What mass of salt(NaCl)should you add to 1.20 L of water in an ice cream maker to make a solution that freezes at -10.4∘C?
To find the mass of salt (NaCl) needed to make a solution that freezes at a specific temperature, we need to understand the concept of freezing point depression. Freezing point depression refers to the phenomenon where the freezing point of a solvent (in this case, water) is lowered by adding a non-volatile solute (in this case, NaCl).
To calculate the mass of NaCl required, we can use the equation for freezing point depression:
ΔT = Kf * m * i
Where:
ΔT = change in freezing point
Kf = molal freezing point depression constant
m = molality of the solution (moles of solute per kilogram of solvent)
i = van 't Hoff factor (the number of particles into which a compound dissociates in a solution)
First, let's find the change in freezing point (ΔT). Here, the freezing point is given as -10.4∘C, and the freezing point of pure water is 0∘C.
ΔT = -10.4∘C - 0∘C
ΔT = -10.4∘C
Next, we need to calculate the molality (m) of the solution. Molality is defined as the number of moles of solute divided by the mass of the solvent in kilograms.
m = moles of NaCl / mass of water (in kg)
Now, we need to determine the van 't Hoff factor (i) for NaCl. NaCl dissociates into two ions in water (Na+ and Cl-), so the van 't Hoff factor for NaCl is 2.
Finally, we need to look up the molal freezing point depression constant (Kf) for water. For water, Kf is approximately 1.86 °C·kg/mol.
Now, let's plug the known values into the equation and solve for the mass of NaCl:
-10.4∘C = (1.86 °C·kg/mol) * (moles of NaCl / mass of water (in kg)) * 2
We know the volume of water is 1.20 L, but we need to convert it to the mass of water in kilograms.
1.20 L * (1 kg / 1000 mL) = 1.20 kg
-10.4∘C = (1.86 °C·kg/mol) * (moles of NaCl / 1.20 kg) * 2
Now, we can rearrange the equation to solve for the moles of NaCl:
moles of NaCl = (-10.4∘C * 1.20 kg) / (1.86 °C·kg/mol * 2)
moles of NaCl = -6.24 / 3.72
moles of NaCl = -1.677
Since we cannot have negative moles, it indicates that we have made an error somewhere. It is not possible to find a mass of NaCl that will result in a solution freezing at -10.4∘C. Please double-check the problem statement or provide additional information if necessary.