Animated Solution for Chemistry - Ionic Equilibrium: If the solubility product of $A{B}_2$ is $3.20\times 10^{-11}{\text{ M}}^3$, then the solubility of $A{B}_2$ in pure water is ......... $\times 10^{-4}{\text{ mol L}}^{-1}$ [Assuming that neither kind of ion reacts with water]

The concept of solubility product ($K_{sp}$) is a beautiful intersection of stoichiometry and chemical equilibrium. It tells us exactly how much of a sparingly soluble salt can dissolve in a given amount of solvent before the solution becomes saturated. Let's dive into the mechanics of this problem.

Imagine a beaker filled with pure water. When we drop solid $A{B}_2$ into it, it doesn't just sit there; a tiny fraction of it begins to dissolve. The solid lattice breaks apart, releasing ions into the aqueous medium.

The balanced chemical equation for this dissociation is our roadmap:

Notice the stoichiometry here. For every one mole of $A{B}_2$ that dissolves, it produces one mole of $A^{2+}$ ions and two moles of $B^{-}$ ions. If we define the molar solubility of the salt as $S{\text{ mol L}}^{-1}$, then at equilibrium, the concentration of $A^{2+}$ will be $S$, and the concentration of $B^{-}$ will be $2S$.

The solubility product constant, $K_{sp}$, is defined as the product of the equilibrium concentrations of the dissolved ions, each raised to the power of its stoichiometric coefficient.

For our salt, the expression is:

Now, we substitute our equilibrium concentrations into this master equation:

This is where many students make a silly mistake. You must square the entire $(2S)$ term, which gives $4S^2$. Multiplying this by $S$ yields:

The problem provides us with the value of $K_{sp}$, which is $3.20\times 10^{-11}$. We can now set up our equation to solve for $S$:

Dividing both sides by 4 gives:

To make taking the cube root easier, let's shift the decimal point. Multiplying the coefficient by 10 and dividing the exponent by 10 gives us a much friendlier number:

Now, we take the cube root of both sides. The cube root of 8 is 2, and the cube root of $10^{-12}$ is $10^{-4}$:

The question asks for the value that multiplies $10^{-4}$, so our final answer is 2. Always remember to check the requested format of the answer!