Solving Equilibrium Problems
Using Tableaux
Aquatic Chemistry
CEE-367
Find the pH of a solution of HAc?
HAc = Acetic or Ethanoic Acid CH3-COOH!
• Recipe:
– [H2O]T = 55.4 M
– [HAc]T = 10-3 M
• Reactions (Nrxn = 2)
• H2O = H+ + OH- Kw
• HAc = H+ + Ac- Ka
• Species (Nspecies = 5)
• H2O, H+, OH-, HAc, Ac-
• Components
• Ncomp = Nspecies – Nrxn = 3
• choose H2O, H+, HAc (since we expect the pH to be acidic!)
Build the tableau:
Need to follow always a consistent method !
H+
H2O
H+
OHHAc
Ac-
HAc
1
-1
-1
Log K
1
0
0
1
1
1
H2O
HAc
H2O
1
1
How to Define the K !
{H+} = K {H+}
-14.0 {OH-} = Kw {H+}-1{H2O}
0
-4.756 {Ac-} = K {H+}-1{HAc}
55.4
10-3
Mass Balance Equations & Equilibrium
• Mass Balances
– TOTH20 = [H2O] +[OH-] = 55.4 M
– TOTHAc = [HAc] + [Ac-] = 10-3 M
– TOTH = [H+] – [OH-] – [Ac-] = 0
• Solve using the TOTH expression
–
–
–
–
Intuition! Acid in water => pH is Acid
The [OH-] should be negligible, then
[H+] = [Ac-]
Use the equilibrium constant Ka & TOTHAc to solve !
Another Example Sodium Acetate: Na-Ac !
• Recipe: Add NaAc to water:
– [H2O]T = 55.4 M
– [NaAc]T = 10-3 M
• Reactions (Nrxn = 2)
• H2O = H+ + OH• HAc = H+ + Ac-
• Species (Nspecies = 6)
• H2O, H+, OH-, HAc, Ac-, Na+
• Components
• Ncomp = Nspecies – Nrxn = 4
• choose H2O, H+, Na+, Ac• Select Ac- since we expect the pH to be basic
A new tableau with a different constant !
H+
OHHAc
AcNa+
H2O
NaAc
H+
1
-1
1
Ac-
Na+
Log K
-14.0
4.756
1
1
1
1
1
55.4
10-3
{HAc} = K {H+}{Ac-}
Tableau Method
• Mass Balances
–
–
–
–
–
–
–
TOTH20 = [H2O] +[OH-] = 55.4 M
TOTAc = [HAc] + [Ac-] = 10-3 M
TOTNa = [Na+] = 10-3 M
TOTH = [H+] – [OH-] + [HAc] = 0
Expect a pH basic => [H+] negligible, then
[OH-] = [HAc]
Use equilibrium constant &TOTAc to solve!
Sodium Bicarbonate in Water
• Recipe: Add Na bicarbonate to water
– [H2O]T = 55.4 M
– [NaHCO3]T = 10-3 M
• Reactions (Nrxn = 3)
• H2O = H+ + OH• H2CO3* = H+ + HCO3• HCO3- = H+ + CO32-
• Species (Nspecies = 7)
• H2O, H+, OH-, Na+, H2CO3*, HCO3-, CO32-
• Components
• Ncomp = Nspecies – Nrxn = 4
• choose H2O, H+, Na+, and HCO3- (likely to be predominant)
The new tableau for this recipe
H+
OHH2CO3*
HCO3CO32Na+
H2O
NaHCO3
H+
1
-1
1
-1
HCO3-
Na+
Log K
-14
6.3
1
1
1
-10.3
1
1
1
55.4
10-3
Tableau Method
• Mass Balances
– TOTHCO3 = [H2CO3*] + [HCO3-] + [CO32-]= 10-3 M
– TOTNa = [Na+] = 10-3 M
– TOTH = [H+] – [OH-] + [H2CO3*] – [CO32-] = 0
• Again: Use TOTH to solve for pH
– TOTH = [H+] – [OH-] + [H2CO3*] – [CO32-] = 0
=> [H2CO3*]  [CO32-]
How to Solve for H+ in TOTH
• Make approximations in TOT expressions
– Based on the predominance of species and how large we think they
are compared to the others!
– The more approximations you make the lower the degree of the
polynomial expression that you have to solve for becomes.
• Solve Numerically or Graphically using a pC-pH diagram
– Numerically: Carry literal expressions until the last step, your
numerical calculation.
• Use trial and error: write [H+] = f([H+]) and iterate until verified starting
from a guessed pH
– Graphically: demonstration in class
• And Verify Your Assumptions
– Check that indeed the species that you have neglected in the TOT
expressions are negligible compared to the others!
– If you do not perform these verifications you cannot expect to
receive full credit for your solution – even if it is correct!