An aqueous CaCl_2 solution has a vapor pressure of 82.1mmHg at 50C. The vapor pressure of pure water at this temperature is 92.6mmHg.
What is the concentration of CaCl_2 in mass percent?
I would approach the problem this way.
You know Psolvent = Xsolvent</sub*Posolvent. This allows you to determine Xsolvent. Now you have X for solvent and X for CaCl2. Convert those to grams and to mass percent.
Can you take it from here?
43.2%
To find the concentration of CaCl2 in mass percent, we need to use the equation:
Mass percent = (mass of solute / mass of solution) * 100%
We can start by finding the mole fraction of CaCl2 in the solution, using the Raoult's law equation:
Psolvent = Xsolvent * Psolvent°
Where Psolvent is the vapor pressure of the solvent in the solution, Xsolvent is the mole fraction of the solvent, and Psolvent° is the vapor pressure of the pure solvent.
Rearranging the equation:
Xsolvent = Psolvent / Psolvent°
In this case, the solvent is water. So:
Xwater = 82.1 mmHg / 92.6 mmHg = 0.886
To find the mole fraction of CaCl2, we can subtract the mole fraction of water from 1:
XCaCl2 = 1 - Xwater = 1 - 0.886 = 0.114
Next, we can find the mole fraction of CaCl2 in terms of mass:
Moles of CaCl2 = XCaCl2 * moles of solute
The mass of the solution is the sum of the mass of water and the mass of CaCl2.
So:
Mass of CaCl2 = (moles of CaCl2 * molar mass of CaCl2) / mass of solution
Finally, we can find the concentration in mass percent:
Concentration in mass percent = (mass of CaCl2 / mass of solution) * 100%
To find the concentration of CaCl2 in mass percent, we need to understand the relationship between vapor pressure and the concentration of a solute in an aqueous solution. In a solution, the vapor pressure of the solvent (water) decreases when a non-volatile solute (CaCl2) is dissolved. This decrease in vapor pressure is determined by the concentration of the solute.
The formula used to calculate the vapor pressure of an aqueous solution relative to that of pure solvent is known as Raoult's law:
P_solvent = X_solvent * P°_solvent,
where P_solvent is the vapor pressure of the solvent in the solution, X_solvent is the mole fraction of the solvent, and P°_solvent is the vapor pressure of the pure solvent.
To find the mole fraction of the solvent, we will assume that the solute (CaCl2) is non-volatile and does not affect the vapor pressure significantly. Therefore, we can consider its vapor pressure to be zero.
Using Raoult's law, we can write the equation for the vapor pressure of the solution:
P_solution = X_water * P°_water,
where P_solution is the vapor pressure of the solution and X_water is the mole fraction of water in the solution.
We can rearrange this equation to solve for the mole fraction of water:
X_water = P_solution / P°_water.
Now, let's substitute the given values into the equation:
P_solution = 82.1 mmHg (vapor pressure of the CaCl2 solution at 50°C),
P°_water = 92.6 mmHg (vapor pressure of pure water at 50°C).
X_water = 82.1 mmHg / 92.6 mmHg.
Calculating this value gives us:
X_water = 0.88607.
The mole fraction of water (X_water) is the ratio of the moles of water to the total moles in the solution.
To find the mole fraction of CaCl2, we can use the fact that the sum of the mole fractions of all components in a solution is equal to 1:
X_water + X_CaCl2 = 1.
Plugging in the value we found for X_water:
0.88607 + X_CaCl2 = 1.
Simplifying this equation:
X_CaCl2 = 1 - 0.88607,
X_CaCl2 = 0.11393.
Now, we need to convert the mole fraction of CaCl2 to mass percent.
Mass percent is the ratio of the mass of the solute to the total mass of the solution, expressed as a percentage.
To calculate the mass percent of CaCl2, we need to know the molar mass of CaCl2 and the mass of the solution.
The molar mass of CaCl2 is:
(1 mol Ca * 40.08 g/mol) + (2 mol Cl * 35.45 g/mol) = 110.98 g/mol.
Let's assume we have 100 g of the solution. Therefore, the mass of CaCl2 in the solution is:
(0.11393) * (110.98 g/mol) = 12.70574 g.
The mass percent of CaCl2 is:
(12.70574 g / 100 g) * 100% = 12.70574%.