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?

Step by step solution

01

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.

02

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.

03

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.

04

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.

05

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.

06

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.

07

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.

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