### Video Transcript

Alum is a hydrated sulfate of
potassium and aluminum with the chemical formula KAl(SO4)2•12H2O. What is the percentage by mass of
water in alum? Give your answer to two decimal
places. The molar mass of potassium is 39
grams per mole, aluminum is 27 grams per mole, sulfur is 32 grams per mole, oxygen
is 16 grams per mole, and hydrogen is one gram per mole.

In this question, we want to know
the percentage by mass of water in alum. The question states that alum is a
hydrated salt. A hydrated salt is a salt that has
water incorporated into its crystal lattice. The presence of water molecules
within the crystal structure of a compound is known as the water of
crystallization.

If we look at the chemical formula
of alum, we can see that for each formula unit of the salt, there are 12 molecules
of water. It is the presence of these
molecules of water which tells us this is a hydrated salt.

If we did an experiment and heated
the hydrated salt, we could find the mass of water that was lost during heating. Since we are not given experimental
data in this question, we need to use the molar masses of the water of
crystallization and the hydrated salt instead. Let’s first calculate the total
molar mass of water present in the hydrated salt.

The molar mass of water can be
calculated by adding together the average molar masses of its constituent atoms. Each water molecule contains two
hydrogen atoms and one oxygen atom. So using the average molar masses
provided in the problem, we find that the molar mass of water is 18 grams per
mole. Since there are 12 molecules of
water in alum, we need to multiply the molar mass of water by 12. The answer is 216 grams per
mole.

Now let’s calculate the molar mass
of the entire hydrated salt. We have already calculated the
molar mass of the water of crystallization, so all that remains is the molar mass of
the potassium aluminum sulfate. The average molar mass of potassium
is 39 grams per mole, and the average molar mass of aluminum is 27 grams per
mole. We need to multiply the average
molar mass of sulfur by two because there are two sulfur atoms in the chemical
formula. Finally, we need to multiply the
average molar mass of oxygen by eight because, excluding water, there are eight
oxygen atoms in the chemical formula.

Now we need to add together the
molar masses of potassium, aluminum, sulfur, oxygen, and water. Therefore, the total molar mass of
alum is 474 grams per mole.

Now we’re ready to calculate the
percentage by mass of water in alum. Let’s divide the molar mass of the
water, which is 216 grams per mole, by the total molar mass of alum, which is 474
grams per mole. After multiplying by 100, our
answer is approximately 45.5696 percent. However, we need to round our
answer to two decimal places. In conclusion, the percentage by
mass of water in alum is 45.57 percent.