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EXAMPLE POGIL ACTIVITY 2:
Solving Problems Involving More than One Equilibrium
Learning Objectives:
•
To learn how a competing acid-base equilibrium affects the solubility of a sparingly
soluble compound in water.
•
To gain experience in setting up mathematical descriptions of equilibria involving
solubility coupled with a weak acid or base.
• To gain confidence in engaging the tools of algebra to attack a non-trivial system of nonlinear equations in multiple unknowns.
Performance Criteria:
•
Be able to predict the effect of a competing acid-base equilibrium on the solubility of a
sparingly soluble compound in water.
•
Be able to prepare a system of mathematical equations describing relevant ionic
concentrations in a solution subject to more than one equilibrium.
•
Be able to calculate the molar solubility of a sparingly soluble compound in a solution
buffered at a particular pH.
Prior Knowledge:
•
Familiarity with the precipitation reaction and the solubility of sparingly-soluble
compounds.
•
Familiarity with the acid-base reaction and dissociation of weak acids.
•
Ability to calculate equilibrium concentrations of ions in saturated solutions or weak
acid-base equilibria taken separately.
•
Familiarity with the concept of activity and ability to calculate it given ionic strength.
Part 1: Why is the solubility of AgCN pH-dependent?
MODEL 1:
Consider the following equilibrium
AgCN (s) <--> Ag+(aq) + CN-(aq)
Ksp = 2.19 x 10-16
Suppose that we prepare a series of saturated solutions of AgCN in water buffered at
several pH values. The buffers are selected so that they contain neither Ag+ nor CN– and don’t
react with either. Suppose further that the buffer solutions are prepared in such a way that all
have a matching ionic strength of 0.1M. The solutions are allowed to stand overnight with slow
stirring to be sure that equilibrium is attained.
The solutions are allowed to settle and a sample of clear supernatant liquid is withdrawn
from each. If we were to determine some ions and compounds of potential interest in these
saturated solutions we’d come up with something like:
TABLE 1:
pH
+
[Ag ], M
[CN–], M
[HCN], M
1.00
1.88 x 10–4
1.17 x 10–12
1.88 x 10–4
2.00
5.96 x 10–5
3.69 x 10–12
5.96 x 10–5
3.00
1.88 x 10–5
1.17 x 10–11
1.88 x 10–5
4.00
5.96 x 10–6
3.69 x 10–11
5.96 x 10–6
5.00
1.88 x 10–6
1.17 x 10–10
1.88 x 10–6
6.00
5.96 x 10–7
3.69 x 10–10
5.95 x 10–7
7.00
1.89 x 10–7
1.16 x 10–9
1.88 x 10–7
8.00
6.14 x 10–8
3.58 x 10–9
5.78 x 10–8
9.00
2.40 x 10–8
9.18 x 10–9
1.48 x 10–8
10.00
1.60 x 10–8
1.38 x 10–8
2.22 x 10–9
11.00
1.50 x 10–8
1.47 x 10–8
2.37 x 10–10
12.00
1.48 x 10–8
1.48 x 10–8
2.39 x 10–11
13.00
1.48 x 10–8
1.48 x 10–8
2.39 x 10–12
1) Where on the pH scale does the most AgCN appear to dissolve?
Here’s another way of looking at some of the data in Table 1 that might give you a better
feeling for what’s going on:
GRAPH 1:
2) With reference to Table 1 and/or Graph 1 describe in words what appears to happen to the
amount of AgCN that dissolves as pH increases (becomes more basic).
3) What happens to the amount of CN– present as pH increases?
4) What happens to the concentration of HCN as pH increases?
Critical Thinking Questions
You’ll recall from General Chemistry that one can actually calculate the [Ag+] one would
expect to observe in a saturated solution of AgCN given its Ksp. At equilibrium a tiny amount of
AgCN dissolves according to the reaction:
AgCN(s) " Ag + + CN –
The degree of this dissociation is described by the Ksp expression just as you learned
previously:
!
K sp = 2.19 "10#16 = [ Ag + ][CN – ]
5) So suppose that we observe a tiny amount of Ag+ to be present at equilibrium in a solution
+
containing only water,
! inert materials and AgCN. Let’s represent it as [Ag ]. Give an
expression for [CN–] in that same solution in terms of [Ag+] (examine the dissociation
reaction stoichiometry above if you’re having trouble with this).
6) Calculate [Ag+] and [CN–] in this solution.
7) Do your [Ag+] and [CN–] results correspond to any of the observed ion concentrations in the
actual solutions in Table 1? Is there a solution or group of solutions there to which your
result comes closest?
8) You probably assumed that [Ag+] = [CN-] in order to do your calculation. Over what pH
range does this appear to be a good assumption?
9) In the very first paragraph of this activity it was claimed that the saturated solutions in Table
1 were prepared using buffers that contained no Ag+ and no CN–. Yet particularly at acidic
pH you’ll notice the presence of a fair amount of HCN. Where is this molecule coming from?
10) Give a reason why the assumption that [Ag+] = [CN–] might not be a good one over some pH
values.
Consider the fact that HCN is a weak acid in water:
11) Write a chemical equation illustrating the acid-base dissociation of HCN that you’d expect in
water.
12) Suppose we have a saturated solution of AgCN in water (H+ is always available to some
extent in water, depending upon the pH We’ll expect some Ag+ and a corresponding amount
of CN– to be formed as a result. LeChatelier’s principle allows you to predict the effect of a
“stressor” on an equilibrium system. In this case, what effect does the acid-base dissociation
reaction you wrote above have on the [CN-]
13) In view of the fact that the dissolution of AgCN in water is also an equilibrium (governed by
the Ksp) what effect would LeChatelier’s Principle predict this change in [CN–] to have on the
[Ag+]?
14) In view of the above, what should we expect to observe happen to [Ag+] in saturated
solutions of AgCN as the pH of the solution is decreased (becomes more acidic)?
15) In one or two complete sentences explain why the solubility of AgCN changes the way it
does as a function of pH.
PART 2: Setting Up Problems Involving Competing Equilibria
METHOD 1: Addressing Problems Involving More Than One Equilibrium
Step 1. Write chemical equations for all
equilibria that might apply.
Examples:
AgCl(s) " Ag + + Cl#
HOAc " H + + OAc#
H 2O " H + + OH#
Step 2. State what’s being sought in terms of a
concentration.
!
Examples:
!
[Ag+], [H+]
!
Step 3. Write mathematical expressions for all
the equilibria in 1.
Examples:
K sp = 1.8 "10#10 = [ Ag + ][Cl# ]
[H ][OAc ]
=
+
#5
K a = 1.76 "10
!
Step 4. Write mass balance equations.
#14
Kw = 1.00 "10
HOAc
= [ H + ][OH # ]
!Examples:
CAgCl = [ Ag + ]
!
CAgCl = [Cl" ]
!
Step 5. Count the number of mathematical
equations and the number of unknowns. They
should be equal. Fix things if they aren’t.
Step 6. Estimate the value of the result.
Step 7. Do what’s necessary to solve the
resulting system of simultaneous equations.
!
!
#
C HOAc = [OAc" ] + [HOAc]
Problem
A solution is prepared by adding solid AgCN to a pH 4.00 buffer that contains no Ag+ or
CN– and doesn’t react with either. The solution is stirred slowly until equilibrium is reached, at
which time there is still solid AgCN present on the bottom of the solution, demonstrating that it
is in fact saturated. Calculate the concentration of silver ion we’ll expect in this solution,
ignoring activity.
It should be clear from the initial parts of this activity that we’re going to have to deal not
only with the solubility equilibrium, governed by Ksp, but also the acid-base behavior of CN–.
The procedure below refers to Method 1:
16) Write chemical equations for all equilibria that might apply to a saturated solution of
AgCN(s) in water. Don’t forget that the water will dissociate to a tiny extent to form H+ and
OH–, as well as the formation of the weak acid HCN. If you enter your result next to Step 1
in the table on page 12 you’ll save yourself some searching when you want these again later
on.
17) What result do we seek? (read the problem! It’s really a drag to work diligently to obtain a
result that wasn’t asked for). Complete Step 2 by listing all the unknown concentrations you
can think of that might apply and then circle the one for which you need to find a value to
solve the problem.
18) Write mathematical equations describing each of the equilibria you wrote for Step 1. Each
will involve an equilibrium constant that you can look up in your textbook. Put them in the
table next to Step 3.
A mass balance expression is a mathematical expression that relates where atoms came
from to where they go. If I dumped one mole of NaCl into a liter of water I’d say the formal
concentration of NaCl, CNaCl , was 1F. Of course, there isn’t actually any NaCl in that solution.
The one mole of NaCl dissociates essentially completely to form one mole of Na+ and one mole
of Cl–, so CNaCl = [Na+] and CNaCl = [Cl–]. These are mass balance equations. In this case the NaCl
solution gives rise to two of them.
19) Here’s a mass balance expression that might apply to a solution containing acetic acid
(HOAc):
C HOAc = [OAc" ] + [HOAc]
Express in words what this mass balance expression tells us about the relationship between
the acetic acid that was originally added to the water and the ions so formed.
!
20) Draw a picture of the beaker, and include the species you would see, along with the forms in
which you might see them.
21) Now suppose that a tiny amount of AgCN dissolves to form the saturated solution. Represent
the formal concentration of AgCN that dissolves as CAgCN. Think about what happens to the
silver ion resulting from the dissociation of AgCN and write a mass balance expression
involving [Ag+]. Put your expression in the table under Step 4.
22) Think about what things might happen to the cyanide ion resulting from the dissociation of
AgCN and write a mass balance expression involving [CN–] as well as possibly other ions or
compounds that contain CN– from the AgCN that dissolved. Put that in the table under Step 4
too.
Part 3: Moving Toward a Solution
In order to solve a system of simultaneous equations it’s necessary that the number of
equations be exactly equal to the number of unknowns. If there aren’t enough equations, it’s
usually because a possible mass balance expression has been neglected. If there are too many
equations it’s possible that an unknown has been overlooked in the count, or that two or more of
the equations boil down algebraically to the same thing (mathematicians would say that they
aren’t independent).
23) Here’s where keeping the table on the next page current pays off. Count the number of
mathematical equations you have so far in that table (Steps 3 and 4).
24) List all of the unknowns in these equations. Don’t forget that we’ve been told the pH so you
can probably assign an explicit value to [H+]. How many unknowns are there?
25) Are the number of independent equations and the number of unknowns equal?
26) Why do you think it’s important or useful to estimate the value of the expected result before
going to the trouble of calculating it?
27) Estimate the value of the molar solubility of AgCN. Don’t forget that you used a simpler
method for obtaining this result earlier which, while wrong, might actually be close in terms
of order of magnitude. At the pH stated in the problem, would you expect the actual molar
solubility of AgCN to be greater than or less than the value you got using this simpler
method?
Table of data for solving the AgCN problem
Step 1. Write chemical equations for all
equilibria that might apply.
Step 2. State what’s being sought in terms of a
concentration.
Step 3. Write mathematical expressions for all
the equilibria in 1.
Step 4. Write mass balance equations.
Step 5. Count the number of mathematical
equations and the number of unknowns. They
should be equal. Fix things if they aren’t.
Step 6. Estimate the value of the result.
The equations in steps 3 and 4 form what is known as a system of simultaneous nonlinear equations. Solving them is a strictly mathematical problem that requires no knowledge of
chemistry (although it can often help). Theoretically at this point, therefore, one could hand off
these equations to a mathematician and instruct that they be solved for the variable circled in
Step 2. In this case however, as in most cases where the pH is known, the solution isn’t difficult
to obtain with a bit of algebra. You might elect to do it yourself, and then boast that you can
solve a system of 5 equations and 5 unknowns rapidly using only a calculator!
Exercise 1
Prepare a system of simultaneous equations whereby one might calculate [La+] in a
saturated solution of lanthanum fluoride, LaF3, in water buffered at pH 3.00. Note in particular
that HF is a weak acid under these conditions.
Exercise 2
Prepare a system of simultaneous equations that one might use to calculate the molar
solubility of silver sulfate, Ag2SO4, in water adjusted to pH 1.00 with HNO3. Note that HSO4– ion
is a weak acid (but the first proton of H2SO4 is strong). The molar solubility of a compound is
just the number of moles of that compound that dissolve in one L of solution. In this case you
will probably represent it as CAg2SO4, the formal concentration of Ag2SO4 that dissolves.
Exercise 3
Here’s one for you to tackle at home, using the solving strategies that we discussed in
class. Prepare a system of simultaneous equations that could be used to obtain the molar
solubility of mercury(I) thiocyanate, Hg2(SCN)2 in water at pH 1.00 taking into account that
hydrogen thiocyanate, HSCN, is a weak acid in water. Then wrestle with the algebra until you
get a real numerical result. The dissociation of this material is typical of mercury(I) compounds:
#
Hg 2 (SCN) 2 ( s) " Hg +2
2 + 2SCN
!