Problem 118 - 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 x 10^-4 mol; (c) C: 70.92%, H: 8.87%, N: 4.59%, O: 15.70%; (d) 39.01 mg of C.

Step 1: Calculate Molar Mass of Capsaicin

To calculate the molar mass, sum the atomic masses of each element in the formula \(\text{C}_{18}\text{H}_{27}\text{NO}_{3}\). The atomic masses are approximately 12.01 g/mol for carbon (C), 1.008 g/mol for hydrogen (H), 14.01 g/mol for nitrogen (N), and 16.00 g/mol for oxygen (O).\[\begin{align*}\text{Molar mass of C}_{18}\text{H}_{27}\text{NO}_{3} &= 18 \times 12.01 \, \text{g/mol} + 27 \times 1.008 \, \text{g/mol} + 1 \times 14.01 \, \text{g/mol} + 3 \times 16.00 \, \text{g/mol} \&= 216.18 \, \text{g/mol} + 27.216 \, \text{g/mol} + 14.01 \, \text{g/mol} + 48.00 \, \text{g/mol} \&= 305.406 \, \text{g/mol}\end{align*}\]

Step 2: Calculate Moles from Mass

To find the moles of capsaicin consumed, divide the mass consumed by the molar mass calculated.\[\text{Moles of capsaicin} = \frac{55 \, \text{mg}}{305.406 \, \text{g/mol}} \times \frac{1 \, \text{g}}{1000 \, \text{mg}} = \frac{55}{305406} \, \text{mol} \approx 1.80 \times 10^{-4} \, \text{mol}\]

Step 3: Calculate Mass Percent of Each Element

To calculate the mass percent of each element, divide the total mass of each element in a mole of capsaicin by the molar mass and multiply by 100.**For carbon (C):**\[\text{Mass percent of C} = \frac{18 \times 12.01}{305.406} \times 100 \approx 70.92\%\]**For hydrogen (H):**\[\text{Mass percent of H} = \frac{27 \times 1.008}{305.406} \times 100 \approx 8.87\%\]**For nitrogen (N):**\[\text{Mass percent of N} = \frac{14.01}{305.406} \times 100 \approx 4.59\%\]**For oxygen (O):**\[\text{Mass percent of O} = \frac{3 \times 16.00}{305.406} \times 100 \approx 15.70\%\]

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

To find the mass of carbon in 55 mg of capsaicin, multiply the total mass consumed by the mass percent of carbon.\[\text{Mass of carbon} = 55 \, \text{mg} \times 0.7092 \approx 39.01 \, \text{mg}\]

Key Concepts

Mass Percent Calculation

Calculating mass percent is a crucial way to understand the composition of a chemical compound. It tells us how much of each element is present in a compound in terms of percentage by mass.
To calculate the mass percent of each element in capsaicin, you start by knowing the molar mass of the entire compound. This is like knowing the whole pie.
Next, we find out how much each piece (element) weighs in this pie. This involves multiplying the atomic mass of each element by the number of atoms it contributes. For example:

• Carbon contributes 18 atoms, each with a mass of 12.01 g/mol.
• Hydrogen has 27 atoms, each 1.008 g/mol.
• Nitrogen is single, with a mass of 14.01 g/mol.
• Oxygen contributes 3 atoms, each with a mass of 16.00 g/mol.

These contributions are divided by the total molar mass of the compound and then multiplied by 100 to get the percentage. This tells us the portion of each element relative to the whole compound.

Chemical Formula

A chemical formula like \( ext{C}_{18} ext{H}_{27} ext{NO}_{3}\) reveals a lot about a compound. It's like a recipe that tells you the ingredients needed to form the substance.
Each letter refers to a different element:

• \( ext{C}\) stands for Carbon.
• \( ext{H}\) is Hydrogen.
• \( ext{N}\) is Nitrogen.
• \( ext{O}\) is Oxygen.

Right behind each letter, you'll find numbers which indicate how many atoms of each are present in a single molecule. So, in capsaicin, there are 18 carbon atoms, 27 hydrogen atoms, 1 nitrogen atom, and 3 oxygen atoms.
This information helps when calculating molar masses or figuring out how the compound might react with others in a chemical reaction.

Moles Calculation

Moles are a basic unit of measure in chemistry, similar to how "dozens" are used in everyday language. They help to measure amounts of a substance in terms of quantity rather than weight.
To calculate moles, you'll need the total mass of the substance and the molar mass of one mole of that substance.
For example, when you consume 55 mg of capsaicin:

• First, convert mg to grams by using the conversion \( ext{1 g} = 1000 ext{ mg}\).
• Then, divide this mass by the molar mass of capsaicin, which is 305.406 g/mol.

This will give you the amount in moles, which shows how many "molecules-worth" of capsaicin you've consumed. Understanding moles helps connect the microscopic world of atoms to the everyday macroscopic world.

Elemental Composition

The elemental composition provides a detailed breakdown of the different elements in a compound, essentially mapping out its chemical makeup.
For capsaicin, understanding its elemental composition involves analyzing the presence of carbon, hydrogen, nitrogen, and oxygen. Each element's contribution to the total formula plays a role in determining properties like molecular weight and reactivity.
This kind of breakdown is vital in many fields:

• In nutrition, it helps to calculate the caloric content of foods.
• In pharmacology, it assists in drug formulation.
• In materials science, it informs the synthesis of new materials.

Elemental composition is the foundational information you need when working with chemical equations or understanding the function and structure of a molecule. It enables chemists to tailor reactions and predict outcomes based on precise elemental arrangements.