Activity

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Activity

• Introduction
• 1.) Hydration
• Ions do not act as independent particles in
  solvent (water)
• Surrounded by a shell of solvent molecules

Oxygen has a partial negative charge and hydrogen
partial positive charge
Oxygen binds cations
Hydrogen binds anions
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Activity
Introduction 2.) H2O exchanges rapidly between
bulk solvent and ion-coordination sites
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Activity

• Introduction
• 3.) Size of Hydration
• Size and charge of ion determines number of bound
  waters
• Smaller, more highly charged ions bind more water
  molecules
• Activity is related to the size of the
  hydrated species

Small Ions bind more water and behave as larger
species in solution
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Activity

• Effect of Ionic Strength on Solubility
• 1.) Ionic Atmosphere
• Similar in concept to hydration sphere
• Cation surrounded by anions and anions are
  surrounded by cations
• - Effective charge is decreased
• - Shields the ions and decreases attraction
• Net charge of ionic atmosphere is less than ion
• - ions constantly moving in/out of ionic
  atmosphere

Each ion-plus-atmosphere contains less net charge
and there is less attraction between any
particular cation and anion
Each ion see less of the other ions charge and
decreases the attraction
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Activity

• Effect of Ionic Strength on Solubility
• 2.) Ionic Strength (m)
• Addition of salt to solution increases ionic
  strength
• - Added salt is inert ? does not interact or
  react with other ions
• In general, increasing ionic strength increases
  salt solubility
• - Opposite of common ion effect

The greater the ionic strength of a solution, the
higher the charge in the ionic atmospheres
More ions added, more ions can be present in
ionic atmospheres
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Activity

• Effect of Ionic Strength on Solubility
• 2.) Ionic Strength (m)
• Measure of the total concentration of ions in
  solution
• - More highly charged an ion is the more it is
  counted
• - Sum extends over all ions in solution

where Ci is the concentration of the ith species
and zi is its charge
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Activity

• Effect of Ionic Strength on Solubility
• 2.) Ionic Strength (m)
• Example What is the ionic strength of a 0.0087
  M KOH and 0.0002 M La(IO3)3 solution? Assume
  complete dissociation and no formation of LaOH2

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Activity

• Effect of Ionic Strength on Solubility
• 3.) Equilibria Involving Ionic Compounds are
  Affected by the Presence of All Ionic Compounds
  in the Solution
• Knowing the ionic strength is important in
  determining solubility
• Example
• Ksp 1.3x10-18

If Hg2(IO3)2 is placed in pure water, up to
6.9x10-7M will dissolve. If 0.050 M KNO3 is
added, up to 1.0x10-6M Hg2(IO3)2 will dissolve.
Occurs Due to Changes in the Ionic Strength
Activity Coefficients
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Activity

• Equilibrium Constant and Activity
• 1.) Typical Form of Equilibrium Constant
• However, this is not strictly correct
• Ratio of concentrations is not constant under all
  conditions
• Does not account for ionic strength differences
• 2.) Activities, instead of concentrations should
  be used
• Yields an equation for K that is truly constant

where AA, AB, AC, AD is activities of A through D
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Activity

• Equilibrium Constant and Activity
• 3.) Activities account for ionic strength effects
• Concentrations are related to activities by an
  activity coefficient (g)
• 4.) Real Equilibrium Constant Using Activity
  Coefficients

where AC is activity of C C is
concentration of C gC is activity coefficient of
C
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Activity

• Equilibrium Constant and Activity
• 4.) Real Equilibrium Constant Using Activity
  Coefficients
• g is always 1
• Activity coefficient measures the deviation from
  ideal behavior
• - If g 1, the behavior is ideal and typical form
  of equilibrium constant is used
• Activity coefficient depends on ionic strength
• - Activity coefficient decrease with increasing
  ionic strength
• - Approaches one at low ionic strength

Activity depends on hydrated radius (a) of the
ion. This includes the ion itself and any water
closely associated with it.
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Activity

• Equilibrium Constant and Activity
• 5.) Activity Coefficients of Ions
• Extended Debye-H?ckel Equation
• Only valid for concentrations 0.1M
• In theory, a is the diameter of hydrated ion

where g is the activity coefficient a is
ion size (pm) z is the ion charge m is the
ionic strength
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Activity

• Equilibrium Constant and Activity
• 5.) Activity Coefficients of Ions
• In practice, a is an empirical value, provide
  agreement between activity and ionic strength
• - sizes can not be taken literally
• - trends are sensible ? small, highly charged
  ions have larger effective sizes
• a Li gt Na gt K gt Rb

Ideal behavior when g 1 - low ionic
strength - low concentration - low charge/large
a
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Activity
Activity Coefficients from Debye-H?ckel Equation
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Activity

• Equilibrium Constant and Activity
• 6.) Example 1
• What is the activity coefficient of Hg22 in a
  solution of 0.033 M Hg2(NO3)2?

Solution Step 1 Determine m
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Activity

• Equilibrium Constant and Activity
• 6.) Example 1
• What is the activity coefficient of Hg22 in a
  solution of 0.033 M Hg2(NO3)2?

Solution Step 2 Identify Activity Coefficient
from table at corresponding ionic strength.
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Activity

• Equilibrium Constant and Activity
• 6.) Example 2
• What is the activity coefficient for H at m
  0.025 M?

Note Values for g at m 0.025 are not listed in
the table.
There are two possible ways to obtain g in this
case a.) Direct Calculation (Debye-H?ckel)
zH
m
a for H from table
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Activity

• Equilibrium Constant and Activity
• 6.) Example 2
• What is the activity coefficient for H at m
  0.025 M?
• b.) Interpolation
• Use values for gH given at m 0.01 and 0.05 m
  from table and assume
• linear change in g with m.

To solve for gH at m 0.025
Fract. Of Interval Between 0.01 and 0.05
Diff. in g values at 0.01 and 0.05
gH at m 0.01
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Activity

• Equilibrium Constant and Activity
• 6.) Example 2
• What is the activity coefficient for H at m
  0.025 M?
• b.) Interpolation
• Use values for gH given at m 0.01 and 0.05 m
  from table and assume
• linear change in g with m.

Note This value is slightly different from the
calculated value (0.88) since it is only an
estimate.
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Activity

• Equilibrium Constant and Activity
• 7.) Activity Coefficients of Gasses and Neutral
  Molecules
• For nonionic, neutral molecules
• - g 1 for m 0.1 M
• - or Ac C
• For gases,
• - g 1 for pressures 1 atm
• - or A P, where P is pressure in atm
• 8.) Limitation of Debye-H?ckel Equation
• Debye-H?ckel predicts g decreases as m increases
• - true up to m 0.10 M
• At higher m, the equation is no longer accurate
• - at m 0.5 M, most ions actually show an
  increase in g
• with an increase in m
• - at higher m, solvent is actually a mixture
  instead of just
• water

Hydration sphere is mixture of water and salt at
high concentration
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Activity

• pH
• 1.) When we measure pH with a pH meter, we are
  measuring the negative logarithm of the hydrogen
  ion activity
• Not measuring concentration
• 2.) Affect of pH with the Addition of a Salt
• Changes ionic strength ? Changes H and OH-
  activity

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Activity

• pH
• 2.) Affect of pH with the Addition of a Salt
• Example
• What is the pH of a solution containing 0.010M
  HCl plus 0.040 M KClO4?

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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 1

What is the Hg22 in a saturated solution of
Hg2Br2 with 0.00100M KCl, where and KCl acts
as an inert salt?
Ksp 5.6x10-23
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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 2

What is the Hg22 in a saturated solution of
Hg2Br2 with 0.00100M KBr?
Note KBr is not an inert salt, since Br- is also
present in the Ksp reaction of Hg2Br2
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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 3

What is the true concentration of Li and F- in a
saturated solution of LiF in water?
Note Only LiF is present in solution. Ionic
strength is only determined by the amount of LiF
that dissolves
Solution Set-up the equilibrium equation in
terms of activities
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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 3

Note Both x and gLi,gF- depend on the final
amount of LiF dissolved in solution
To solve, use the method of successive of
approximation
Solution Assume gLi gF- 1. Solve for x.
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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 3

Solution Step 2 use the First Calculated Value
of Li and F- to Estimate the Ionic
Strength and g Values.
Obtained by using m0.041 and interpolating data
in table
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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 3

Solution Step 3 use the calculated values for
gF and gLi to re-estimate Li and F-.
substitute
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Activity

• Using Activity Coefficients
• 1.) Activity Coefficients Need to be Considered
  for Accurate Answers Involving Equilibrium
  Constants
• Example 3

Solution Repeat Steps 2-3 Until a Constant
Value for x is obtained
For this example, this occurs after 3-4 cycles,
where x 0.050M
gF, gLi
Use g to calculate new concentrations.
Use concentrations to calculate new m and g.
F-, Li
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Equilibrium

• Systematic Treatment of Equilibrium
• 1.) Help Deal with Complex Chemical Equilibria
• Set-up general equations
• Simplify using approximations
• Introduce specific conditions ? number of
  equations number of unknowns
• 2.) Charge Balance
• The sum of the positive charges in solution
  equals the sum of the negative charges in
  solution.

(positive charge)
(negative charge)
where C is the concentration of a cation n
is the charge of the cation A is the
concentration of an anion m is the charge of
the anion
A solution will not have a net charge!
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Equilibrium

• Systematic Treatment of Equilibrium
• 2.) Charge Balance
• If a solution contains the following ionic
  species H, OH-,K,H2PO4-,HPO42- and PO43-, the
  charge balance would be

The coefficient in front of each species
always equals the magnitude of the charge
on the ion.
For a solution composed of 0.0250 mol of KH2PO4
and 0.0300 mol of KOH in 1.00L
H 5.1x10-12M H2PO4- 1.3x10-6 M K
0.0550 M HPO42- 0.0220M OH-
0.0020M PO43- 0.0030M
Charge balance
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Equilibrium

• Systematic Treatment of Equilibrium
• 3.) Mass Balance
• Also called material balance
• Statement of the conservation of matter
• The quantity of all species in a solution
  containing a particular atom must equal the
  amount of that atom delivered to the solution

Acetic acid
Acetate
Mass balance for 0.050 M in water
Include ALL products in mass balance H3PO4?
H2PO4-,HPO42-, PO43-
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Equilibrium

• Systematic Treatment of Equilibrium
• 3.) Mass Balance
• Example 1

Write the mass balance for a saturated solution
of the slightly soluble salt Ag3PO4, which
produces PO43- and Ag when it dissolves.
Solution If phosphate remained as PO43-,
then but, PO43- reacts with water
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Equilibrium

• Systematic Treatment of Equilibrium
• 3.) Mass Balance
• Example 2

Write a mass balance for a solution of Fe2(SO4)3,
if the species are Fe3, Fe(OH)2, Fe(OH)2,
Fe2(OH)24, FeSO4, SO42- and HSO4-.
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Equilibrium

• Systematic Treatment of Equilibrium
• 1.) Write all pertinent reactions.
• 2.) Write the charge balance equation.
• Sum of positive charges equals the sum of
  negative charges in solution
• 3.) Write the mass balance equations. There may
  be more than one.
• Conservation of matter
• Quantity of all species in a solution containing
  a particular atom must equal the amount of atom
  delivered to the solution
• 4.) Write the equilibrium constant expression for
  each chemical reaction.
• Only step where activity coefficients appear
• 5.) Count the equations and unknowns
• Number of unknowns must equal the number of
  equations
• 6.) Solve for all unknowns

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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 1.) Example 1
• Ionization of water

Kw
Kw 1.0x10-14 at 25oC
Step 1 Pertinent reactions
Step 2 Charge Balance
Step 3 Mass Balance
H2O, H, OH- determined by Kw
Not True!
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 1.) Example 1
• Ionization of water

Step 4 Equilibrium constant expression
Step 5 Count equations and unknowns
Two equations
(1)
(2)
Two unknowns
(1)
(2)
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 1.) Example 1
• Ionization of water

Step 6 Solve
Ionic strength (m) of pure water is very low, gH
and gOH- 1
substitute
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 1 Pertinent reactions
This information is generally given
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 2 Charge Balance
Step 3 Mass Balance
Doesnt matter what else happens to these ions!
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 4 Equilibrium constant expression (one
for each reaction)
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 5 Count equations and unknowns
Seven Equations
(1)
(CB)
(2)
(MB)
(3)
(4)
(6)
(5)
(7)
Seven Unknowns
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 6 Solve (Not Easy!) - dont know ionic
strength ? dont know activity coefficients -
where to start with seven unknowns

• Make Some Initial Assumptions
• At first, set all activities to one to calculate
  ionic strength

• HOH-1x10-7, remaining chemical reactions
  are independent of water

• At first, ignore equations with small
  equilibrium constants

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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 6 Solve (Not Easy!)

• Assumptions Reduce Number of Equations and
  Unknowns
• Three unknowns
• Three equations

Mass balance and charge balance reduces to
Charge balance
Mass balance
H OH-
Low concentrations ? small equilibrium constant
Low concentrations ? small equilibrium constant
Simple Cancellation
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 6 Solve (Not Easy!)
So, CaSO4 is known
Therefore, only two equations and two unknowns
substitute
and
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 6 Solve (Not Easy!)
Given
Determine Ionic Strength
Determine Activity Coefficients
From table
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 6 Solve (Not Easy!)
Use activity coefficients and Ksp equation to
calculate new concentrations
Use new concentrations to calculate new ionic
strength and activity coefficients
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 6 Solve (Not Easy!)
Repeat process until calculated numbers converge
to a constant value
Stop, concentrations converge
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 2
• Solubility of Calcium Sulfate
• Find concentrations of the major species in a
  saturated solution of CaSO4

Step 7 Check Assumptions
With
Both HSO4- and CaOH are 5 times less than
Ca2 and SO42- ? assumption is reasonable
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 1 Pertinent reactions
Ksp
Ksp 7.1x10-12
K1
K1 3.8x102
Kw
Kw 1.0x10-14
Step 2 Charge Balance
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 3 Mass Balance (tricky)
OH- 2Mg2
But, two sources of OH-, OH- H
Account for both sources of OH-
Species containing OH-
Species containing Mg
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 4 Equilibrium constant expression (one
for each reaction)
Proper to write equilibrium equations using
activities, but complexity of manipulating
activity coefficients is a nuisance.
Most of the time we will omit activity
coefficients
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 5 Count equations and unknowns
Four equations
(1)
CBMB
(2)
(3)
(4)
Four unknowns
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 6 Solve (Not Easy!)

• Assumption to Reduce Number of Equations and
  Unknowns
• Solution is very basic OH- gtgt H, neglect
  H

CBMB
Rearrange K1 (ignore activity coefficients)
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 6 Solve (Not Easy!)
Substitute K1 into Mass or Charge Balance
Solve for Mg2
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Equilibrium

• Applying the Systematic Treatment of Equilibrium
• 2.) Example 3
• Solubility of Magnesium Hydroxide
• Find concentrations of the major species in a
  saturated solution of Mg(OH)2

Step 6 Solve (Not Easy!)
Substitute Mg2 into Ksp equation
Reduces to a single equation with a single
variable
Solve using spreadsheet, vary OH- until obtain
correct value for Ksp (7.1x10-12)
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Excel Demo of Goal Seek