### How Equilibrium Calculations Can Be Applied to Complex Systems

```Jess Sproul
Chapter 10 Notes
Chapter 10
How Equilibrium Calculations Can
Be Applied to Complex Systems
10A: Solving Multiple-Equilibrium
Problems by a Systematic Method
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Aqueous solutions encountered in the lab often contain
several species that interact with one another and water
to yield two or more equilibria that function
simultaneously.
When water is saturated with barium sulfate, three
equilibria evolve:
BsSO4(s)  Ba2+ + SO42SO42- + H3O+  HSO4- + H2O
2H2O  H3O+ + OHThese equilibria will shift accordingly if ions are added.
Solution of a multiple-equilibrium problem requires as
many independents equations as there are participants in
the system being studied. In the above system, there are
five species: Ba2+, SO42-, HSO4-, H3O+, and OH-. In order
to calculate the solubility of barium sulfate, there needs to
be five independent algebraic equations.
10A-1: Mass-Balance Equation
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Mass-balance equations relate the equilibrium
concentrations of various species in a solution to one
another and to the analytical concentrations of the
various solutes.
Example 10-1
Write mass-balance expressions for a 0.0100 M solution of HCl that is in equilibrium with an excess of solid
BaSO4.
From our general knowledge of the behavior of aqueous solutions, we can write equations for three
equilibria that must be present in this solution.
BsSO4(s)  Ba2+ + SO42SO42- + H3O+  HSO4- + H2O
2H2O  H3O+ + OHBecause the only source for the two sulfate species is the dissolved BsSO4, the barium ion concentration
must equal the total concentration of sulfate-containing species, and a mass-balance equation can be
written that expresses this equality. Thus
[Ba2+] = [SO42-] + [HSO4-]
The hydronium ion concentration in this solution has two sources: one from the HCl and the other from the
dissociation of the solvent. A second mass-balance equation expression is this
[H3O+] + [HSO4-] = cHCl + [OH-] = 0.0100 + [OH-]
Since the only source of hydroxide is the dissociation of water, [OH-] is equal to the hydronium ion
concentration from the dissociation of water.
10A-2: Charge-Balance Equation
Solutions are neutral because the molar concentration of positive
charge in an electrolyte solution always equals the molar
concentration of negative charge.
# mol/L positive charge = # mol/L negative charge
The concentration of charge contributed to a solution by an ion is
equal to the molar concentration of that ion multiplied by its charge.
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For Na+
mol positive charge/L = (1 mol positive charge/mol Na+) X (mol Na+/L)
= 1 X [Na+]
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For Mg2+
mol positive charge/L = (2 mol positive charge/mol Mg2+) X (mol Mg2+/L)
= 2 X [Mg2+]
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For PO43-
mol positive charge/L = (3 mol positive charge/mol PO43-) X (mol PO43- /L)
= 3 X [PO43-]
10A-2: Charge-Balance Equation
(cont.)
•
Example 10-3
Write a charge-balance equation for the system in
Example 10-2.
[Ag+] + [Ag(NH3)2+] + [H3O+] + [NH4+] = [OH-] + [Br -]
•
Example 10-4
Neglecting the dissociation of water, write a chargebalance equation for a solution that contains NaCl,
Ba(ClO4)2, and Al2(SO4)3.
[Na+] + 2[Ba2+] + 3[Al3+] = [Cl-] + [ClO4-] + 2[SO42-]
10A-3: Steps for Solving Problems Involving
Several Equilibria
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Write a set of balanced chemical equations for all pertinent
equilibria.
Sate in terms of equilibrium concentrations what quantity is being
sought.
Write equilibrium-constant expressions for all equilibria developed
in step 1 and find numerical values for the constants in tables of
equilibrium constants.
Write mass-balance expressions for the system.
If possible, write a charge-balance expression for the system.
If the number of unknown concentrations is equal to the number
of equations at this point, proceed on to step 7. If the number of
unknown concentrations is greater than the number of equations,
seek additional equations. If there are no possible equations to
be found and suitable assumptions regarding the unknowns
cannot be made, the problem cannot be solved.
10A-3: Steps for Solving Problems Involving
Several Equilibria (cont.)
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Make suitable approximations to simplify the algebra.
Sole the algebraic equations for the equilibrium
concentrations needed to give a provisional answer as
defined in step 2.
Check the validity of the approximations made in step 7
using the provisional concentrations computed in step
8.Crucibles are used as either containers or filtering
devices that allow the supernatant to pass through while
retaining the precipitate.
Sometimes, filter paper is used and it must be burned off.
If this is the case, be sure to handle the heated crucible
with care.
10A-4: Making Approximations to
Solve Equilibrium Equations
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When step 6 is completed, there is the mathematical
problem of solving several nonlinear simultaneous
equations.
Without computer help, solving this complex system
is formidable, tedious, and time consuming.
Approximations can allow a complex system to be
solved much more easily.
Only the mass-balance and charge-balance
equations can be simplified because they involve only
sums and differences, not products and quotients.
With enough knowledge of the chemistry of a system,
it is possible to assume that a given term in a massbalance or charge-balance equation is sufficiently
small that it can be neglected.
10B-2: How pH Influences Stability
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The solubility of a
precipitate
containing an anion
with basic
properties, a cation
with acidic
properties, or both,
will depend on pH.
10B-3: The Solubility of Precipitates in the
Presence of Complexing Agents
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Complex Formation with a Common Ion
 Many precipitates react with the
precipitating agent to form soluble
complexes.
 Increases in solubility caused by large
excesses of a common ion are not
unusual.
Quantitative Treatment of the Effect of
Complex Formation on Solubility
 Solubility calculations for a precipitate in
the presence of a complexing reagent are
similar in principle to those discussed in
the previous section.
10C: Separating Ions by pH
Control: Sulfide Separations
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Several precipitating agents permit
separation of ions based on solubility
differences. This requires close control
over the reacting agent at a suitable and
predetermined lever. This is often done
by controlling the pH with suitable
buffers.
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