A solid consists of mixture of NaNO3 and Mg(NO3)2. When 6.50 g of this solid is dissolved in 50.0 g of water, the freezing point is lowered by 5.40⁰C. What is the composition of the solid (in mass percent)? Assume ideal behavior but watch out for the van’t Hoff factor.
To find the composition of the solid in mass percent, we will use the formula:
Mass percent = (mass of component / total mass of the solid) * 100
First, let's calculate the molality of the solution using the formula:
ΔTf = -Kf * m * i
Where:
ΔTf is the change in freezing point = 5.40 ⁰C
Kf is the freezing point depression constant for water = 1.86 ⁰C/m
m is the molality of the solution
i is the van't Hoff factor
Rearranging the equation, we can solve for m:
m = -ΔTf / (Kf * i)
Substituting the given values, we get:
m = -5.40 ⁰C / (1.86 ⁰C/m * i)
Next, let's calculate the molality of the solution. We know that molality is defined as moles of solute per kilogram of solvent, so:
molality = moles of solute / mass of solvent (water)
To find the moles of solute, we need to determine the number of moles of NaNO3 and Mg(NO3)2 in the solution.
Let's assume that the mass of NaNO3 in the solid is x g.
Therefore, the mass of Mg(NO3)2 in the solid would be (6.50 g - x g).
The molar mass of NaNO3 = 85 g/mol
The molar mass of Mg(NO3)2 = 148 g/mol
Using the molality formula, we have:
molality = (moles of NaNO3 + moles of Mg(NO3)2) / (mass of water in kg)
As the moles of NaNO3 and Mg(NO3)2 differ by a constant factor (stoichiometric ratio or moles ratio), we can represent the moles of NaNO3 as n and express the moles of Mg(NO3)2 as 0.5n.
Therefore,
molality = (n + 0.5n) / (50.0 g water / 1000)
Simplifying,
molality = 1.5n / 0.05 kg
Rearranging the equation to solve for n:
n = (0.05 kg * molality) / 1.5
Now, we need to find the total mass of the solid, which is the sum of the masses of NaNO3 and Mg(NO3)2:
Total mass of solid = mass of NaNO3 + mass of Mg(NO3)2
Total mass of solid = x g + (6.50 g - x g)
Total mass of solid = 6.50 g
Finally, we can calculate the mass percent of NaNO3 and Mg(NO3)2 in the solid:
mass percent of NaNO3 = (x g / 6.50 g) * 100
mass percent of Mg(NO3)2 = [(6.50 g - x g)/ 6.50 g] * 100
To find the composition of the solid in mass percent, we need to calculate the number of moles of NaNO3 and Mg(NO3)2 in the solid. Here's how you can do it:
1. Calculate the amount of substance present in the water solution:
- Calculate the molality (m) of the solution using the formula:
m = (ΔTf) / Kf
- ΔTf is the freezing point depression, which is given as 5.40⁰C.
- Kf is the cryoscopic constant for water, which is 1.86⁰C/m.
- Substitute the values to calculate the molality.
2. Calculate the moles of solute in the solution:
- Calculate the mass of water present in the solution using the formula:
mass of water = mass of solution - mass of solute
- Mass of solution is the sum of the mass of water (50.0 g) and the mass of the solid.
- Mass of solute is given as 6.50 g.
- Calculate the moles of water using the formula:
moles of water = mass of water / molar mass of water
- The molar mass of water is approximately 18.015 g/mol.
- Calculate the moles of solute using the formula:
moles of solute = moles of water * (m / 1000)
- Convert the molality to molality per kilogram (m/kg) by dividing m by 1000.
3. Calculate the moles of NaNO3 and Mg(NO3)2 in the solid:
- Assume the van't Hoff factor (i) for NaNO3 is 2 and for Mg(NO3)2 is 3.
- The total moles of solute in the solution is given by:
total moles of solute = moles of NaNO3 + moles of Mg(NO3)2
- Use the van't Hoff equation to relate the moles of solute to the moles of solid:
total moles of solute = moles of solid * i
- Rearrange the equation to solve for the moles of solid:
moles of solid = total moles of solute / i
4. Calculate the mass percent composition of the solid:
- Calculate the mass of NaNO3 and Mg(NO3)2 separately using the moles of each compound:
mass of NaNO3 = moles of NaNO3 * molar mass of NaNO3
mass of Mg(NO3)2 = moles of Mg(NO3)2 * molar mass of Mg(NO3)2
- The molar masses of NaNO3 and Mg(NO3)2 are approximately 85.00 g/mol and 148.32 g/mol, respectively.
- Determine the mass percent composition of NaNO3 and Mg(NO3)2 in the solid using the following formulas:
mass percent of NaNO3 = (mass of NaNO3 / mass of solid) * 100%
mass percent of Mg(NO3)2 = (mass of Mg(NO3)2 / mass of solid) * 100%
By following these steps and substituting the given values, you can solve for the mass percent composition of the solid (in this case, NaNO3 and Mg(NO3)2).