Problem 40 - Chemistry The Central Science

Question

The molecular formula of aspartame, the artificial sweetener marketed as NutraSweet, is \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\) . (a) What is the molar mass of aspartame? (b) How many moles of aspartame are present in 1.00 \(\mathrm{mg}\) of aspartame? (c) How many molecules of aspartame are present in 1.00 \(\mathrm{mg}\) of aspartame? (d) How many hydrogen atoms are present in 1.00 \(\mathrm{mg}\) of aspartame?

Step-by-Step Solution

Answer

(a) The molar mass of aspartame is 294.31 g/mol. (b) There are 3.40 × 10^{-6} moles of aspartame in 1.00 mg. (c) There are 2.05 × 10^{18} molecules of aspartame in 1.00 mg. (d) There are 3.69 × 10^{19} hydrogen atoms in 1.00 mg of aspartame.

Step 1: Find the molar mass of aspartame

To find the molar mass of aspartame, we need to consider the molecular formula, which is \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\). We'll first find the molar mass of each of the individual elements and then multiply by the number of atoms of each element present in the molecule. Molar Mass of Carbon (C) = 12.01 g/mol Molar Mass of Hydrogen (H) = 1.01 g/mol Molar Mass of Nitrogen (N) = 14.01 g/mol Molar Mass of Oxygen (O) = 16.00 g/mol Molar Mass of aspartame = (14 × 12.01 g/mol) + (18 × 1.01 g/mol) + (2 × 14.01 g/mol) + (5 × 16.00 g/mol)

Step 2: Calculate the number of moles in 1.00 mg of aspartame

First, convert the mass of aspartame from mg to g: 1.00 mg = 0.00100 g Next, use the molar mass of aspartame that we found earlier to calculate the number of moles present in 1.00 mg of aspartame: Number of moles = mass / molar mass

Step 3: Calculate the number of molecules in 1.00 mg of aspartame

To find the number of molecules in 1.00 mg of aspartame, we can use the number of moles we found in the previous step and multiply it by Avogadro's number (which is 6.022 × 10^23): Number of molecules = number of moles × Avogadro's number

Step 4: Calculate the number of hydrogen atoms in 1.00 mg of aspartame

To find the number of hydrogen atoms in 1.00 mg of aspartame, we can multiply the total number of molecules we found in the previous step by the number of hydrogen atoms in each molecule of aspartame: Number of hydrogen atoms = number of molecules × number of hydrogen atoms per aspartame molecule

Key Concepts

Molecular Formula

The molecular formula is a representation of the actual number of atoms of each element in a molecule. In the case of aspartame, which is an artificial sweetener, the molecular formula is \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\). This tells us that a single molecule of aspartame consists of 14 carbon (C) atoms, 18 hydrogen (H) atoms, 2 nitrogen (N) atoms, and 5 oxygen (O) atoms.

To visualize this, imagine building a tiny model using different colored balls to represent each atom. You'd need 14 black balls for carbon, 18 white balls for hydrogen, 2 blue balls for nitrogen, and 5 red balls for oxygen. The molecular formula helps chemists understand the composition and structure of the molecule, which is crucial for predicting its properties and behaviors.

Molar Mass

Molar mass is the weight of one mole of a substance, expressed in grams per mole (g/mol). It's a measure that combines the atomic masses of all atoms in a molecule. To calculate the molar mass of aspartame, we add up the atomic masses of its constituent atoms, following its molecular formula \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}\).

Calculating Aspartame's Molar Mass:

For carbon, with an atomic mass of about 12.01 g/mol, multiplying by 14 gives us the total mass of carbon in aspartame. We apply this same reasoning for the hydrogen, nitrogen, and oxygen atoms using their respective atomic masses. The sum of these calculations gives us the molar mass of aspartame. It's like tallying the total weight of our previously mentioned tiny ball model, where the weight of each ball type (atom) contributes to the overall model's (molecule's) weight.

Avogadro's Number

Avogadro's number, \(6.022 \times 10^{23}\), is the number of units (usually atoms or molecules) in one mole of any substance. This constant allows chemists to count the incredibly large number of particles in a sample by relating mass to number of particles. When we determine the number of molecules in 1.00 mg of aspartame, we first convert the mass to moles, and then multiply by Avogadro's number to count the actual molecules.

Avogadro's number is foundational to the concept of moles and bridging the gap between the atomic scale and the macroscopic world we interact with. Just like knowing the number of eggs in a dozen helps you understand how many cakes you can bake, knowing Avogadro's number helps chemists understand how much of a substance they're working with in tangible terms.