calculate molality

The Watcher · Dream Team

Problem: An aqueous antifreeze solution is 40.0% ethylene glycol (C2H6O2) by mass. The density of the solution is 1.05 g/cm3. Calculate the molality, molarity, and mole fraction of the ethylene glycol.

Solution:

First, let's calculate molality, which is defined by the following expression...

Molality = m = moles of solute
kg of solvent

Let's assume that we have 100 g of the solution and, hence, 40.0 grams of ethylene glycol and 60.0 grams of water (sort of like what we do when we figure out empirical formula). In this case, we can easily figure out how many moles of solute we have...

40.0 g EG x 1 mol EG = 0.645 moles EG (solute)
62 g EG

...and how many kilograms of water we have...

60.0 g H2O x 1 kg H2O = 0.06 kg H2O (solvent)
1000 g H2O

Divide one by the other, and we have our molality.

m = moles of solute = 0.645 mol EG = 10.8 m (pronounced "10.8 molal")
kg of solvent 0.06 kg H2O

Next, let's find the molarity. First, let's look at the expression for molarity...

Molarity = M = moles of solute
L of solution

...and figure out what information we already have. Again, let's assume our 100 g sample of this solution. That way, we already know how many moles of solute we have (0.645 moles EG, from the molality example) and only need to calculate the volume of the solution in liters (using the given density of the solution to relate mass and volume).

100 g solution x 1 cm3 solution x 1 L solution = 0.095 L solution
1.05 g solution 1000 cm3 solution

Finally, divide...

M = moles of solute = 0.645 mol EG = 6.79 M
L of solution 0.095 L solution

Now, let's figure out mole fraction...

Mole fraction of EG = cEG = moles of EG
moles of EG + moles H2O

Again, let's assume the 100 g sample. We already know the number of moles of EG, so the only unknown factor is the number of moles of H2O. But we can get this easily enough by converting 60.0 g of H2O to moles of H2O...

60.0 g H2O x 1 mol H2O = 3.33 moles H2O
18 g H2O

Finally, plug the numbers in...

cEG = moles of EG
moles of EG + moles of H2O

= 0.645 mol EG
0.0645 mol EG + 3.33 mol H2O

= 0.162

Notice that there aren't units associated with mole fraction...it's simply a ratio.