Lesson Explainer: Solubility Product Chemistry

In this explainer, we will learn how to explain and calculate the solubility product.

There are many ionic substances that we would classify as being soluble, such as sodium chloride. However, there are also many substances that we would consider insoluble, such as barium sulfate or calcium carbonate. There are even some ionic substances that we may consider to be slightly soluble, such as calcium hydroxide.

When something is described as insoluble, it is often taken to mean that the substance does not dissolve at all in water. However, for insoluble substances such as barium sulfate or copper(II) hydroxide, the reality is that these substances can dissolve in water but only in very small amounts.

If we were to express this dissolution as an equilibrium for a general metal ion, M+, and a general anion, A–, we could write the dissolution equation as follows: MA()M()+A()saqaq+–

Here, we can imagine that the equilibrium lies very far to the left-hand side with only a very small number of ions actually being in solution.

This equilibrium can be expressed in the form of an equilibrium constant known as the solubility product, 𝐾sp. 𝐾sp for this general equation can be written as 𝐾=.sp+–[M][A]

Example 1: Constructing the Equation for the Solubility Product of a Generic Inorganic Compound

What is the equation for the solubility product of a generic inorganic compound with the formula MA?

Answer

The solubility product of a compound involves the product of the concentration of the ions raised to the power of their respective stoichiometric coefficients. This generic compound, MA, dissociates into a single positive ion of M+ and a single negative ion of A–. Square brackets are used to indicate that this is the concentration, and it is their product that is equal to 𝐾sp.

The equation is, therefore, 𝐾sp+–=[M][A].

The square brackets here indicate concentration, measured in mol⋅dm−3 or mol/L, and as such, the units to this particular solubility product equation would be mol2⋅dm−6, as calculated below: moldmmoldmmoldmmoldm⋅×⋅=⋅=⋅.

This particular type of equilibrium constant is different from the others where some value of the products is usually divided by some value of the reactants. For example, the equilibrium constant for concentration is calculated by multiplying together the concentrations of the products and then dividing that value by the product of the concentrations of the reactants. In the case of solubility products, the reactants in this heterogeneous equilibrium are the insoluble solid, and the amount that changes between the initial start of the dissolution and the equilibrium is so small that it can be considered constant and is therefore incorporated into 𝐾sp.

Definition: Solubility Product

The solubility product 𝐾sp of a compound is the product of the concentrations of the ions in a saturated solution raised to the power of their respective stoichiometric coefficients.

The following is an example: MA()M()+A()[M][A]saqaq<=>𝑚𝑎𝐾=.+–sp+–

When 𝐾sp equals the product of the ion concentration, the solution is said to be saturated: 𝐾=.sp+–[M][A]

However, when the product of the ion concentration becomes greater than the value of 𝐾sp, then a precipitate starts to form: 𝐾<.sp+–[M][A]

Finally, when 𝐾sp is greater than the product of the ion concentration, then more of the solid will dissolve before reaching equilibrium: 𝐾>.sp+–[M][A]

Furthermore, we all know that it is easier to dissolve more sugar in hot rather than cold tea, and as such, it should be no surprise that values of 𝐾sp are temperature dependent and vary according to the temperature at which they were measured.

Before looking at how to calculate the solubility product, it is important to correctly write the equation for 𝐾sp for ionic substances with more than one of the same ion: MA()M()+A()[M][A]2+–sp+–2saqaq2𝐾=.

Nickel phosphate has the formula Ni(PO)342, and so 𝐾sp can be written as follows: 𝐾=.sp2+343–2[Ni][PO]

The units can be calculated as follows: ⋅×⋅=⋅=⋅.moldmmoldmmoldmmoldm

Notice in the examples above how the different powers each of the concentrations were raised to affect the units of 𝐾sp.

Example 2: Constructing the Equation for the Solubility Product of Vanadium(III) Carbonate

What is the equation for the solubility product of vanadium(III) carbonate  (V(CO))233?

Answer

The solubility product can be defined as the product of the concentrations of the ions in a saturated solution raised to the power of their respective stoichiometric coefficients. In this instance, the two ions in solution will be vanadium 3+ and carbonate 2−. When vanadium carbonate dissolves in water, two vanadium 3+ ions and three carbonate 2− ions are produced: V(CO)()2V()+3CO()2333+32–saqaq

Therefore, we must raise the concentration of vanadium 3+ ions to the power of two and raise the concentration of carbonate ions to the power of three, giving us our final equation: 𝐾=.sp3+232–3[V][CO]

Questions related to 𝐾sp either rely on calculating values of 𝐾sp from given concentrations or on calculating concentrations and related pieces of information from values of 𝐾sp. We know that, for insoluble ionic substances, the equilibrium lies very much toward the reactants, and so, unsurprisingly, the values of 𝐾sp are very small.

For example, the value of 𝐾sp for silver bromide is 5.35×10 mol2⋅dm−6. Using this value, we can determine how much silver bromide, in grams, will actually dissolve in 1 dm3 (1‎ ‎000 mL) of water.

The equilibrium for the dissolution of silver bromide can be written as follows: AgBr()Ag()+Br()saqaq+–.

This gives us the equation for 𝐾sp: 𝐾=5.35×10⋅.sp+–(AgBr)[Ag][Br]moldm

We know that when the ionic substance dissolves, equal quantities of silver and bromide ions are produced. This allows us to state the following: [Ag][Br]+–=𝑥.

We can then substitute 𝑥 into the equilibrium equation: 𝐾=5.35×10⋅=𝑥.sp+–(AgBr)[Ag][Br]moldm

And we can solve for 𝑥 by taking the square root: 𝑥=√5.35×10=7.31×10⋅.moldm

We can therefore see that 7.31×10 moles of silver bromide has dissolved in 1 dm−3 (1‎ ‎000 mL) of water. Finally, we can use the molar mass of silver bromide (188 g/mol) and the number of moles to determine the mass that has dissolved: massmolgmolg=𝑛×𝑀=7.31×10×188/=1.37×10.

The final value tells us that 0.000137 g of silver bromide is dissolved in 1 dm3 (1‎ ‎000 mL) of water at 298 K.

Example 3: Calculating the Mass of Zinc Carbonate That Will Dissolve in 1000 mL of Water

Taking the solubility product of zinc carbonate to be 1.46×10 mol2⋅L−2 at 298 K, how many grams of zinc carbonate with molar mass 125.38 g/mol will dissolve in 1‎ ‎000 mL of water? Give your answer in scientific notation to 2 decimal places.

Answer

We can begin by writing out the equation for the dissolution of zinc carbonate at equilibrium: ZnCO()Zn()+CO()32+32–saqaq

From this, we can write out the equation for 𝐾sp: 𝐾==1.46×10.sp32+32–(ZnCO)[Zn][CO]

From this equation, we can also determine the units of 𝐾sp to be mol2⋅L−2.

We know that at equilibrium the concentrations of zinc ions and carbonate ions are equal, and we can symbolize this as 𝑥: [Zn][CO]2+32–=𝑥.

We can now substitute in the value given to us in the question for 𝐾sp: [Zn][CO]2+32–=1.46×10=𝑥.

We can solve for 𝑥 by taking the square root: 𝑥=√1.46×10=3.82×10⋅.molL

We can therefore see that 3.82×10 moles of zinc carbonate has dissolved in 1‎ ‎000 mL of water.

Finally, we use the molar mass to calculate the grams of zinc carbonate that will dissolve: MassofZnCOdissolvedmolgmolg3=3.82×10×125.38⋅=4.79×10.

We can also do this approach in reverse and calculate the value of 𝐾sp from the amount of substance that has dissolved in water to create a saturated solution.

Example 4: Calculating the Value of 𝐾𝐬𝐩 for a Saturated Solution of Copper(II) Hydroxide

A saturated solution of copper(II) hydroxide Cu(OH)2 contains 1.72×10 g of (Cu(OH))2 in every 1 dm3 (1 L) of water. Taking the molar mass of (Cu(OH))2 to be 97.56 g/mol, answer the following questions.

  1. What is the value of 𝐾sp without units, to two decimal places in scientific notation?
  2. What is the unit of 𝐾sp for this hydroxide?
    1. mol5⋅dm−15
    2. mol4⋅dm−12
    3. mol⋅dm−3
    4. mol2⋅dm−6
    5. mol3⋅dm−9

Answer

Part 1

Firstly, we can write the equation for 𝐾sp: 𝐾.sp22+–2(Cu(OH))[Cu][OH]

In order to calculate 𝐾sp, we firstly need to calculate the concentration of the ions at equilibrium using concentration=𝑛𝑉 where 𝑛 is the number of moles and 𝑉 is the volume.

We can calculate the number of moles using the molar mass and mass of copper(II) hydroxide: 𝑛=𝑚𝑀=1.72×1097.56⋅=1.763…×10.ggmolmol

As the volume is equal to 1 dm3, then the concentration is simply equal to 𝑛. Therefore, the concentration of ions at equilibrium is 1.763…×10 mol.

Next, we set the concentration of the ions to equal 𝑥: 1.763…×10=𝑥.mol

We know that, at equilibrium, the concentration of hydroxide ions is double the concentration of copper ions: Cu(OH)()Cu()+2OH()22+–saqaq

We equivalate this to 𝑥 and simplify it as follows: 𝐾=[𝑥][2𝑥]=4𝑥.sp22+–2(Cu(OH))[Cu][OH]

We then solve for 𝐾sp using the concentration we calculated earlier for 𝑥: 4𝑥=4×1.763…×10=2.19×10.

Part 2

If we look at the equation for 𝐾sp, we can see that there are three instances of concentration: 𝐾sp22+–2(Cu(OH))[Cu][OH]. Each set of square brackets indicates a concentration in mol⋅dm−3. As such, we have moldmmoldm⋅×⋅, which, when simplified, gives us mol3⋅dm−9.

For silver bromide, it was necessary for us to use the square root operation to resolve 𝑥 and determine the concentration. However, for some ionic substances that contain more than one anion or cation, more complicated roots may be necessary.

Consider an aqueous solution of aluminum hydroxide. If we know that the value of 𝐾=3×10⋅spmoldm, how can we calculate the concentration of Al3+ in a saturated solution of aluminum hydroxide?

Initially, we would write out the equation for the equilibrium and 𝐾sp: Al(OH)()Al()+OH()(Al(OH))[Al][OH]33+–sp33+–3saqaq3𝐾.

We can then substitute in 𝑥: [Al]3+=𝑥×(3𝑥)=27𝑥.

We can then substitute in our value for 𝐾sp and solve for 𝑥: 𝐾=3.0×10⋅=27𝑥3.0×10⋅27=𝑥3.0×10⋅27=𝑥.spmoldmmoldmmoldm

We are now able to solve for 𝑥 to give the concentration of Al3+ ions in the saturated solution: 𝑥=1.83×10⋅.moldm

Key Points

  • The solubility product is a heterogeneous equilibrium constant.
  • The solubility product can be defined as the product of the concentrations of the ions in a saturated solution raised to the power of their respective stoichiometric coefficients.
  • When the solubility product equals the product of the ion concentration, the solution is saturated.
  • Solubility products are temperature dependent with values usually quoted at 298 K.
  • Certain ionic substances may necessitate the use of cubed roots or fourth roots if calculating the concentration from 𝐾sp.

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