A solution of the sugar mannitol ( molar mass 182.2 g/mol ) is prepared by adding 54.66 g of mannitol to 1.000 kg of water. The vapor pressure of pure liquid water is 17.54 torr at 20o C. Mannitol is nonvolatile and does not ionize in aqueous solution.

a.) Assuming that aqueous mannitol solutions behave ideally, calculate the vapor-pressure lowering for the above solution at 20o C.

b.) The observed vapor-pressure lowering of the mannitol solution above is 0.0930 torr. Calculate the activity coefficient ( based on mole fraction ) of water in this solution.

c.) Calculate the osmotic pressure of the mannitol solution of part ( b ) when it is measured against pure water, and compare it with the osmotic pressure of the ideal solution.

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To solve these questions, we need to apply the following formulas and concepts:

1. Raoult's Law: According to Raoult's Law, the vapor pressure of a solvent in a solution is equal to the vapor pressure of the pure solvent multiplied by its mole fraction in the solution.
Mathematically: P_solution = X_solvent * P_solvent

2. Mole fraction (X): Mole fraction is the ratio of moles of a component to the total moles of all components in a solution.
Mathematically: X = Moles of component / Total moles in solution

3. Ideal solution: An ideal solution is one in which the interactions between different molecules and different phases are negligible.

Now, let's solve the questions step by step:

a.) To calculate the vapor-pressure lowering for the mannitol solution, we need to calculate the mole fraction of mannitol in the solution and use Raoult's Law.

Step 1: Calculate the moles of mannitol in the solution.
Molar mass of mannitol = 182.2 g/mol
Mass of mannitol = 54.66 g
Moles of mannitol = Mass of mannitol / Molar mass of mannitol

Step 2: Calculate the moles of water in the solution.
Mass of water = 1000 g (given)
Molar mass of water = 18 g/mol
Moles of water = Mass of water / Molar mass of water

Step 3: Calculate the total moles in the solution.
Total moles = Moles of mannitol + Moles of water

Step 4: Calculate the mole fraction of mannitol (X_mannitol).
X_mannitol = Moles of mannitol / Total moles

Step 5: Calculate the vapor pressure lowering (ΔP).
P_solvent = Vapor pressure of pure water = 17.54 torr (given)
ΔP = X_mannitol * P_solvent

b.) To calculate the activity coefficient of water, we need to use the vapor-pressure lowering observed and the mole fraction of water in the solution.

Step 1: Calculate the mole fraction of water (X_water).
X_water = 1 - X_mannitol

Step 2: Calculate the activity coefficient (γ).
ΔP_observed = Observed vapor-pressure lowering = 0.0930 torr (given)
γ = ΔP_observed / (X_water * P_solvent)

c.) To calculate the osmotic pressure, we can use the formula for osmotic pressure of an ideal solution and compare it with the osmotic pressure of the mannitol solution.

The osmotic pressure (π) of the mannitol solution can be calculated using the formula:
π = MRT
Where M is the molarity of the solute, R is the ideal gas constant, and T is the temperature in Kelvin.

For an ideal solution, the osmotic pressure is directly proportional to the mole fraction of the solute.

In this case, we are comparing the osmotic pressure of the mannitol solution with pure water as the solvent. Thus, the molarity of the solute (mannitol) will be equal to the mole fraction of mannitol in the solution (X_mannitol).

Therefore, the osmotic pressure of the mannitol solution (π_solution) can be calculated using the formula:
π_solution = X_mannitol * RT

Then, we can compare it with the osmotic pressure of the ideal solution (π_ideal).

Note: Make sure to convert the temperature to Kelvin for all calculations.