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The concept of equivalence points (EP) runs like a golden thread through acid-base theory and applications. There are different types of equivalence points. This article provides a classification of EPs and semi-EPs. This is done for the general case of N-protic acids (based on simple mathematical equations).
The concept of equivalence points (EP) runs like a golden thread through acid-base theory and applications. There are different types of equivalence points. This article provides a classification of EPs and semi-EPs. This is done for the general case of N-protic acids (based on simple mathematical equations).
2.
The set of Equivalence Points
(EP & semi-EP) is the unique barcode
of an acid-base system.
Motivation
- 2 -
3.
An Equivalence Point is a special
equilibrium state at which chemical
equivalent quantities of acids and bases
are mixed:
EP: [acid] = [base]
square brackets indicate
molar concentrations
Common Definition
- 3 -
4.
This definition will be extended to
N-protic acid systems in two ways:
Part 1
Simplified Approach General Approach
Part 2
simple & nice equations
(but without self-ionization
of H2O)
based on integer and
half-integer values of
equivalent fraction n = CB /CT
- 4 -
5.
Simplified Approach
(valid for High-Concentrated Acids)
Part 1
6.
The Simplified Approach
is based on the equivalence of
acid species concentrations.
more precisely:
conjugated pairs of acid species
What is a conjugated acid-base pair?
- 6 -
7.
acid
(proton donor)
conj. base
(proton acceptor)
conjugate pair
HA = H+ + A-
- 7 -
8.
HA + H2O = A- + H3O+
conjugate pair
conjugate pair
acid base conj. base conj. acid
- 8 -
9.
Diprotic Acid
H2A = H+ + HA-
HA- = H+ + A-2
conj. acid base
acid conj. base
1st dissociation step:
2nd dissociation step:
HA- is the conj. base of acid H2A, and
HA- is the conj. acid of base A-2
- 9 -
10.
HA- is the conj. base of acid H2A, and
HA- is the conj. acid of base A-2
EP of HA-: [H2A] = [A-2]
Diprotic Acid
Notation:
The diprotic acid has 3 species: [H2A], [HA-] and [A-2].
They add up to the total amount: CT = [H2A] + [HA-] + [A-2] [H2A]T
11.
Triprotic Acid (H3A)
EP of H2A-: [H3A] = [HA-2]
EP of HA-2: [HA-2] = [A-3]
In addition, there are much more types of
equivalence points. Let‘s systematize it.
- 11 -
12.
Equivalence &
Semi-Equivalence Points of H3A
Part 1a
Triprotic Acid
13.
Triprotic Acid (H3A)
The 3-protic acid dissolves into 3+1 species:
H3A0 H2A-1 HA-2 A-3
There are several ways/equations
to define equivalence points
(including semi-equivalence points).
at least there are 23+1 EPs:
EPn with n = 0, ½, 1, ... 3
Note: We abbreviate the dissolved, neutral acid-species H3A0 also by H3A.
15.
Triprotic Acid
1st diss. step: H3A = H+ + H2A- K1 = [H+][H2A-]/[H3A]
2nd diss. step: H2A- = H+ + HA-2 K2 = [H+][HA-2]/[H2A-]
3rd diss. step: HA-2 = H+ + A-3 K3 = [H+][HA-3]/[HA-2]
stepwise equilibrium constants
K1 = K1 = [H+] pH = pK1
[H2A-] = [H3A]
(semi-EP)
[H+][H2A-]
[H3A]
K1K2 = K1K2 = [H+]2 pH = ½(pK1+pK2)
[HA-2] = [H3A]
(EP)
[H+]2[HA-2]
[H3A]
Each EP (or semi-EP) is characterized by one specific pH value
that relies directly on the acidity constants K: EP pH pK
pH = lg [H+]
pK = lg K
16.
EP0 [H+] = [H2A-]
EP1/2 [H3A] = [H2A-] pH1/2 = pK1
EP1 [H3A] = [HA-2] pH1 = ½(pK1+pK2)
EP3/2 [H2A-] = [HA-2] pH3/2 = pK2
EP2 [H2A-] = [A-3] pH2 = ½(pK2+pK3)
EP5/2 [HA-2] = [A-3] pH5/2 = pK3
EP3 [HA-2] = [OH-]
Each EP (or semi-EP) is characterized by one pH value
that relies directly on the acidity constants K: EP pH pK
The two “external EPs” EP0 and EP3 are non-trivial;
they depend, in addition, on the total amount of acid, CT.
- 16 -
21.
Unified Notation
for Internal EPs
EPn
pHn = ½ (pKn +pKn+1)
pHn = pKn+1/2
[n-1] = [n+1]
[n-½] = [n+½]
integer n = j
(n = 1,2, ... N-1)
half-integer n = j-½
(n = ½, ³/₂, ... N-½)
semi-EPs:
Note: j is integer and indicates the acid-species [j] and acidity constants Kj;
n is integer and half-integer and labels the EPs and semi-EPs.
the true, chemical meaning of n
becomes clear in Part 2
22.
External EPs
EPN [N-1] = [OH-] pHN 14
EP0 [H+] = [1] pH0 0
There is no simple relationship between EP pH.
pH of EP0 and EPN depend on K1 to KN and on CT (= total
amount of acid).
However,
for CT the following asymptotic behavior exists:
- 22 -
23.
pH
CT[M]CT[M]
n=0 n=0.5 n=1.5n=1
n=2
n=0
n=0.5
n=1.5n=1 n=2
n=2.5
n=3
H3A (phosphoric acid)
H2A (carbonic acid)
Examples
Internal EPs:
red lines
(independent of CT)
External EPs:
blue & green lines
(CT-dependent)
Curves as dashed (and not solid) lines remind us that the simplified approach is an approximation.
24.
Caution:
The simplified approach fails
for highly diluted acids
This restriction is
removed in the General Approach Part 2
(because self-ionization of H2O is ignored)
25.
General Approach
based on Equivalent Fraction n
Part 2
n = CB/CT
26.
(B+ = Na+, K+, NH4
+, ...)
n = =
CB amount of base
CT amount of acid
HNA BOH+ n
Titration of
weak acid HNA with strong base BOH
Equivalent Fraction n
- 26 -
27.
EP1 1 = CB /CT
EP½ ½ = CB /CT
EP1: [acid]T = [base]T
Generalization to other EPs:
EPn n = CB /CT
⁞
CT CB T = total amount
- 27 -
28.
equivalent fraction n =
CB
CT
Equivalent Points are special states
where the
n = 0, 1, ... N EPn
n = ½, 3/2, ... N-½ semi-EPn
Definition
becomes an integer or half-integer
value:
- 28 -
29.
Basic Equation for HNA
n = Y1(x) +
w(x)
CT
self-ionization H2O
x
x
K
w w
N211 Naa2aY
0j
j
j a
x
k
a
1
N
N
2
21
0
x
k
...
x
k
x
k
1a
ionization fractions (j = 0, 1, ... N)
with
cumulative equilibrium constants: k1 = K1, k2 = K1K2, ...
total amount of acid
x [H+] = 10-pH
Ref: www.aqion.de/file/acid-base-systems.pdf
30.
Plots of EPn in pH-CT Diagrams
for an integer or half-integer n
you get from
one curve CT = CT(n,x) with x = 10-pH
)x(Yn
)x(w
C
1
T
n
Given: N-protic acid HNA 2N+1 curves
Note: You can perform the calculations by Excel, for example.
- 30 -
31.
General Relationship
EPn pHn
pH
n (Equivalent Fraction)
pH1/2 pH3/2pH0 pH1 pH2
EP1/2
EP3/2
EP0
EP1
EP2
Titration Curve of Diprotic Acid (Example: 100 mM H2CO3)
n = Y1(pH) +
w(pH)
CT
32.
n=0 n=0.5 n=1
n=0 n=0.5 n=1 n=1.5
n=2
HA (acetic acid)
H2A (carbonic acid)
CT[M]
n=0
0.5 1.51 2 2.5
n=3
H3A (phosphoric acid)
n=0 0.5 1.51 2 2.5
n=3
H3A (citric acid)
pH
CT[M]
pH
EP & semi-EP
internal EP
external EP0
external EPN
dashed lines refer to the
“pure-acid case” in Part 1
33.
n = Y1(x) +
w(x)
CT
General Approach
n = Y1(x)
Simplified Approach
(Pure-Acid Case)
either w = 0 (ignoring self-ionization)
or CT 0 (high amount of acid)
EPn pHn =
½ (pKn +pKn+1) for integer n
pKn+1/2 for half-integer n
34.
comprises two subsystems (as limiting cases):
pure H2O defined by w(x) = 0 for CT = 0
pure HNA defined by n – Y1(x) = 0 for CT
n = Y1(x) +
w(x)
CT
The general approach
Alternative Interpretation
... and each subsystem has its own EPs
- 34 -
35.
The general approach
decouples into two subsystems:
CT = 0 CT
pure H2O pure HNA
with one single EP at pH 7 with EPn at pHn = ½(pKn +pKn+1)
and semi-EPn at pHn = pKn+1/2
w = 0 n – Y1 = 0 (poles of CT)
)x(Yn
)x(w
C
1
T
EPn
- 35 -
36.
The diagrams on the next two slides illustrate the situation
before and after both subsystems are joined together:
1st Diagram
The two uncoupled subsystems are located at both ends of the
CT scale:
• pure H2O at CT = 0 with one EP at pH 7
• pure acid at CT with EPs at pHn
2nd Diagram
Starting at pH 7 the curves fan out when CT increases until they
fit the ‘pure acid’ values at the top of the chart.
- 36 -
37.
Example H3PO4
CT[M]
pure acid (H3A)
pure H2O
two uncoupled subsystems
pH
pH=7
EP1/2 EP3/2EP1 EP2 EP5/2EP0 EP3
- 37 -
38.
Example H3PO4
CT[M]
acid + H2O
pH
EP1/2 EP3/2EP1 EP2 EP5/2EP0 EP3
)x(Yn
)x(w
C
1
T
Coupling of two Subsystems
- 38 -
40.
An N-protic acid has 2N+1 equivalence points:
EPn n = CB /CT for n = 0, ½, 1, ... N
Equivalence points are special equilibrium states where
the equivalent fraction n = CB/CT becomes an integer or
half-integer value.
The relationship EPn pHn is given by
where Y1 describes the acid and w the water.
n = Y1(pH) + w(pH)/CT
For high-concentrated acids (CT ) the relationship
simplifies to: n = Y1(pH)
which yields the direct
link to acidity constants:
½ (pKn +pKn+1) EP
pKn+1/2 semi-EP
pHn =
41.
For high-concentrated acids (CT ) there is
an alternative definition of EPs
based on equal species concentrations:
EPn: [n-1] = [n+1]
semi-EPn: [n-½] = [n+½]
Example: In carbonate systems EP1 is often introduced
as state where [H2CO3] = [CO3
-2].
The equivalent fraction n = Y1+ w/CT (titration curve)
describes the buffer capacity. Its pH-derivative is the
buffer intensity β = dn/dpH. EPs are extrema of β:
EPn minimum buffer intensity
semi-EPn maximum buffer intensity
43.
Simplified Approach
HNA Subsystem
General Approach
HNA + H2O
valid for large CT only (>10-3 M)
definition
based on acid species
EPn: [n-1] = [n+1]
semi-EPn: [n-½] = [n+½]
outcome
½ (pKn +pKn+1) EP
pKn+1/2 semi-EP
pHn =
CT
(nY1 = 0)
)x(Yn
)x(w
C
1
T
n
based on total amount of
compounds
EPn: n = =
[HNA]T
[strong base]T
CT
CB
- 43 -