The solubility of insoluble substances can be decreased by the presence of a common ion. AgCl will be our example.

Present in silver chloride are silver ions (Ag^{+}) and chloride ions (Cl¯). Silver nitrate (which is soluble) has silver ion in common with silver chloride. Sodium chloride (also soluble) has chloride ion in common with silver chloride.

In fact, mixing sufficiently concentrated solutions of AgNO_{3} and NaCl will produce a precipitate of AgCl. In order to be sufficiently concentrated, the product of the [Ag^{+}] and the [Cl¯] must exceed the K_{sp} of 1.77 x 10¯^{10}.

**Example #1:** AgCl will be dissolved into a solution with is ALREADY 0.0100 M in chloride ion. What is the solubility of AgCl?

By the way, the source of the chloride is unimportant (at this level). Let us assume the chloride came from some dissolved sodium chloride, sufficient to make the solution 0.0100 M. So, on to the solution . . .

The dissociation equation for AgCl is:

AgCl (s) <===> Ag^{+}(aq) + Cl¯ (aq)

The K_{sp} expression is:

K_{sp}= [Ag^{+}] [Cl¯]

This is the equation we must solve. First we put in the K_{sp} value:

1.77 x 10¯^{10}= [Ag^{+}] [Cl¯]

Now, we have to reason out the values of the two guys on the right. The problem specifies that [Cl¯] is already 0.0100. I get another 'x' amount from the dissolving AgCl. Of course, [Ag^{+}] is 'x.'

Substituting, we get:

1.77 x 10¯^{10}= (x) (0.0100 + x)

This will wind up to be a quadratic equation which is solvable via the quadratic formula. However, there is a chemical way to solve this problem. We reason that 'x' is a small number, such that '0.0100 + x' is almost exactly equal to 0.0100. If we were to use 0.0100 rather than '0.0100 + x,' we would get essentially the same answer and do so much faster. So the problem becomes:

1.77 x 10¯^{10}= (x) (0.0100)

and

x = 1.77 x 10¯^{8}M

There is another reason why neglecting the 'x' in '0.0100 + x' is OK. It turns out that measuring K_{sp} values are fairly difficult to do and, hence, have a fair amount of error already built into the value. So the very slight difference between 'x' and '0.0100 + x' really has no bearing on the accuracy of the final answer. Why not? Because the K_{sp} already has significant error in it to begin with. Our "adding" a bit more error is insignificant compared to the error already there.

**Example #2:** The K_{sp} for silver carbonate is 8.4 x 10¯^{12}. The concentration of carbonate ions in a saturated solution is 1.28 x 10¯^{4} M. What is the concentration of silver ions?

**Solution:**

Dissociation equation:

Ag_{2}CO_{3}(s) <===> 2 Ag^{+}(aq) + CO_{3}^{2}¯ (aq)

K_{sp} expression:

K_{sp}= [Ag^{+}]^{2}[CO_{3}^{2}¯]

Let us substitue into the K_{sp} expression:

8.4 x 10¯^{12}= (x)^{2}(1.28 x 10¯^{4})

Note: I could have used (1.28 x 10¯^{4} + 0.5x) for the carbonate. See above discussion for why the 0.5x can be dropped.

Divide both sides by 1.28 x 10¯^{4} and then take the square root:

[Ag^{+}] = x = 2.56 x 10¯^{4}M

**Example #3:** The molar solubility of a generic substance, M(OH)_{2} in 0.10 M KOH solution is 1.0 x 10¯^{5} mol/L. What is the K_{sp} for M(OH)_{2}?

**Solution:**

In this case, we are being asked for the K_{sp}, so that is where our 'x' will be. That means the right-hand side of the K_{sp} expression (where the concentrations are) cannot have an unknown.

Dissociation equation:

M(OH)_{2}(s) <===> M^{2+}(aq) + 2 OH¯ (aq)

K_{sp} expression:

K_{sp}= [M^{2+}] [OH¯]^{2}

Let us substitue into the K_{sp} expression:

x = (1.0 x 10¯^{5}) (0.10)^{2}

The 1.0 x 10¯^{5} comes from the molar solubility information, coupled with the fact that for every one M(OH)_{2}, one M^{2+} is produced.

Also, we could have used (0.10 + 2.0 x 10¯^{5}) M for the [OH¯]. However, the 2.0 x 10¯^{5} M, being much smaller than 0.10, is generally ignored.

The answer:

K_{sp}= 1.0 x 10¯^{7}

**Example #4:** What is the solubility of AgI in a 0.274-molar solution of NaI. (K_{sp} of AgI = 8.52 x 10¯^{17})

**Solution:**

Dissociation equation:

AgI (s) <===> Ag^{+}(aq) + I¯ (aq)

K_{sp} expression:

K_{sp}= [Ag^{+}] [I¯]

Let us substitue into the K_{sp} expression:

8.52 x 10¯^{17}= (x) (0.274)

The answer:

[Ag^{+}] = 3.11 x 10¯^{16}M

**Example #5:** What is the solubility of Ca(OH)_{2} in 0.0860 M Ba(OH)_{2}?

**Solution:**

Ba(OH)_{2}is a strong base so [OH¯]= 2 times 0.0860 = 0.172 M

Dissociation equation:

Ca(OH)_{2}(s) <===> Ca^{2+}(aq) + 2OH¯ (aq)

K_{sp} expression:

K_{sp}= [Ca^{2+}] [OH¯]^{2}

The K_{sp} for Ca(OH)_{2} is known to be 4.68 x 10¯^{6}. We set [Ca^{2+}] = x and [OH¯] = (0.172 + 2x). Substituting into the K_{sp} expression:

4.68 x 10¯^{6}= (x) (0.172 + 2x)^{2}

Ignoring the "2x," we find x = 1.58 x 10¯^{4} M

Comment: There are several different values floating about the Internet for the K_{sp} of Ca(OH)_{2}. I got mine from the CRC Handbook, 73rd Edition, pg. 8-43.

**Example #6:** The solubility product of Mg(OH)_{2} is 1.2 x 10¯^{11}. What minimum OH¯ concentration must be attained (for example, by adding NaOH) to decrease the Mg^{2+} concentration in a solution of Mg(NO_{3})_{2} to less than 1.1 x 10¯^{10} M?

**Solution:**

K_{sp} expression:

K_{sp}= [Mg^{2+}] [OH¯]^{2}

We set [Mg^{2+}] = 1.1 x 10¯^{10} and [OH¯] = x. Substituting into the K_{sp} expression:

1.2 x 10¯^{11}= (1.1 x 10¯^{10}) (x)^{2}x = 0.33 M

Any sodium hydroxide solution greater than 0.33 M will reduce the [Mg^{2+}] to less than 1.1 x 10¯^{10} M.

**Example #7:** Calculate the pH at which zinc hydroxide just starts to precipitate from a 0.00857 M solution of zinc nitrate. K_{sp} for zinc hydroxide = 3.0 x 10^{-17}

**Solution:**

1) K_{sp} expression:

K_{sp}= [Zn^{2+}] [OH¯]^{2}

2) Substitute and solve for [OH¯]:

3.0 x 10^{-17}= (0.00857) (x)^{2}x = 5.91657 x 10

^{-8}M (I kept a few guard digits.)

3) Compute the pH:

pOH = 7.228pH = 6.772

Note how zinc hydroxide would precipitate even when the solution is slightly acidic.

**Example #8:** What is the solubility, in moles per liter, of AgCl (K_{sp} = 1.77 x 10^{-10}) in 0.0300 M CaCl_{2} solution?

**Solution:**

1) Concentration of chloride ion from calcium chloride:

0.0300 M x 2 = 0.0600 Mfrom here:

CaCl_{2}(s) ---> Ca^{2+}(aq) + 2Cl¯(aq)

2) Calculate solubility of Ag^{+}:

K_{sp}= [Ag^{+}] [Cl¯]1.77 x 10

^{-10}= (x) (0.0600)x = 2.95 x 10

^{-9}M

Since there is a 1:1 ratio between the moles of aqueous silver ion and the moles of silver chloride that dissolved, 2.95 x 10^{-9} M is the molar solubility of AgCl in 0.0300 M CaCl_{2} solution.

**Example #9:** Calculate the number of moles of Ag_{2}CrO_{4} that will dissolve in 1.00 L of 0.010 M K_{2}CrO_{4} solution. K_{sp} for Ag_{2}CrO_{4} = 9.0 x 10^{-12}.

**Solution:**

1) Concentration of dichromate ion from potassium chromate:

0.010 M

2) Calculate solubility of Ag^{+}:

K_{sp}= [Ag^{+}]^{2}[CrO_{4}^{2}¯]9.0 x 10

^{-12}= (x)^{2}(0.010)x = 3.0 x 10

^{-5}M

Since there is a 2:1 ratio between the moles of aqueous silver ion and the moles of silver chromate that dissolved, 1.5 x 10^{-5} M is the molar solubility of Ag_{2}CrO_{4} in 0.010 M K_{2}CrO_{4} solution.

Since we were asked for the moles of silver chromate that would disolve in 1.00 L, the final answer is:

1.5 x 10^{-5}mol

**Example #10:** What is the maximum concentration of Mg^{2+} ion that can remain dissolved in a solution that contains 0.7147 M NH_{3} and 0.2073 M NH_{4}Cl? (K_{sp} for Mg(OH)_{2} is 1.2 x 10¯^{11}; K_{b} for NH_{3} is 1.77 x 10¯^{5})

**Solution:**

1) Use the acid base data supplied to calculate [OH¯]:

K_{b}= ([NH_{4}^{+}] [OH¯]) / [NH_{3}]1.77 x 10¯

^{5}= [(0.2073) (x)] / 0.7147x = 6.10 x 10¯

^{5}M

2) Use the K_{sp} expression to calculate the [Mg^{2+}]:

K_{sp}= [Mg^{2+}] [OH-]^{2}1.2 x 10¯

^{11}= (x) (6.10 x 10¯^{5})^{2}x = 3.2 x 10¯

^{3}M

**Example #11:** 1.1 x 10^{-4} g of Cr(OH)_{3} is added to 120 L of water at 25 °C. Will it all dissolve? (K_{sp} = 6.7 x 10^{-31})

**Solution:**

1) Solve K_{sp} expression for the molar solubility:

K_{sp}= [Cr^{3+}] [OH¯]^{3}6.7 x 10

^{-31}= (x) (3x)^{3}x = 1.255 x 10

^{-8}M

2) Convert to grams per liter:

1.255 x 10^{-8}mol/L times 103.0 g/mol = 1.29 x 10^{-6}g/L

3) Check the problem's data:

1.1 x 10^{-4}g / 120 L = 9.17 x 10^{-7}g/LAll the Cr(OH)

_{3}dissolves.

**Example #11 - Part 2:** 4.0 x 10^{-4} g of NaOH is added. Will a precipitate form?

**Solution:**

1) Convert g/L to mol/L:

Cr(OH)_{3}9.17 x 10^{-7}g/L divided by 103 g/mol = 8.90 x 10^{-9}MNaOH

4.0 x 10^{-4}g / 120 L = 3.33 x 10^{-6}g/L3.33 x 10

^{-6}g/L divided by 40.0 g/mol = 8.33 x 10^{-8}M

2) Calculate a Q_{sp}:

x = (8.90 x 10^{-9}) (8.33 x 10^{-8})^{3}x = 5.15 x 10

^{-30}Cr(OH)

_{3}precipitates.

**Example #11 - Part 3:** Calculate the molar solubility of Cr(OH)_{3} in a solution buffered at pH = 11.00

**Solution:**

1) Determine the [OH¯]:

pOH = 14.00 - 11.00 = 3.00[OH¯] = 10¯

^{pOH}= 10¯^{3.00}= 1.0 x 10¯^{3}M

2) Determine the molar solubility:

6.7 x 10^{-31}= (x) (1.0 x 10¯^{3})^{3}x = 6.7 x 10

^{-22}M