A solution is prepared by dissolving 20.2 {\rm mL} of methanol ({\rm{CH}}_3 {\rm{OH}}) in 100.0 {\rm mL} of water at 25 ^\circ {\rm C}. The final volume of the solution is 118 {\rm mL}. The densities of methanol and water at this temperature are 0.782 {\rm g/mL} and 1.00 {\rm g/mL}, respectively. For this solution, calculate each of the following.

find 1 ) molality
2) percent by mass
3) moles fraction
4 )mole fraction
5) molarity

20.2 mL CH3OH + 100 mL H2O. Final volume = 118 mL.

1).
Use density to convert 20.2 mL CH3OH to grams. mass = volume*density = 20.2*0.782 = about 15 g(but you need to do it more accurately than that).
Convert 15g to mols. mol = g/molar mass = 15/32 = about 0.5 mol.
Then molality = mols/kg solvent = 0.5/0.1 = ?

2)percent w/w = (grams solute/g soln)*100.
mass solute = about 15g
mass solvent = 100 g
mass solution = 100 + 15 = 115 g
%w/w = (15/115)*100 = ?

3).
mole fraction CH3OH = XCH3OH = mols CH3OH from #1/total mols.
XH2O = mols H2O/total mols.
mols H2O = grams/molar mass = 100/molar mass.

4. Same as 3?
5.M = mols/L soln. You know mols from #1, L soln = 118 mL = 0.118 L.

NOTE: I assume you are same poster under various screen names of aa, anonymous, and others who insist on making this difficult for you to type and for us to answer by continuing to use symbols that, in my opinion, have no place in the question or the subject. 20.2 mL is 20.2 mL no matter how you slice it; writing it as 20.2{\rm
mL} is not only confusing but, in my opinion, is gibberish. You could save yourself a lot of typing time and you could make it lot easier for us to understand and respond by keeping this rubbish out of your posts.

1) molality:

To find the molality (m) of the solution, we need to first calculate the number of moles of methanol present.
Given that the density of methanol is 0.782 g/mL, and we have 20.2 mL of methanol, we can calculate the mass of methanol as:
Mass of methanol = Volume of methanol * Density of methanol
Mass of methanol = 20.2 mL * 0.782 g/mL

Now, we convert the mass of methanol to moles:
Moles of methanol = Mass of methanol / Molar mass of methanol

Finally, we calculate the molality using the equation:
Molality (m) = Moles of solute / Mass of solvent (in kg)

2) percent by mass:
To find the percent by mass, we need to calculate the mass of methanol and water separately and then find the percentage of methanol in the total mass of the solution:
Mass of methanol = Volume of methanol * Density of methanol
Mass of water = Volume of water * Density of water

Percentage by mass = (Mass of methanol / Total mass of solution) * 100

3) mole fraction:
The mole fraction (X) is the ratio of the moles of one component to the total moles of the solution.
First, calculate the moles of methanol and water separately, then find the total moles of the solution.

Mole fraction of methanol = Moles of methanol / Total moles of solution
Mole fraction of water = Moles of water / Total moles of solution

4) mole fraction (Repeat of previous step)

5) molarity:
The molarity (M) of a solution is the ratio of the moles of solute to the volume of the solution in liters.
First, calculate the moles of methanol, then divide it by the total volume of the solution in liters:

Molarity (M) = Moles of methanol / Volume of solution (in liters)

To calculate the values requested, we'll need to follow these steps:

1) Calculate the mass of methanol and water.
2) Calculate the molality.
3) Calculate the percent by mass.
4) Calculate the mole fraction.
5) Calculate the molarity.

Let's calculate each value step-by-step.

1) Mass of methanol:
Given that the density of methanol is 0.782 g/mL, and we have 20.2 mL,
mass of methanol = volume * density = 20.2 mL * 0.782 g/mL

mass of methanol = 15.8044 g (rounded to four decimal places)

2) Mass of water:
Given that the density of water is 1.00 g/mL, and we have 100.0 mL,
mass of water = volume * density = 100.0 mL * 1.00 g/mL

mass of water = 100.0 g

3) Molality:
Molality is defined as the moles of solute per kilogram of solvent.

Moles of solute (methanol) = mass of methanol / molar mass of methanol

Molar mass of methanol (CH3OH) = 12.01 g/mol (C) + 3 * (1.01 g/mol) (H) + 16.00 g/mol (O)

Moles of solute = 15.8044 g / (12.01 g/mol + 3 * 1.01 g/mol + 16.00 g/mol)

Moles of solute = 0.385 mol (rounded to three decimal places)

Mass of solvent (water) = mass of water / 1000 (to convert grams to kilograms)

Mass of solvent = 100.0 g / 1000 = 0.100 kg

Molality = moles of solute / mass of solvent

Molality = 0.385 mol / 0.100 kg

Molality = 3.85 mol/kg (rounded to two decimal places)

4) Percent by mass:
Percent by mass is the ratio of the mass contribution of a specific component to the total mass of the solution, expressed as a percentage.

Total mass of the solution = mass of methanol + mass of water = 15.8044 g + 100.0 g

Total mass of the solution = 115.8044 g

Percent by mass of methanol = (mass of methanol / total mass of the solution) * 100

Percent by mass of methanol = (15.8044 g / 115.8044 g) * 100

Percent by mass of methanol = 13.64% (rounded to two decimal places)

5) Mole fraction:
Mole fraction is the ratio of the number of moles of a component to the total number of moles in the mixture.

Moles of methanol = 0.385 mol

Moles of water = (mass of water / molar mass of water)

Molar mass of water (H2O) = 2 * (1.01 g/mol) (H) + 16.00 g/mol (O)

Moles of water = 0.100 kg / [(2 * 1.01 g/mol) + 16.00 g/mol]

Moles of water = 5.547 mol (rounded to three decimal places)

Mole fraction of methanol = moles of methanol / (moles of methanol + moles of water)

Mole fraction of methanol = 0.385 mol / (0.385 mol + 5.547 mol)

Mole fraction of methanol = 0.064 (rounded to three decimal places)

6) Molarity:
Molarity is defined as the number of moles of a solute per liter of solution.

Volume of solution = 118 mL

Volume of solution = 118 mL / 1000 (to convert mL to L)

Volume of solution = 0.118 L

Molarity = moles of solute (methanol) / volume of solution

Molarity = 0.385 mol / 0.118 L

Molarity = 3.27 mol/L (rounded to two decimal places)

To summarize:

1) The molality of the solution is 3.85 mol/kg.
2) The percent by mass of methanol is 13.64%.
3) The mole fraction of methanol is 0.064.
4) The molarity of the solution is 3.27 mol/L.

To calculate the properties of the solution, we need to find the mass of methanol and water in the solution, as well as the moles of methanol and water. Once we have this information, we can calculate the molality, percent by mass, mole fraction, and molarity.

1) Molality:
Molality (m) is defined as moles of solute per kilogram of solvent.

To find the molality, we first need to calculate the moles of methanol and water in the solution. We can use the equation:

moles = mass / molar mass

For methanol (CH3OH):
moles of methanol = (volume of methanol x density of methanol) / molar mass of methanol

moles of methanol = (20.2 mL x 0.782 g/mL) / 32.05 g/mol

Similarly, for water (H2O):
moles of water = (volume of water x density of water) / molar mass of water

moles of water = (100.0 mL x 1.00 g/mL) / 18.02 g/mol

Now, we need to convert the mass of water to kilograms:
mass of water = volume of water x density of water

mass of water = 100.0 mL x 1.00 g/mL = 100.0 g

mass of water = 100.0 g / 1000 = 0.100 kg

Therefore, the molality of the solution is:

molality = (moles of methanol + moles of water) / mass of water

2) Percent by mass:
Percent by mass is defined as the mass of solute divided by the mass of the solution, multiplied by 100.

To find the percent by mass, we need to calculate the mass of methanol in the solution and the total mass of the solution.

mass of methanol = volume of methanol x density of methanol

mass of methanol = 20.2 mL x 0.782 g/mL

mass of methanol = 15.7864 g

Total mass of the solution = mass of methanol + mass of water

Total mass of the solution = 15.7864 g + 100.0 g = 115.7864 g

Therefore, the percent by mass of methanol in the solution is:

percent by mass = (mass of methanol / total mass of the solution) x 100

3) Mole fraction:
The mole fraction is defined as the ratio of moles of a component to the total moles in the solution.

To find the mole fraction of methanol and water, we can use the following equations:

mole fraction of methanol = moles of methanol / (moles of methanol + moles of water)

mole fraction of water = moles of water / (moles of methanol + moles of water)

4) Molarity:
Molarity is defined as moles of solute per liter of solution.

To find the molarity, we first need to calculate the moles of methanol and water. We already calculated these values in step 1.

moles of methanol = (20.2 mL x 0.782 g/mL) / molar mass of methanol

moles of water = (100.0 mL x 1.00 g/mL) / molar mass of water

Now, we need to convert the volume of the solution to liters:

volume of solution = 118 mL = 118 mL / 1000 = 0.118 L

Therefore, the molarity of the solution is:

molarity = (moles of methanol + moles of water) / volume of solution

Now that we have explained how to calculate each of the properties, you can substitute the values given in the problem to find the respective answers.