Aspartame is an artificial sweetener that is 160 times sweeter than sucrose (table sugar) when dissolved in water. It is marketed as Nutra-Sweet. The molecular formula of aspartame is C14H18N2O5. a) Calculate the molar mass of aspartame. b) What amount (moles) of molecules are present in 10.0 g aspartame? c) Calculate the mass in grams of 1.56 mol aspartame.
Aspartame is an artificial sweetener that is 160 times sweeter than sucrose (table sugar) when dissolved in water. It is marketed as Nutra-Sweet. The molecular formula of aspartame is C14H18N2O5.
a) Calculate the molar mass of aspartame.
b) What amount (moles) of molecules are present in 10.0 g aspartame?
c) Calculate the mass in grams of 1.56 mol 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 ×109 molecules of aspartame?
g) What is the mass in grams of one molecule of aspartame??
The Correct Answer and Explanation is :
Let’s go step by step to answer each part of the question, using the given molecular formula for aspartame (C14H18N2O5).
a) Molar Mass of Aspartame
The molar mass is calculated by adding up the atomic masses of each element in the molecular formula. Here is the breakdown:
- Carbon (C): 14 atoms × 12.01 g/mol = 168.14 g/mol
- Hydrogen (H): 18 atoms × 1.008 g/mol = 18.144 g/mol
- Nitrogen (N): 2 atoms × 14.01 g/mol = 28.02 g/mol
- Oxygen (O): 5 atoms × 16.00 g/mol = 80.00 g/mol
Now, sum these values:
[
\text{Molar mass of aspartame} = 168.14 \, \text{g/mol} + 18.144 \, \text{g/mol} + 28.02 \, \text{g/mol} + 80.00 \, \text{g/mol} = 294.30 \, \text{g/mol}
]
So, the molar mass of aspartame is 294.30 g/mol.
b) Moles of Molecules in 10.0 g Aspartame
To find the number of moles, use the formula:
[
\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}
]
Substituting the given values:
[
\text{moles} = \frac{10.0 \, \text{g}}{294.30 \, \text{g/mol}} = 0.0339 \, \text{mol}
]
So, the amount of aspartame in 10.0 g is 0.0339 mol.
c) Mass in Grams of 1.56 mol Aspartame
To find the mass, use the formula:
[
\text{mass} = \text{moles} \times \text{molar mass}
]
Substituting the given values:
[
\text{mass} = 1.56 \, \text{mol} \times 294.30 \, \text{g/mol} = 459.51 \, \text{g}
]
So, the mass of 1.56 mol of aspartame is 459.51 g.
d) Number of Molecules in 5.0 mg Aspartame
First, convert 5.0 mg to grams:
[
5.0 \, \text{mg} = 0.0050 \, \text{g}
]
Next, calculate the moles of aspartame:
[
\text{moles} = \frac{0.0050 \, \text{g}}{294.30 \, \text{g/mol}} = 1.70 \times 10^{-5} \, \text{mol}
]
Now, calculate the number of molecules using Avogadro’s number ((6.022 \times 10^{23}) molecules/mol):
[
\text{molecules} = 1.70 \times 10^{-5} \, \text{mol} \times 6.022 \times 10^{23} \, \text{molecules/mol} = 1.02 \times 10^{19} \, \text{molecules}
]
So, there are 1.02 × 10¹⁹ molecules in 5.0 mg of aspartame.
e) Number of Nitrogen Atoms in 1.2 g Aspartame
First, calculate the moles of aspartame in 1.2 g:
[
\text{moles} = \frac{1.2 \, \text{g}}{294.30 \, \text{g/mol}} = 0.00408 \, \text{mol}
]
Each molecule of aspartame has 2 nitrogen atoms, so the number of nitrogen atoms is:
[
\text{atoms of nitrogen} = 0.00408 \, \text{mol} \times 2 \, \text{atoms/molecule} \times 6.022 \times 10^{23} \, \text{atoms/mol} = 4.92 \times 10^{21} \, \text{atoms}
]
So, there are 4.92 × 10²¹ atoms of nitrogen in 1.2 g of aspartame.
f) Mass in Grams of (1.0 \times 10^9) Molecules of Aspartame
To find the mass of (1.0 \times 10^9) molecules, first calculate the moles:
[
\text{moles} = \frac{1.0 \times 10^9 \, \text{molecules}}{6.022 \times 10^{23} \, \text{molecules/mol}} = 1.66 \times 10^{-15} \, \text{mol}
]
Now, calculate the mass:
[
\text{mass} = 1.66 \times 10^{-15} \, \text{mol} \times 294.30 \, \text{g/mol} = 4.89 \times 10^{-13} \, \text{g}
]
So, the mass of (1.0 \times 10^9) molecules of aspartame is 4.89 × 10⁻¹³ g.
g) Mass of One Molecule of Aspartame
The mass of one molecule is:
[
\text{mass of one molecule} = \frac{\text{molar mass}}{\text{Avogadro’s number}} = \frac{294.30 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1}} = 4.89 \times 10^{-22} \, \text{g}
]
So, the mass of one molecule of aspartame is 4.89 × 10⁻²² g.
Summary:
- a) 294.30 g/mol
- b) 0.0339 mol
- c) 459.51 g
- d) 1.02 × 10¹⁹ molecules
- e) 4.92 × 10²¹ atoms of nitrogen
- f) 4.89 × 10⁻¹³ g
- g) 4.89 × 10⁻²² g