Answer
(a) 0.00204
(b) 0.0101
Work Step by Step
(a)
1. Molarity, by definition:
0.112 M $C_6H_{12}O_6$: 0.112 mol $C_6H_{12}O_6$ for 1 L of solution.
2. Find the mass of $C_6H_{12}O_6$
$ C_6H_{12}O_6 $ : ( 1.008 $\times$ 12 )+ ( 12.01 $\times$ 6 )+ ( 16.00 $\times$ 6 )= 180.16 g/mol
- Using the molar mass as a conversion factor, find the mass in g:
$$ 0.112 \space mole \times \frac{ 180.16 \space g}{1 \space mole} = 20.2 \space g$$
3. Calculate the mass of solution, and the mass of water.
1 L of solution: $$1 \space L \space solution \times \frac{1.006 \space g}{1 \space mL} \times \frac{1000 \space mL}{1 \space L} = 1006 \space g$$
Mass of water = 1006 g - 20.2 g = 986 g
4. Find the amount of moles of water.
$ H_2O $ : ( 1.008 $\times$ 2 )+ ( 16.00 $\times$ 1 )= 18.02 g/mol
- Using the molar mass as a conversion factor, find the amount in moles:
$$ 986 \space g \times \frac{1 \space mole}{ 18.02 \space g} = 54.7 \space moles$$
5. Calculate the mole fraction.
$$Mole \space fraction \space solute = \frac{0.112 \space mol}{0.112 \space mol + 54.7 \space mol} = 0.00204$$
(b)
1. By definition:
3.20% ethanol by volume: 3.20 mL of ethanol for each 100 mL of solution.
2. Calculate the amount of moles of ethanol
$ CH_3CH_2OH $ : ( 1.008 $\times$ 6 )+ ( 12.01 $\times$ 2 )+ ( 16.00 $\times$ 1 )= 46.07 g/mol
$$3.20 \space mL \times \frac{0.789 \space g}{1 \space mL} \times \frac{1 \space mol}{46.07 \space g} = 0.0548 \space mol$$
3. Calculate the amount of moles of water.
Mass of solution: $$100 \space mL \times \frac{0.993 \space g}{1 \space mL} = 99.3 \space g$$
Mass of ethanol: $$3.20 \space mL \times \frac{0.789 \space g}{1 \space mL} = 2.52 \space g$$
Mass of water = 99.3 g - 2.52 g = 96.8 g
$ H_2O $ : ( 1.008 $\times$ 2 )+ ( 16.00 $\times$ 1 )= 18.02 g/mol
$$96.8 \space g \times \frac{1 \space mol}{18.02 \space g} = 5.37 \space mol$$
4. Calculate the mole fraction:
$$Mole \space fraction \space ethanol = \frac{0.0548 \space mol}{0.0548 \space mol + 5.37 \space mol} = 0.0101 $$
