Video Transcript
Taking the solubility product of
aluminum hydroxide to be 1.90 times 10 to the negative 33 at 298 kelvin, what is the
concentration of Al3+ ions in a saturated solution? Give your answer in scientific
notation to two decimal places. And the answer options are (A) 2.90
times 10 to the negative nine moles per dm cubed. (B) 5.02 times 10 to the negative
nine moles per dm cubed. (C) 6.60 times 10 to the negative
nine moles per dm cubed. (D) 5.95 times 10 to the negative
12 moles per dm cubed. Or (E) 2.58 times 10 to the
negative 11 moles per dm cubed.
We are given a solubility product,
or Ksp value, of 1.9 times 10 to the negative 33. This is for aluminum hydroxide or
Al(OH)3. And we are asked to find the
concentration of aluminum three plus ions in a saturated solution. Now aluminum hydroxide is sparingly
soluble. It only partially dissolves in
water to give Al3+ ions and OH− or hydroxide ions according to this balanced
equation. We are told that the solution is
saturated, so we know that it is at equilibrium. We can write the Ksp expression for
this sparingly soluble salt. Ksp or the solubility product
constant is equal to the molar concentration of the products multiplied with each
other. So we can write both ions in square
brackets.
Now the stoichiometric coefficient
of Al3+ is one. And so we raise the concentration
of Al3+ to the power of one. And the stoichiometric coefficient
of the hydroxide ion is three. So we raise the concentration of
the hydroxide ion to a power of three. Now we are looking for the
concentration of the Al3+ ion. So putting in some data that we
know, specifically the Ksp value of 1.90 times 10 to the negative 33 — and by the
way, in this example, we are not given the units of Ksp though Ksp does have units —
and knowing that according to the balanced equation for every one mole of the solid
ionic substance, one mole of aluminum ions is formed and three moles of hydroxide
ions.
So if we make the concentration of
Al3+ ions equal to 𝑥, we could even write let the concentration of Al3+ equal 𝑥,
then the concentration of hydroxide ions must be three 𝑥 because of the
stoichiometric coefficients. You can write this information down
on the side if you wish. We must remember to raise the
concentration of hydroxide ions three 𝑥 to a power of three. You don’t have to expand three 𝑥
to the three, but I have here just to make it clear what numbers we are dealing with
and to do our calculation correctly. Multiplying everything out, we get
27𝑥 to the power of four. Let’s solve for 𝑥. So simplifying, we get 1.90 times
10 to the negative 33 divided by 27 equals 𝑥 to the four. To get rid of the fourth power, we
take the fourth root of both sides, and this gives 𝑥 equals 2.90 times 10 to the
negative nine.
Now 𝑥 is equal to the
concentration of Al3+, which is what the question was asking for. Since the concentrations of the
ions in a Ksp expression are given as molar concentrations, we can write the unit
for 𝑥 as moles per dm cubed. The answer is in two decimal places
and in scientific notation. So the concentration of Al3+ ions
in the saturated solution is 2.9 times 10 to the negative nine moles per dm
cubed.
