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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a.) Well, if we assume that the solution behaves ideally, then we can use Raoult's law to calculate the vapor-pressure lowering. According to Raoult's law, the vapor pressure of a component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution.
So, the mole fraction of water (solvent) can be calculated by dividing the molar mass of water by the sum of the molar mass of water and mannitol:
Mole fraction of water = (molar mass of water) / [(molar mass of water) + (molar mass of mannitol)]
Plugging in the values, we get:
Mole fraction of water = (18.0 g/mol) / [(18.0 g/mol) + (182.2 g/mol)]
Now, we can calculate the vapor-pressure lowering using Raoult's law:
Vapor-pressure lowering = (mole fraction of water) * (vapor pressure of pure water)
Plugging in the values, we get:
Vapor-pressure lowering = (mole fraction of water) * (17.54 torr)
I hope that helps! Now let's move on to part b.
b.) Alrighty then, we have the observed vapor-pressure lowering, which is 0.0930 torr. Now we can use this value to find the activity coefficient of water in the solution.
The activity coefficient of a component in a solution is defined as the ratio of its activity (effective concentration) to its mole fraction.
So, we can write:
Activity of water = (mole fraction of water) * (activity coefficient of water)
Since the activity of water is directly proportional to the vapor-pressure lowering, we can write:
Vapor-pressure lowering = (mole fraction of water) * (activity coefficient of water) * (vapor pressure of pure water)
Now we can rearrange this equation to solve for the activity coefficient of water:
Activity coefficient of water = (Vapor-pressure lowering) / [(mole fraction of water) * (vapor pressure of pure water)]
Simply plug in the values to get the activity coefficient.
And now, onto part c!
c.) Alrighty, so we need to calculate the osmotic pressure of the mannitol solution when measured against pure water, and compare it with the osmotic pressure of the ideal solution.
The osmotic pressure can be calculated using the equation:
Osmotic pressure = (Molarity of solute) * (R) * (Temperature)
But since we don't have the molarity of mannitol, we need to find it first. The molarity can be calculated by dividing the number of moles of mannitol by the volume of solution in liters:
Molarity of mannitol = (mass of mannitol) / [(molar mass of mannitol) * (volume of solution in liters)]
Plugging in the values, we get:
Molarity of mannitol = (54.66 g) / [(182.2 g/mol) * (1.000 kg / 1000 g)]
Now we have the molarity of mannitol, we can calculate the osmotic pressure using the equation mentioned earlier.
For comparison, we can also calculate the osmotic pressure of the ideal solution using the equation:
Osmotic pressure of ideal solution = (Molarity of mannitol) * (R) * (Temperature)
Again, plug in the values and calculate the osmotic pressure.
And voila! You have calculated the osmotic pressure of the mannitol solution and compared it with the osmotic pressure of the ideal solution.
To solve these problems, we can use the equations related to colligative properties of solutions.
a.) To calculate the vapor-pressure lowering, we can use Raoult's law which states that the vapor pressure of a solvent above a solution is equal to the mole fraction of the solvent multiplied by the vapor pressure of the pure solvent. The equation is given by:
ΔP = Xsolvent * Psolvent
Where:
ΔP = vapor-pressure lowering
Xsolvent = mole fraction of the solvent
Psolvent = vapor pressure of the pure solvent
First, we need to calculate the mole fraction of the solvent (water) in the solution. The mole fraction of water can be calculated using the total moles of water and the total moles of solute (mannitol):
Mole fraction (water) = Moles of water / (Moles of water + Moles of mannitol)
Moles of water = mass of water / molar mass of water
Moles of mannitol = mass of mannitol / molar mass of mannitol
Given:
Mass of water = 1.000 kg
Mass of mannitol = 54.66 g
Molar mass of water = 18.015 g/mol
Molar mass of mannitol = 182.2 g/mol
Moles of water = (1.000 kg / 1000 g/kg) / 18.015 g/mol
Moles of mannitol = 54.66 g / 182.2 g/mol
Calculate the mole fraction of water:
Mole fraction (water) = Moles of water / (Moles of water + Moles of mannitol)
Now, plug the values into the equation:
ΔP = Xsolvent * Psolvent
b.) To calculate the activity coefficient (based on mole fraction) of water in the solution, we can use the equation:
ΔP = -(Xsolute^2) * Psolvent * gamma
Where:
Xsolute = mole fraction of solute (mannitol)
Psolvent = vapor pressure of the pure solvent
gamma = activity coefficient of water
Since mannitol does not ionize in aqueous solution, Xsolute can be considered negligible, so we can ignore it in the equation.
Now, rearrange the equation to solve for gamma:
gamma = -(ΔP) / (Xsolute^2 * Psolvent)
Plug in the values for ΔP and Psolvent:
c.) To calculate the osmotic pressure of the mannitol solution when measured against pure water, we can use the equation:
Π = MRT
Where:
Π = osmotic pressure
M = molarity of the solute (mannitol)
R = ideal gas constant (0.0821 L*atm/mol*K)
T = temperature in Kelvin
We will calculate the molarity of the solution using the moles of mannitol and the total volume of the solution. The total volume can be calculated by adding the mass of water and the mass of mannitol and converting to liters.
Molarity (mannitol) = Moles of mannitol / Total volume in liters
Now, solve for the osmotic pressure using the calculated values:
Π = MRT
Compare the calculated osmotic pressure with the osmotic pressure of an ideal solution where the osmotic pressure is given by the following equation:
Πideal = nRT / V
Where:
n = moles of solute (mannitol)
V = volume of solution in liters
By comparing the calculated osmotic pressure to the osmotic pressure of an ideal solution, we can determine if the solution deviates from ideality.
To answer these questions, we need to apply the concepts of ideal solutions, Raoult's law, and osmotic pressure.
a.) The vapor-pressure lowering for the solution can be calculated using Raoult's law, which states that the vapor pressure of a component in a solution is proportional to its mole fraction in the solution. The equation to calculate the vapor-pressure lowering (ΔP) is:
ΔP = P° - P
Where P° is the vapor pressure of the pure solvent (water) and P is the vapor pressure of the solution.
Given:
Molar mass of mannitol (M) = 182.2 g/mol
Mass of mannitol (m1) = 54.66 g
Mass of water (m2) = 1.000 kg = 1000 g
Vapor pressure of pure water (P°) = 17.54 torr
First, we need to calculate the mole fraction of mannitol (X1) in the solution:
X1 = (moles of solute) / (moles of solute + moles of solvent)
Since mannitol is nonvolatile and does not ionize, it remains as a solute in the solution. The number of moles of mannitol can be calculated as follows:
moles of solute = mass of mannitol / molar mass of mannitol
moles of solute = 54.66 g / 182.2 g/mol
Now, let's calculate the mole fraction of mannitol in the solution:
X1 = (54.66 g / 182.2 g/mol) / ((54.66 g / 182.2 g/mol) + (1000 g / 18.015 g/mol))
Next, we can use Raoult's law to calculate the vapor pressure of the solution (P):
P = X1 * P°
Finally, we can calculate the vapor-pressure lowering (ΔP):
ΔP = P° - P
b.) The activity coefficient (γ) of water in the solution can be calculated using the observed vapor-pressure lowering (ΔP) and the ideal vapor-pressure lowering (ΔP_ideal). The relationship is given by:
ΔP = ΔP_ideal * γ
We are given the observed vapor-pressure lowering (ΔP) and the mole fraction of water (X2) can be calculated as 1 - X1. The ideal vapor-pressure lowering (ΔP_ideal) can be calculated using Raoult's law:
ΔP_ideal = X2 * P°
Now, we can solve for the activity coefficient (γ):
γ = ΔP / ΔP_ideal
c.) The osmotic pressure (π) of a solution can be calculated using the formula:
π = MRT
Where M is the molarity of the solution, R is the ideal gas constant, and T is the temperature in Kelvin.
We need to calculate the molarity (M) of the mannitol solution. The number of moles of mannitol (n1) can be calculated as:
n1 = mass of mannitol / molar mass of mannitol
Next, we need to calculate the volume (V) of the solution. The volume can be calculated using the mass of water (m2) and the density of water (ρ):
V = m2 / ρ
Finally, we can calculate the molarity (M) using the equation:
M = n1 / V
Now, we have all the values required to calculate the osmotic pressure of the mannitol solution.