Question

The solubility of `KNO_3` is 155 g per 100 g of water at 75° Cand 38.0 g at 25°C. What mass (in grams) of `KNO_3` will crystallize out of solution if exactly 100.0 g of its saturated solution at 75° C is cooled to 25°C?

Answer 1

`"45.9 g"`

Explanation:

So, you know that potassium nitrate, `"KNO"3`, has a solubility of `"155 g"` per `"100 g"` of water at `75^@"C"` and of `"38.0 g"` at `25^@"C"`.

What that means is that at `75^@"C"`, you can dissolve as much as `"155 g"` of solid potassium nitrate in water without the solution becoming saturated.

Once you hit that `"155 g"` mark, the solution becomes saturated, which means that the solution can't dissolve any more solid.

Now, the same thing can be said for the solution at `25^@"C"`. At this temperature, dissolving less than `"38.0 g"` of potassium nitrate per `"100 g"` of water will result in an unsaturated solution.

At the `"38.0 g"` mark, the solution will become saturated.

Now, take a look at how the solubility graph for potassium nitrate looks like

So, you are starting with `"100 g"` of saturated solution at `75^@"C"`. Mind you, you have `"100 g"` of solution that contains as much dissolved potassium nitrate as possible.

This solution will contain

`100.0 color(red)(cancel(color(black)("g solution"))) * "155 g KNO"_3/((155 + 100)color(red)(cancel(color(black)("g solution")))) = "60.78 g KNO"_3`

Now, potassium nitrate's solubility is given per `"100 g"` of water. Calculate how much water you have in this initial solution

`m_"water" = "100.0 g" - "60.78 g" = "39.22 g"`

Next, determine how much potassium nitrate can be dissolved in `"39.22 g"` of water at `26^@"C"` in order to make a saturated solution, i.e. have the maximum amount of dissolved potassium nitrate possible

`39.22 color(red)(cancel(color(black)("g water"))) * "38.0 g KNO"_3/(100color(red)(cancel(color(black)("g water")))) = "14.9 g KNO"_3`

This means that when the initial solution is cooled from `75^@"C"` to `25^@"C"`, the amount of water that it contained will only hold `"14.9 g"` of dissolved potassium nitrate.

The rest will crystallize out of solution

`m_"crystallize" = "60.78 g" - "14.9 g" = color(green)("45.9 g")`

The answer is rounded to three sig figs.