Problem 171 - Chemistry Matter and Change

Question

Artificial Sweetener Determine the percent composition of aspartame \(\left(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\right),\) an artificial sweetener.

Step-by-Step Solution

Answer

The percent composition of aspartame \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\) is approximately: Carbon: \(57.10\%\) Hydrogen: \(6.16\%\) Nitrogen: \(9.52\%\) Oxygen: \(27.22\%\)

Step 1: Find the molar mass of aspartame

First, we need to find the molar mass of the aspartame molecule. We can do this by adding up the molar masses of each element in the molecule, taking into account the number of each element. Molar mass of aspartame = \(\mathrm{C} * \mathrm{H} * \mathrm{N} * \mathrm{O}\) The molar masses of each element are: Carbon (C) = 12.01 g/mol Hydrogen (H) = 1.008 g/mol Nitrogen (N) = 14.01 g/mol Oxygen (O) = 16.00 g/mol Molar mass of aspartame = (14 * 12.01 g/mol) + (18 * 1.008 g/mol) + (2 * 14.01 g/mol) + (5 * 16.00 g/mol)

Step 2: Calculate the mass of individual elements

Next, we will find the mass of each element in aspartame. Mass of Carbon = 14 * 12.01 g/mol = 168.14 g/mol Mass of Hydrogen = 18 * 1.008 g/mol = 18.144 g/mol Mass of Nitrogen = 2 * 14.01 g/mol = 28.02 g/mol Mass of Oxygen = 5 * 16.00 g/mol = 80.00 g/mol

Step 3: Determine the total molar mass of aspartame

Now, we will sum up the mass of each element to find the total molar mass of aspartame. Total Molar Mass = Mass of Carbon + Mass of Hydrogen + Mass of Nitrogen + Mass of Oxygen Total Molar Mass = 168.14 g/mol + 18.144 g/mol + 28.02 g/mol + 80.00 g/mol Total Molar Mass = 294.304 g/mol

Step 4: Calculate the percent composition of each element

Finally, we will determine the percent composition of each element in aspartame. To do this, we will divide the mass of each element by the total molar mass and then multiply the result by 100. Percent composition of Carbon = \(\frac{168.14}{294.304} * 100\% = 57.10\%\) Percent composition of Hydrogen = \(\frac{18.144}{294.304} * 100\% = 6.16\%\) Percent composition of Nitrogen = \(\frac{28.02}{294.304} * 100\% = 9.52\%\) Percent composition of Oxygen = \(\frac{80.00}{294.304} * 100\% = 27.22\%\) Thus, the percent composition of aspartame is approximately: Carbon: \(57.10\%\) Hydrogen: \(6.16\%\) Nitrogen: \(9.52\%\) Oxygen: \(27.22\%\)

Key Concepts

Molar Mass Calculation

Calculating the molar mass of a compound is key to understanding its composition. Here, the molar mass of aspartame needs calculation by summing up the individual molar masses of each of its elements. Aspartame's formula is \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\), indicating the number of each type of atom it contains.

To find the molar mass:

• Multiply the number of each element by its atomic mass (found on the periodic table).
• Carbon (C) has an atomic mass of \(12.01\, \text{g/mol}\).
• Hydrogen (H) is \(1.008\, \text{g/mol}\).
• Nitrogen (N) is \(14.01\, \text{g/mol}\).
• Oxygen (O) is \(16.00\, \text{g/mol}\).

For aspartame, this calculation goes as follows:

• Carbon: \(14 \times 12.01 = 168.14\, \text{g/mol}\)
• Hydrogen: \(18 \times 1.008 = 18.144\, \text{g/mol}\)
• Nitrogen: \(2 \times 14.01 = 28.02\, \text{g/mol}\)
• Oxygen: \(5 \times 16.00 = 80.00\, \text{g/mol}\)

Adding these together provides the total molar mass: \(168.14 + 18.144 + 28.02 + 80.00 = 294.304\, \text{g/mol}\).

This calculation lays the groundwork for deeper analysis like percent composition.

Chemical Formula Interpretation

Understanding a chemical formula helps decipher what a compound is composed of. The chemical formula provides a snapshot of how many atoms of each element are in a molecule.

For aspartame, the formula \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\) tells us the following:

• There are 14 carbon atoms.
• There are 18 hydrogen atoms.
• There are 2 nitrogen atoms.
• There are 5 oxygen atoms.

Each subscript in a formula communicates the number of atoms immediately preceding it. When no subscript is visible, it implies the presence of one atom. This interpretation is crucial when performing calculations like determining the molar mass or conducting an elemental analysis, as it directly affects the numerical values involved.

Elemental Analysis

Elemental analysis involves determining how much of each element is present in a compound compared to the whole. The goal is to find the percent composition, which indicates the percentage of the compound's total mass made up by each element.

To find the percent composition, you follow these steps:

• Calculate the mass contribution of each element using its molar mass from the formula \(\text{Mass of element} = \text{Number of atoms} \times \text{Atomic mass}\).
• Use the total molar mass of the compound as the denominator.
• For each element, divide its individual mass by the total molar mass.
• Multiply the result by 100 to convert it into a percentage.

Applying this to aspartame:

• Carbon: \(\frac{168.14}{294.304} \times 100\% = 57.10\%\)
• Hydrogen: \(\frac{18.144}{294.304} \times 100\% = 6.16\%\)
• Nitrogen: \(\frac{28.02}{294.304} \times 100\% = 9.52\%\)
• Oxygen: \(\frac{80.00}{294.304} \times 100\% = 27.22\%\)

The result gives a detailed view of the composition, allowing one to understand an artificial sweetener's chemical makeup more scientifically.