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+ HAc H2O Log K

H2O 1 0

H+ 1 0

OH- -1 1 -14.0

HAc 1 0

Ac- -1 1 -4.756

H2O 1 55.4

HAc 1 10-3

{OH-} = Kw {H+}-1{H2O}

{H+} = K {H+}

{Ac-} = K {H+}-1{HAc}

How to Define the K !

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+ Ac- Na+ Log K

H+ 1

OH- -1 -14.0

HAc 1 1 4.756

Ac- 1

Na+ 1

H2O 55.4

NaAc 1 1 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+ + CO3

2-

• Species (Nspecies = 7)• H2O, H+, OH-, Na+, H2CO3

*, HCO3-, CO3

2-

• Components• Ncomp = Nspecies – Nrxn = 4• choose H2O, H+, Na+, and HCO3

- (likely to be predominant)

The new tableau for this recipe

H+ HCO3- Na+ Log K

H+ 1

OH- -1 -14

H2CO3* 1 1 6.3

HCO3- 1

CO32- -1 1 -10.3

Na+ 1

H2O 55.4

NaHCO3 1 1 10-3

Tableau Method• Mass Balances

– TOTHCO3 = [H2CO3*] + [HCO3

-] + [CO32-]= 10-3 M

– TOTNa = [Na+] = 10-3 M

– TOTH = [H+] – [OH-] + [H2CO3*] – [CO3

2-] = 0

• Again: Use TOTH to solve for pH

– TOTH = [H+] – [OH-] + [H2CO3*] – [CO3

2-] = 0

=> [H2CO3*] [CO3

2-]

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!