Problem 76 - 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

(a) 305.406 g/mol; (b) 1.80 × 10⁻⁴ moles; (c) C: 70.83%, H: 8.86%, N: 4.59%, O: 15.72%; (d) 38.96 mg C.

Step 1: Calculating the Molar Mass of Capsaicin

The molecular formula for capsaicin is \( \mathrm{C}_{18} \mathrm{H}_{27} \mathrm{NO}_{3} \). We will calculate the molar mass by summing the atomic masses of all atoms in the formula:\[\text{Molar mass} = (18 \times 12.01) + (27 \times 1.008) + (1 \times 14.01) + (3 \times 16.00) \]\[ = 216.18 + 27.216 + 14.01 + 48.00 = 305.406 \text{ g/mol} \]

Step 2: Calculating Moles Consumed

To find the moles of capsaicin consumed, we use its molar mass and the mass consumed:\[\text{Moles of capsaicin} = \frac{\text{mass consumed (g)}}{\text{molar mass (g/mol)}}\]First, convert 55 mg to grams: 55 mg = 0.055 g. \[\text{Moles of capsaicin} = \frac{0.055}{305.406} = 1.80 \times 10^{-4} \, \text{moles} \]

Step 3: Calculating Mass Percent of Each Element

The mass percent of each element is calculated as follows:\[\% ext{C} = \left(\frac{18 \times 12.01}{305.406}\right) \times 100 = 70.83\% \]\[\% ext{H} = \left(\frac{27 \times 1.008}{305.406}\right) \times 100 = 8.86\% \]\[\% ext{N} = \left(\frac{14.01}{305.406}\right) \times 100 = 4.59\% \]\[\% ext{O} = \left(\frac{3 \times 16.00}{305.406}\right) \times 100 = 15.72\% \]

Step 4: Calculating the Mass of Carbon in 55 mg of Capsaicin

First, convert the mass of capsaicin to grams (55 mg = 0.055 g), then use the mass percent of carbon:\[\text{Mass of C} = 0.055 \times \frac{70.83}{100} = 0.03896 \, \text{g}\]Convert this mass back to milligrams:\[\text{Mass of C} = 0.03896 \, \text{g} \times 1000 = 38.96 \, \text{mg} \]

Key Concepts

Mass Percent Calculation

Mass percent is a way to express the composition of a compound in terms of the contribution of each element to the compound's total mass. To calculate mass percent, you need to know two things:

• The molar mass of each element in the compound.
• The total molar mass of the compound.

For example, in capsaicin \[\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3} \], the molar masses contribute as follows:To find the mass percent of an element, like carbon, you take the total mass of carbon in the compound divided by the molar mass of capsaicin and multiply by 100. In formula terms, it's:\[\% \text{Element} = \left(\frac{\text{mass of element in one mole}}{\text{molar mass of compound}}\right) \times 100\] So, the mass percent gives you a straightforward measure of how much of the compound's mass comes from each element.

Chemical Formula

A chemical formula tells us the number and type of atoms in a molecule. For capsaicin, the chemical formula is \( \mathrm{C}_{18} \mathrm{H}_{27} \mathrm{NO}_{3} \). This tells us that in one molecule of capsaicin:

• There are 18 carbon atoms.
• There are 27 hydrogen atoms.
• There is 1 nitrogen atom.
• There are 3 oxygen atoms.

Each element has a specific role in the chemical and physical properties of the compound.Thus, the formula not only helps us understand the composition but is key to calculating other important properties, such as molar mass and mass percent. Knowing how to read and interpret chemical formulas is foundational to exploring more complex chemical concepts.

Mole Conversion

Mole conversion is a method to translate between mass and the number of molecules based on the concept of Avogadro's number. The mole is a standard scientific unit for measuring large quantities of very small entities like atoms or molecules.When you have a sample of a compound, knowing its mass allows you to find out how many moles of that compound are present using its molar mass.The formula for the mole conversion is:\[\text{Moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\]In the case of eating capsaicin, if you consumed 55 mg, you first convert that to grams (0.055 g), then use the mole conversion formula to find out how many moles you have consumed.Moles help us to scale from a tangible quantity like mass to the number of molecules, offering insight into chemical reactions and properties.

Atomic Mass

Atomic mass is the weight of a single atom of a chemical element. It is typically measured in atomic mass units (amu), where one amu is defined as one twelfth the mass of a carbon-12 atom. Every element on the periodic table has an atomic mass based on isotopic composition and relative abundances. For example, the atomic mass of carbon is approximately 12.01 amu, hydrogen 1.008 amu, nitrogen 14.01 amu, and oxygen 16.00 amu. These individual atomic masses are used to calculate the molar mass of a compound by summing the atomic masses of the atoms in its chemical formula. Understanding atomic mass is vital for calculations involving chemical reactions, stoichiometry, and molecular weight, directly impacting how we predict and manipulate chemical processes.