To solve this problem, we can use the formula for boiling point elevation:
ΔTb = Kb * m
where ΔTb is the change in boiling point, Kb is the molal boiling point elevation constant (0.512 °C/m for water), and m is the molality of the solution.
First, we need to calculate the change in boiling point:
ΔTb = Tb2 - Tb1 = (150.0 - 100.0) °C = 50.0 °C
Next, we need to convert the volume of water to mass, assuming a density of 1.00 g/mL:
mass of water = volume of water * density = 120.0 g
Now we can calculate the molality of the solution:
molality = moles of solute / mass of solvent (in kg)
We want to find the mass of sugar needed to achieve the desired boiling point elevation, so we can rearrange the equation:
moles of solute = molality * mass of solvent (in kg)
mass of solute = moles of solute * molar mass of sugar
The molar mass of sugar (C12H22O11) is 342.3 g/mol.
Now we need to find the molality of the solution. To do this, we need to calculate the mass of sugar needed to add to the water to achieve the desired boiling point elevation.
ΔTb = Kb * m
m = ΔTb / Kb = 50.0 / 0.512 = 97.7 mol/kg
Now we can calculate the mass of sugar needed to achieve this molality:
mass of sugar = moles of solute * molar mass of sugar = (0.0977 mol/kg) * (342.3 g/mol) = 33.4 g
Therefore, you need to add 33.4 grams of sugar to 1/2 cup (120.0mL) of water in order to raise the boiling point up to 300° F (150.° C).