have you consumed? (c) Calculate the mass percent of each element in the compound. (d) What mass of carbon (in milligrams) is there in \(55 \mathrm{mg}\) of capsaicin?

Step-by-Step Solution

Answer

Molar mass of capsaicin is 305.46 g/mol. You consumed 1.8x10^-4 moles of capsaicin. The mass percentages are approximately 70.78% C, 8.93% H, 4.59% N, and 15.71% O. Mass of carbon in 55 mg of capsaicin is 38.93 mg.

Step 1: Calculate the Molar Mass of Capsaicin

To calculate the molar mass, add up the atomic masses of all the atoms in the formula \(\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3}\). Atomic masses are approximately: Carbon (C) = 12.01 g/mol, Hydrogen (H) = 1.01 g/mol, Nitrogen (N) = 14.01 g/mol, and Oxygen (O) = 16.00 g/mol.Calculating gives:\[ M = (18 \times 12.01) + (27 \times 1.01) + (1 \times 14.01) + (3 \times 16.00) = 216.18 + 27.27 + 14.01 + 48.00 = 305.46 \text{ g/mol} \]

Step 2: Calculate the Moles of Capsaicin Consumed

To find the number of moles consumed, use the formula:\[ n = \frac{m}{M} \]where \( n \) is the number of moles, \( m \) is the mass in grams (0.055 grams), and \( M \) is the molar mass (305.46 g/mol).By substituting the known values:\[ n = \frac{0.055}{305.46} = 1.800 \times 10^{-4} \text{ moles} \]

Step 3: Calculate the Mass Percent of Each Element

First, calculate the mass of each element in one mole of capsaicin, using their atomic masses:- Carbon: \( 18 \times 12.01 = 216.18 \text{ g} \)- Hydrogen: \( 27 \times 1.01 = 27.27 \text{ g} \)- Nitrogen: \( 1 \times 14.01 = 14.01 \text{ g} \)- Oxygen: \( 3 \times 16.00 = 48.00 \text{ g} \)Then calculate mass percent:- Carbon: \( \frac{216.18}{305.46} \times 100\% \approx 70.78\% \)- Hydrogen: \( \frac{27.27}{305.46} \times 100\% \approx 8.93\% \)- Nitrogen: \( \frac{14.01}{305.46} \times 100\% \approx 4.59\% \)- Oxygen: \( \frac{48.00}{305.46} \times 100\% \approx 15.71\% \)

Step 4: Calculate Mass of Carbon in 55 mg of Capsaicin

To find the mass of carbon, first determine the mass of carbon per mg of capsaicin using the mass percent calculated in the previous step. Using the mass percent:\[ \text{Mass of } \text{C} = 55 \times \frac{70.78}{100} = 38.93 \text{ mg} \]

Key Concepts

Molar Mass

The concept of molar mass is essential in chemistry for converting between the mass of a substance and the number of moles. Molar mass is expressed in grams per mole (g/mol) and is calculated by adding the atomic masses of all the atoms in a molecule. For capsaicin (\(\text{C}_{18}\text{H}_{27}\text{NO}_{3}\)), we add:

• Carbon: 18 atoms \(\times\) 12.01 g/mol = 216.18 g/mol
• Hydrogen: 27 atoms \(\times\) 1.01 g/mol = 27.27 g/mol
• Nitrogen: 1 atom \(\times\) 14.01 g/mol = 14.01 g/mol
• Oxygen: 3 atoms \(\times\) 16.00 g/mol = 48.00 g/mol

Adding these together, the molar mass of capsaicin is 305.46 g/mol. This value helps us understand how much one mole of the substance weighs and allows for easier conversion from grams to moles and vice versa.

Mass Percent Composition

Mass percent composition provides insight into the contribution of each element to the overall mass of a compound. By calculating the mass percent, we can see how much of each element is present in a molecule in terms of percentage.For capsaicin, we use the formula:\[\text{Mass percent of an element} = \frac{\text{Mass of the element in one mole}}{\text{Molar mass of the compound}} \times 100\%\]Here's how it works out:

• Carbon: \(\frac{216.18}{305.46} \times 100\% \approx 70.78\%\)
• Hydrogen: \(\frac{27.27}{305.46} \times 100\% \approx 8.93\%\)
• Nitrogen: \(\frac{14.01}{305.46} \times 100\% \approx 4.59\%\)
• Oxygen: \(\frac{48.00}{305.46} \times 100\% \approx 15.71\%\)

This breakdown shows how each element proportionally constitutes the compound, giving another layer of understanding to its chemical structure.

Mole Calculation

Mole calculations connect the mass of a substance to its amount in terms of moles. This conversion is crucial for understanding how much of a substance is present at the molecular level. To calculate the number of moles from a given mass, use the formula:\[ n = \frac{m}{M} \]where \( n \) is the number of moles, \( m \) is the mass in grams, and \( M \) is the molar mass. For capsaicin, when 55 mg (0.055 g) is consumed, the moles can be calculated as:\[ n = \frac{0.055}{305.46} = 1.800 \times 10^{-4} \text{ moles}\]This calculation helps us understand just how many molecules are in that small quantity. The concept demystifies the relationship between the weight of a substance and its molecular scale, facilitating deeper chemical comprehension.