Problem 38 - Chemistry: The Central Science

Question

The molecular formula of aspartame, the artificial sweetener marketed as NutraSweet \(^{0}\), 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 \(294.31 \mathrm{~g/mol}\). (b) There are \(3.40 \times 10^{-6} \mathrm{~moles}\) of aspartame in 1.00 mg. (c) There are \(2.04 \times 10^{18}\) molecules of aspartame in 1.00 mg. (d) There are \(3.67 \times 10^{19}\) hydrogen atoms in 1.00 mg of aspartame.

Step 1: Calculate the molar mass of aspartame

To find the molar mass of aspartame, we will add up the molar masses of each element in the molecular formula. Aspartame: \(\mathrm{C}_{14}\mathrm{H}_{18}\mathrm{N}_{2}\mathrm{O}_{5}\) 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) + (18 × 1.01) + (2 × 14.01) + (5 × 16.00)

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

To calculate the number of moles in 1.00 mg of aspartame, we will use the molar mass we found in step 1 and the given mass of aspartame. 1.00 mg of aspartame = 1.00 × 10^{-3} g of aspartame (since 1 mg = 10^{-3} g) Number of moles = mass / molar mass = (1.00 × 10^{-3} g) / (molar mass of aspartame in g/mol)

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 will use the number of moles calculated in step 2 and Avogadro's number. Avogadro's number = \(6.022 \times 10^{23} \underline{\phantom{xxx}} \mathrm{molecules/mol}\) Number of molecules = (number of moles) × (Avogadro's number)

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

To calculate the number of hydrogen atoms in 1.00 mg of aspartame, we will use the number of molecules calculated in step 3 and the molecular formula of aspartame. Aspartame: \(\mathrm{C}_{14}\mathrm{H}_{18}\mathrm{N}_{2}\mathrm{O}_{5}\) Number of hydrogen atoms = (number of molecules) × (number of hydrogen atoms per molecule) Here, the number of hydrogen atoms per molecule is 18, as given by the molecular formula.

Key Concepts

Molecular Formula

The molecular formula is a crucial concept in chemistry that provides insight into the composition of molecules. It represents the actual number of each type of atom present in a molecule. For aspartame, its molecular formula is \(\mathrm{C}_{14}\mathrm{H}_{18}\mathrm{N}_{2}\mathrm{O}_{5}\). This means:

• It contains 14 carbon (C) atoms
• 18 hydrogen (H) atoms
• 2 nitrogen (N) atoms
• 5 oxygen (O) atoms

The molecular formula not only helps in understanding the molecular structure but is also the first step in calculating the molar mass, which, in turn, is essential for any stoichiometric calculations. Each element’s molar mass contributes to the overall molar mass of the compound.

Moles Calculation

The concept of moles is central to chemical calculations. A mole represents a specific quantity of particles, usually atoms or molecules, and is defined using the amount of a substance. To calculate the number of moles in a given mass of a substance, you can use the formula:

• \( \text{Number of moles} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}} \)

In the problem, when dealing with 1.00 mg of aspartame, you first convert this mass into grams (\(1.00 \times 10^{-3} \text{ g}\)).
Then, divide by the molar mass of aspartame (as calculated from its molecular formula) to find the number of moles present.

Avogadro's Number

Avogadro's number is a fundamental constant that serves as a bridge between the macroscopic and microscopic worlds in chemistry. It defines the number of particles, usually atoms or molecules, in one mole of a substance:

• \( 6.022 \times 10^{23} \text{ particles/mol} \)

This astonishingly large number helps us convert the number of moles into discrete particles, such as molecules. For example, if you calculate the number of moles of aspartame in 1.00 mg, you can then multiply by Avogadro's number to find how many actual molecules there are.
This conversion is crucial for understanding chemical reactions and processes at the molecular level.

Number of Atoms

The molecular formula not only indicates the types of atoms in a molecule but also their quantities. When dealing with chemical substances, knowing the number of specific atoms is important for understanding its properties. For aspartame, each molecule contains 18 hydrogen atoms.
After determining the total number of aspartame molecules in your sample using Avogadro's number, you multiply by the number of hydrogen atoms per molecule to find the total number of hydrogen atoms.

• If there are \(\text{X}\) molecules and 18 hydrogen atoms per molecule, the total hydrogen atoms = \(\text{X} \times 18\).

This approach can be applied to any element in the molecule and is key in analyzing and predicting the behavior of chemical compounds.