An aqueous salt solution is formed by adding 11.67 g sodium sulfate (solute) to water (solvent). What mass (in g) of water is used if the freezing point of the solution is -12.9 oF.
Kf H2O = 1.86 oC/m
I know to use delta T= i*Kf*m
I then solve for m. But, then I'm confused on what numbers to plug into the other parts of the equation. I've tried 3 different answers and don't know what I'm doing wrong or how far I'm off.
moles Na2SO4 = grams/molar mass.
Solve for moles Na2SO4.
Convert -12.9F to C.
delta T = i*Kf*m
i = 3, you know Kf and dT, solve for m
m = moles Na2SO4 from above/kg solvent.
Solve for kg solvent.
To determine the mass of water used, you can follow these steps:
1. Calculate the freezing-point depression (∆T) using the freezing point depression formula:
∆T = i * Kf * m
Given: Kf H2O = 1.86 oC/m, ∆T = -12.9 oF
2. Convert the freezing point depression (∆T) from oF to oC:
∆T(oC) = (∆T(oF) - 32) / 1.8
∆T(oC) = (-12.9 - 32) / 1.8 = -23.83 oC
3. Rearrange the freezing point depression formula to solve for the molality (m):
m = ∆T / (i * Kf)
4. Determine the van't Hoff factor (i) for the solute. In this case, sodium sulfate (Na2SO4) dissociates into three ions in water (Na^+ + Na^+ + SO4^2-), so i = 3.
5. Substitute the values of ∆T, i, and Kf into the molality formula to calculate the molality (m):
m = -23.83 oC / (3 * 1.86 oC/m) ≈ -4.0328 m
6. Convert the mass of solute (11.67 g sodium sulfate) to moles:
Moles of solute = mass / molar mass
Molar mass of Na2SO4 = 22.99 g/mol (2 Na) + 32.06 g/mol (S) + (4 * 16.00 g/mol) (4 O) = 142.04 g/mol
Moles of solute = 11.67 g / 142.04 g/mol ≈ 0.0822 mol
7. Use the molality (m) to calculate the moles of solute (Na2SO4) dissolved in the solution:
Moles of solute = m * kg solvent
To convert the result from step 6 to kg, divide by 1000:
Moles of solute = -4.0328 m * (mass of solvent in kg) / 1000
8. Solve for the mass of water (solvent):
Mass of solvent = (Moles of solute / moles of solvent) * molar mass of solvent
The molar mass of water is 18.015 g/mol
Mass of solvent = (0.0822 mol / (0.0822 mol + moles of water)) * 18.015 g/mol
Rearrange the equation to solve for moles of water:
moles of water = (0.0822 mol / 0.0322 mol) - 0.0822 mol
Simplifying, we have:
moles of water = 0.0822 mol * (1 - (0.0322 mol / 0.0822 mol))
Mass of water in g = moles of water * molar mass of water
Calculate the values to find the mass of water used in grams.
To find the mass of water used in the solution, you will need to solve the equation for molality, which is the ratio of moles of solute to the mass of the solvent (water in this case). Here's how you can solve it step by step:
1. Calculate the moles of sodium sulfate (Na2SO4):
- You are given the mass of sodium sulfate, which is 11.67 grams.
- To calculate the moles, you need to divide the given mass by the molar mass of sodium sulfate (142.04 g/mol).
Mass of Na2SO4 = 11.67 g
Molar mass of Na2SO4 = 22.99 g/mol + 32.06 g/mol + (4 × 16.00 g/mol) = 142.04 g/mol
Moles of Na2SO4 = 11.67 g / 142.04 g/mol
2. Calculate the molality (m) of the solution:
- The molality (m) is defined as the moles of solute divided by the mass of the solvent (water in this case).
Mass of water = ?
Molality (m) = moles of solute / mass of solvent
We know the molality equation ΔT = i * Kf * m, and we are looking for the mass of the solvent (water), so let's rearrange the equation to solve for m:
ΔT / (i * Kf) = m
The freezing point depression (ΔT) is given as -12.9 °F and the freezing point depression constant (Kf) for water is 1.86 °C/m.
You need to convert -12.9 °F to °C before using it in the equation:
- (-12.9 °F - 32) × (5/9) = ?
Now you can plug in the values into the rearranged equation to find m (molality):
m = (-12.9 °C / (1 * 1.86 °C/m))
3. Calculate the mass of the solvent (water):
- Now that you have the molality (m), you can use it to find the mass of the solvent (water) with the given moles of solute.
Mass of solvent (water) = moles of solute / molality
Substitute the values to calculate the mass of the solvent:
Mass of solvent = moles of solute / molality
= (11.67 g / 142.04 g/mol) / m
Finally, substitute the value of m (from step 2) into the equation to find the mass of the solvent.
Hopefully, this step-by-step explanation helps you solve the problem correctly.