Titrations of polyprotic acids:
Polyprotic acids and Ka values:
H3PO4 + H2O
H3O+ + H2PO4H2PO4- + H2O
H3O+ + HPO42H3O+ + PO43HPO42- + H2O
__________________________________
3H3O+ + PO43-
H3PO4 + 3H2O
[
]
[
]
K a1
[ H 3 O + ] H 2 PO4−
=
= 7.11x10 −3
[H 3 PO4 ]
Ka2
[ H 3O + ] HPO42 −
−8
=
=
6
.
32
x
10
H 2 PO4−
[
[
]
Ka1=7.11x10-3
Ka2=6.32x10-8
Ka3=4.5x10-13
Ka1Ka2Ka3
]
[ H 3O + ] PO43−
K a3 =
= 4.5 x10−13
2−
HPO4
[
]
When consecutive equilibria are added, the Ka values
are multiplied:
H3PO4 + 2H2O
2H3O+ + HPO42[ H 3O + ]2 HPO42−
K a1 K a 2 =
= 4.49 x10 −10
[H 3 PO4 ]
[
H3PO4 + 3H2O
K a1 K a 2 K a 3
]
[
3H3O+ + PO43-
]
[ H 3O + ]3 PO43−
=
= 2.0 x10 −22
[H 3 PO4 ]
Titrations of polyprotic acids:
multiple endpoints observable when Ka,n/Ka,n+1>103
Titration curve of a weak diprotic acid H2A:
1. pH before titration
2. pH before first equiv. point
3. pH at first equiv. pt.
4. pH between equiv. pts.
5. pH at second equiv. pt.
6. pH after second equiv. pt.
H2A + H2O
HA- + H2O
H3O+ + HAH3O+ + A2-
Ka1
Ka2
1. pH prior to titration:
for a strong diprotic acid, same as strong acid
for a weak diprotic acid,
if Ka1 > 103 Ka2, second equilibrium makes little
contribution
assuming autoprotolysis contributes little
+
[ H ] ≅ K a1CH 2 A or
2.
− K a1 + K a21 + 4 K a C H 2 A
[H ] =
2
+
pH prior to first equiv. pt., 1st buffer region
1st buffer region, both H2A and HA- present
if Ka1 > 103 Ka2, second equilibrium makes little
contribution, pH calculated like a normal buffer
solution
half way to equivalence, CH2A = CHA-
[H+] = Ka1
3.
pH at first equiv. pt.
solution is like that of a salt of a diprotic acid
(e.g., NaHA)
[H ] =
+
[
[
]
]
K a 2 HA − + K w
1 + HA − K a1
If it can be assumed that [HA-] ≈ CNaHA
[H ] ≈
+
K a 2C NaHA + K w
1 + C NaHA K a1
[ ]
+
If CNaHAKa1 > 10-13 and CNaHA/Ka1 > 100, H ≈ K a1K a 2
4.
pH in 2nd buffer region
2nd buffer region, both HA- and A2- present
if Ka1 > 103 Ka2, first equilibrium makes little
contribution, pH calculated like a normal buffer
solution
half way to equivalence, CHA- = CA2-
[H+] = Ka2
5.
pH at the second equivalence point
Like a salt of A2-, main equilibrium is
A2- + H2O
OH- + HA-
[
][
Kw
OH − HA−
K b1 =
=
Ka2
A 2−
[ ]
]
[OH − ] ≅ K b1C A2 −
or a more sophisticated relationship, if necessary
6.
pH beyond 2nd equiv. pt.
treated like the addition of strong base to water
Two common types of titration curves are used to
determine equivalence points for any kind of titration:
sigmoidal curves and linear-segment curve:
linear segment curve
Depends upon difference in instrument response
between reactants and products. Intersection of
response lines before and after equivalence point
determined location of equivalence point. Data
typically collected far from equivalence point.
E.g. titration leading to complex formation
Analyte
+ Reagent
→
Complex
lo response
hi response
lo response
Analyte
+
lo response
Reagent
→
lo response
Complex
hi response
Sigmoidal curve
Plot(s) of p-function of analyte (e.g., pH or pOH)
versus reagent volume. Careful measurements made
near the equivalence point.
Acid-base titrations usually make use of this
approach. Reagent solutions are almost always a
standardized solution of a strong acid or strong base
because they give sharper end points than do weak
acids or bases.
Identifying equivalence points:
1.
titration with indicators (sigmoidal curve)
2.
titration with linear-segment curve
3.
titration monitored with a pH meter
4.
Gran plot (see feature 14-5, text)
Titration monitored with a pH meter:
1st derivative shows point of greatest slope – eq. pt.
2nd derivative indicates inflection point – eq. pt.
A bit about indicators:
Characteristics of analytically useful chemical
reactions:
1.
2.
3.
Reactants and products are easily distinguished
The reaction provides useful information
The reaction proceeds at high rates/efficiencies
most acids and their conjugate bases are
“transparent” to visible radiation
pH indicators are exceptions, proton transfer
reactions involving indicators meet the criteria for an
analytically useful reaction
pH transition range for an indicator:
[H3O+] + [In-]
HIn + H2O
[H O ][In ]
=
+
Ka
−
3
[HIn]
[ H 3O]+ = K a
[HIn]
[In ]
−
color changes at ratios
and
≤ [HIn]/In-] = 0.1
≥ [HIn]/[In-] = 10.0
cannot be discerned by eye
Hence, useful range for a pH indicator is:
pH = pKa±1
Note: concentration of indicator must be minimized
to avoid introduction of systematic error