Question

Aspartame is an artificial sweetener that is 160 times sweeter than sucrose (table sugar) when dissolved in water. It is marketed as NutraSweet. The molecular formula of aspartame is \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5}.\) a. Calculate the molar mass of aspartame. b. What amount (moles) of molecules are present in \(10.0 \mathrm{g}\) aspartame? c. Calculate the mass in grams of 1.56 mole of aspartame. d. What number of molecules are in \(5.0 \mathrm{mg}\) aspartame? e. What number of atoms of nitrogen are in \(1.2 \mathrm{g}\) aspartame? f. What is the mass in grams of \(1.0 \times 10^{9}\) molecules of aspartame? g. What is the mass in grams of one molecule of aspartame?

Short answer

a. The molar mass of aspartame is 294.30 g/mol. b. There are 0.0340 mol of aspartame in 10.0 g. c. The mass of 1.56 mol of aspartame is 458.908 g. d. There are 1.024 × 10^{19} molecules in 5.0 mg aspartame. e. There are 4.914 × 10^{22} nitrogen atoms in 1.2 g aspartame. f. The mass of 1.0 × 10^9 molecules of aspartame is 4.883 × 10^{-13} g. g. The mass of one molecule of aspartame is 4.883 × 10^{-22} g.

Step-by-step solution

a. Calculate the molar mass of aspartame.

First, we need to determine the molar mass of aspartame. We can do this by summing the molar mass of each element in the molecular formula, multiplied by the number of atoms of each element. Molar mass of aspartame = (14 × Molar mass of C) + (18 × Molar mass of H) + (2 × Molar mass of N) + (5 × Molar mass of O) Using the molar mass of elements: C = 12.01 g/mol, H = 1.01 g/mol, N = 14.01 g/mol, and O = 16.00 g/mol, we get: Molar mass of aspartame = (14 × 12.01) + (18 × 1.01) + (2 × 14.01) + (5 × 16.00) = 294.30 g/mol.

b. What amount (moles) of molecules are present in 10.0 g aspartame?

To find the moles of aspartame in 10.0 g, we can use the formula: Moles = mass / molar mass Moles of aspartame = (10.0 g) / (294.30 g/mol) = 0.0340 mol.

c. Calculate the mass in grams of 1.56 mole of aspartame.

We can use the same relationship between mass and moles, solving for mass: Mass = moles × molar mass Mass of 1.56 mol of aspartame = (1.56 mol) × (294.30 g/mol) = 458.908 g.

d. What number of molecules are in 5.0 mg aspartame?

First, we need to convert the mass of aspartame from mg to g: 5.0 mg = 0.0050 g Next, we find the moles of aspartame: Moles of aspartame = (0.0050 g) / (294.30 g/mol) = 1.700 × 10^{-5} mol Now we can use Avogadro's number to find the number of molecules: Number of molecules = moles × Avogadro's number = (1.700 × 10^{-5} mol) × (6.022 × 10^{23} mol^{-1}) = 1.024 × 10^{19} molecules.

e. What number of atoms of nitrogen are in 1.2 g aspartame?

First, find the moles of aspartame in 1.2 g: Moles of aspartame = (1.2 g) / (294.30 g/mol) = 0.00408 mol Since there are 2 nitrogen atoms in each aspartame molecule, we find the moles of nitrogen: Moles of nitrogen = (0.00408 mol) × 2 = 0.00816 mol Now we use Avogadro's number to find the number of nitrogen atoms: Number of nitrogen atoms = moles × Avogadro's number = (0.00816 mol) × (6.022 × 10^{23} mol^{-1}) = 4.914 × 10^{22} nitrogen atoms.

f. What is the mass in grams of 1.0 × 10^9 molecules of aspartame?

First, find the moles of aspartame: Moles of aspartame = (1.0 × 10^9 molecules) / (6.022 × 10^{23} mol^{-1}) = 1.660 × 10^{-15} mol Now, find the mass: Mass of aspartame = (1.660 × 10^{-15} mol) × (294.30 g/mol) = 4.883 × 10^{-13} g.

g. What is the mass in grams of one molecule of aspartame?

First, we need to find the moles of one molecule of aspartame: Moles of one aspartame molecule = 1 molecule / (6.022 × 10^{23} mol^{-1}) = 1.660 × 10^{-24} mol Now, calculate the mass of one molecule: Mass of one aspartame molecule = (1.660 × 10^{-24} mol) × (294.30 g/mol) = 4.883 × 10^{-22} g.

Key concepts

Molecules Calculation

When it comes to calculating molecules in a given mass of substance, we follow a systematic approach. The goal is to understand how many individual molecules are present within a specified amount of material. To achieve this, we start by determining the number of moles in the sample, because moles provide a bridge between the macroscopic world and the microscopic realm of atoms and molecules.

Here's how you can calculate molecules from mass:

• First, convert the mass of the substance from its given unit to grams, if necessary.
• Next, find the moles of the substance using the formula: \[ \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
• Finally, multiply the number of moles by Avogadro's Number to find the number of molecules: \[ \text{Number of molecules} = \text{moles} \times 6.022 \times 10^{23} \text{ molecules/mol} \]

By following this process, you can determine the number of molecules present in any mass of a compound, as demonstrated in the step involving 5.0 mg of aspartame.

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry that links the amount of substance and the microscopic entities it is composed of, such as atoms, molecules, or ions. It is defined as the number of atoms in exactly 12 grams of carbon-12, which is approximately 6.022 × 10²³ entities per mole. Avogadro's Number allows chemists to translate between atomic scale quantities and macroscopic amounts we can measure in a lab.

Key uses of Avogadro's Number include:

• Converting moles of a substance to molecules or atoms: This is crucial in bridging measurements taken in moles to particulate counts.
• Understanding the concept of a mole: A mole is simply a quantity that contains Avogadro's Number of entities, making it easier to compute macroscopic properties of reactions and compounds.

This number serves as a conversion factor in countless chemical calculations, like determining the quantity of nitrogen atoms in a mass of aspartame, by helping us tally up microscopic components like molecules or atoms.

Nitrogen Atoms

The calculation of individual types of atoms, such as nitrogen atoms, in a compound follows directly from an understanding of the molecule's composition. In a substance like aspartame, each molecule has a defined number of nitrogen atoms specified in its chemical formula, \( \mathrm{C}_{14} \mathrm{H}_{18} \mathrm{N}_{2} \mathrm{O}_{5} \), where 'N' represents nitrogen.

To find the number of nitrogen atoms in a specific mass of aspartame, follow these steps:

• Calculate the moles of aspartame using the conversion from mass.
• Determine the moles of nitrogen, taking into account that each molecule of aspartame contains two nitrogen atoms. Multiply the moles of aspartame by 2 to find the moles of nitrogen.
• Finally, multiply the moles of nitrogen by Avogadro's Number to obtain the total number of nitrogen atoms: \[ \text{Number of nitrogen atoms} = \text{moles of nitrogen} \times 6.022 \times 10^{23} \text{ atoms/mol} \]

This approach ensures you can isolate a particular atom type and predict its quantity in any given amount of a chemical substance.