Animated Solution for Chemistry - Ionic Equilibrium: The stoichiometry and solubility product of a salt with the solubility curve given below is, respectively :

When tackling a physical chemistry problem, the units are often just as important as the numbers themselves. In this question, a quick glance at the options reveals that the unit for the solubility product, $K_{sp}$, is ${\text{M}}^3$ (or ${\text{mol}}^3{\text{L}}^{-3}$).

Why is this a massive hint? Because the solubility product is calculated by multiplying the molar concentrations of the constituent ions. If the final unit is ${\text{M}}^3$, it mathematically guarantees that the salt dissociates into exactly three ions.

Knowing that the salt produces three ions, we can narrow down its general formula to two possibilities:

Case 1: The salt is $X{Y}_2$. Upon dissociation: $X{Y}_2\rightleftharpoons X^{2+}+2Y^{-}$ The solubility product expression is:

Case 2: The salt is $X_{2}Y$. Upon dissociation: $X_{2}Y\rightleftharpoons 2X^{+}+Y^{2-}$ The solubility product expression is:

Now, we turn our attention to the provided solubility curve. To find the exact value of $K_{sp}$, we need to pick a clear, unambiguous point on the graph where the curve intersects the grid lines perfectly.

Looking closely, the curve passes exactly through the coordinate where: $[X]=1\text{ mM}$ $[Y]=2\text{ mM}$

Since our options are in Molar ($\text{M}$), we must convert these millimolar values: $[X]=1\times 10^{-3}\text{ M}$ $[Y]=2\times 10^{-3}\text{ M}$

Let's test our two possible stoichiometries using these extracted concentrations.

Testing $X{Y}_2$:

Testing $X_{2}Y$:

Comparing our calculated values with the given options, we can confidently conclude that the salt must be $X{Y}_2$ with a solubility product of $4\times 10^{-9}{\text{ M}}^3$. This perfectly matches option (d).