# If I cool a solution containing 62 g of ethylene glycol in 250 g water to -9.3 °C, what mass of ice will separate out?

May 2, 2017

#### Answer:

The amount of ice that will separate out is 50 g.

#### Explanation:

The formula for freezing point depression ΔT_text(f) is

color(blue)(bar(ul(|color(white)(a/a)ΔT_text(f) = iK_text(f)bcolor(white)(a/a)|)))" "

where

$i =$ the van't Hoff $i$ factor
${K}_{\textrm{f}} =$ the freezing point depression constant for the solvent
$b =$ the molality of the solute

We can rearrange this expression to get

b = (ΔT_text(f))/(iK_text(f)

In this problem,

ΔT_text(f) = "9.3 °C"
$i = 1$, because ethylene glycol is a nonelectrolyte
${K}_{\textrm{f}} = \text{1.86 °C·kg·mol"^"-1}$

b = (9.3 color(red)(cancel(color(black)("°C"))))/(1 × 1.86 color(red)(cancel(color(black)("°C")))·"kg·mol"^"-1") = "5.00 mol·kg"^"-1"

$\text{Moles of ethylene glycol" = 62 color(red)(cancel(color(black)("g EG"))) × "1 mol EG"/(62.07 color(red)(cancel(color(black)("g EG")))) = "0.999 mol EG}$

So, the allowed amount of water is

$\text{Mass of water" = 0.999 color(red)(cancel(color(black)("mol EG"))) × "1 kg water"/(5 color(red)(cancel(color(black)("mol EG")))) = "0.200 kg water" = "200 g water}$

You started with 250 g of water, so the other 50 g must be present as ice.