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

Identify element molar masses

Find the molar mass of each element in the formula (\(C = 12.01\text{g/mol}, H = 1.01\text{g/mol}, N = 14.01\text{g/mol}, O = 16\text{g/mol\))

02

Calculate Aspartame's Molar Mass

Multiply each element's molar mass by the number of atoms in the molecular formula and add the results: \(M = 14(12.01) + 18(1.01) + 2(14.01) + 5(16) = 294.30\text{ g/mol}\) The molar mass of aspartame is 294.30 g/mol. #b. What amount (moles) of molecules are present in 10.0 \(\mathrm{g}\) aspartame?#

03

Use the mass to find moles

Use the formula: moles = mass / molar mass moles = \(10.0\text{g} / 294.30\text{g/mol} = 0.0340\) moles There are 0.0340 moles of aspartame in 10.0 g. #c. Calculate the mass in grams of 1.56 mole of aspartame#

04

Convert moles to mass

Use the formula: mass = moles × molar mass mass = \(1.56\text{ mol} \times 294.30\text{ g/mol} = 458.91\text{ g}\) The mass of 1.56 moles of aspartame is 458.91 grams. #d. What number of molecules are in 5.0 mg aspartame?#

05

Convert mass to moles

5.0 mg = 0.005 g, so the moles = 0.005 g / 294.30 g/mol = \(1.70 \times 10^{-5}\) moles

06

Calculate molecules

Number of molecules = moles × Avogadro's number molecules = \(1.70 \times 10^{-5}\text{ mol} \times 6.022 \times 10^{23}\text{ mol}^{-1} = 1.02 \times 10^{19}\) molecules There are \(1.02 \times 10^{19}\) molecules in 5.0 mg of aspartame. #e. What number of atoms of nitrogen are in 1.2 g aspartame?#

07

Convert mass to moles and molecules

moles = 1.2 g / 294.30 g/mol = \(4.08 \times 10^{-3}\) moles molecules = \(4.08 \times 10^{-3}\text{ mol} \times 6.022 \times 10^{23}\text{ mol}^{-1} = 2.46 \times 10^{21}\) molecules

08

Calculate nitrogen atoms

Since there are 2 nitrogen atoms in one aspartame molecule: N atoms = \(2.46 \times 10^{21}\) molecules × 2 = \(4.92 \times 10^{21}\) N atoms There are \(4.92 \times 10^{21}\) nitrogen atoms in 1.2 g of aspartame. #f. What is the mass in grams of \(1.0 \times 10^{9}\) molecules of aspartame?#

09

Convert molecules to mass

moles = number of molecules / Avogadro's number moles = \((1.0 \times 10^{9}\) molecules) / \((6.022 \times 10^{23}\) molecules/mol) = \(1.66 \times 10^{-15}\) mol mass = moles × molar mass = \((1.66 \times 10^{-15}\mathrm{mol}) \times (294.30\mathrm{g/mol}) = 4.89 \times 10^{-13}\) grams The mass of \(1.0 \times 10^{9}\) molecules of aspartame is \(4.89 \times 10^{-13}\) grams. #g. What is the mass in grams of one molecule of aspartame?#

10

Find the mass of one molecule of aspartame.

mass/molecule = molar mass / Avogadro's number = \(294.30\text{ g/mol}\) / \(6.022 \times 10^{23}\) molecules/mol = \(4.89 \times 10^{-22}\) g The mass of one molecule of aspartame is \(4.89 \times 10^{-22}\) grams.

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