(2011-07-17) Dilute Aqueous Solutions
The activity of water is assumed to be constant (55.51 mol/kg).
Under normal pressure (1 atm) at 25°C, the
density of water
(SMOW)
is 997.0479 g/L
and its molar mass is 18.015268g/mol.
If the presence of hydronium and hydroxide ions is neglected
(there's about 0.1 ppm of each species in pure water)
this corresponds to a molar concentration of water molecules equal to:
The quantity -log(55) = -1.74 is thus a lower bound for the
pH of a "solution" of water where nearly every molecule has been protonated
into an hydronium ion (H3O+).
(2011-07-17) pH (= pondus hydrogenii )
of aqueous solutions
The pH scale was introduced by S.P.L. Sørensen in 1909.
Although the "p" in pH (or also in pK, pKd, pKw, pKa or pKb)
is properly the abbreviation of a Latin term (pondus)
it's often given a common mnemonic
interpretation in whatever vehicular language is used... For example,
proportion d'hydrogène in French or
power of hydrogen in English.
The pH scale is primarily used to quantify the acidity
of diluted aqueous solutions containing a number of solutes that
interfer with the concentrations of hydronium or hydroxide ions normally
present in pure water due to the aforementioned autoprotolysis.
pH = - log10 h =
- log10 ( [ H3O+ ] )
Assume that water contains m monoprotic acids and n bases
partially ionized according to the following m+n+1
equilibrium relations. For i=1 to m and j=1 to n,
ai and bj
are the respective concentrations of the i-th acid and the j-th base.
The K coefficients
(Kw, KA, KB)
are the relevant equilibrium constants expressed
in the practical system where the chemical activity of water is unity
and other activities are equal to the concentrations expressed
in moles per liter (mol/L)
as is consistent with the pH definition.
Thus, for example,
ai = [ HAi ] + [ Ai- ]
and
KAi = h [ Ai- ]
/ [ HAi ]
Those two relations give the value of
[ Ai- ] stated below.
HAi + H2O
«
H3O+ + Ai-
[ Ai- ] =
ai/ (1 + h/KAi )
2 H2O
«
H3O+ + OH -
[ OH - ] =
Kw / h
Bj OH
«
Bj+ + OH -
[ Bj+ ] =
bj/ (1 + Kw / h KBj )
As the liquid is electrically neutral,
the total concentration of negative ions is equal to the total concentration
of positive ions and we obtain immediately:
General equation for h = [ H3O+ ] with
monoprotic acids and bases
Kw / h +
å iai/ (1 + h/KAi )
= h +
å jbj/ (1 + Kw / h KBj )
Solving this equation for h = 10- pH
(by successive approximations and/or dichotomy) gives the pH of
such an aqueous solution, without restrictions.
In the special case of very weak acids and bases,
the concentration of every ion is very small compared to the concentration
of the matching un-ionized species and the above boils down to the following approximation:
Kw / h +
å iai KAi / h
= h +
å jbj KBj h / Kw
That equation is linear in
h2 = 10- 2 pH
and gives the following formula:
pH of an aqueous solution of monoprotic
weak acids and weak bases
pH = ½ log
(
Kw +
å jbj KBj
)
- ½ log ( Kw )
Kw +
å iai KAi
Commonly, one base and one acid will respectively dominate the numerator
and the denominator of the above fraction (the dominant acid and/or the dominant base
can be ampholytic
water, which is always at concentration 1 by convention).
In this case, either sum is well approximated by its largest term and the following
simple approximation holds:
pH = ½ log ( b KB/ a KA KW )
Example :Acetic acid
(CH3COOH) at a mol/L in pure water:
pH = ½ log ( 1 / a KA )
= ( pKA - log a ) /2
= ( 4.76 - log a ) /2
Acetic acid at moderate or low concentrations ( 1 M = 1 mol/L )
a
1 M
0.1 M
0.01 M
1 mM
0.1 mM
0.01 mM
1 mM
0.1 mM
pH
2.38
2.88
3.38
3.88
4.38
4.88
5.38
5.88
At very low concentrations, the simplified formula is no longer valid but the
general formula for weak acids is not much more complicated, namely:
pH = - ½ log ( KW + a KA )
= 7 - 0.5 log ( 1 + a 106.24)
Acetic acid at extremely low concentrations :
a
0.01 mM
1 nM
0.1 nM
0.01 nM
1 pM
pH
6.368
6.781
6.965
6.996
6.9996
For the extreme dilutions where a 106.24
becomes very small, we may even expand our last formula as a
power series which gives,
in the main:
pH = 7.000 - 0.377 ( a / nM )
That approximation becomes usable when a is below 0.1 nM.
It's excellent when
a is 0.01 nM or less.
(2011-08-04) Diprotic acids
Some diprotic acids can donate two distinguishable protons.
All diprotic acids have the same type of mathematical
influence on the pH,
whether the two protons they can donate are distinguishable or not.
As shown below, the real thermodynamical equilibria result in the same acidity as
would be obtained by considering only the total concentration of all singly ionized species
(governed by fictitious global dissociation constants whose values depend
on the actual equilibrium constants).
As we previously observed, the influence of a particular acid on
the pH of the solution is entirely determined by the concentration of
negative charges carried by its anions.
In this case, that's given by the following expression:
A simpler formula is closer to what we used for monoprotic acids (K2 = 0):
a
(1 + 2 K2 / h )
/ (h / K1 + 1 + K2/ h )
Acidity doesn't depend on the ratio
[AH - ] / [HA- ].
For example, with a mol/L of our weak diprotic acid
and b mol/L of some strong monoprotic base, we obtain
an aqueous solution with a pH equal to -log h,
where h verifies the following equation of degree 4:
Kw / h + a
(1 + 2 K2 / h )
/ (h / K1 + 1 + K2/ h )
= h + b
(2011-08-06) Polyprotic acids & polyprotic bases
Formally, electrical neutrality is the key to pH computation.
Our remarks for diprotic acids generalize to all polyprotic
acids or bases:
We may lump together all the ions that have donated or received
the same number of protons. Some chemical properties
may depend on the relative proportions among ions that have gained or lost
the same number of protons, but acidity isn't one of them...
Formally, instead of spelling out the successive relevant coefficients
(K1, K2, K3, etc.)
we may introduce the following dissociation polynomial :
With this notation, if the hydronium concentration is h
then the total concentration (a)
of a weak polyprotic acid is
KA (1/h) times the concentration of its un-ionized form.
The concentration of all the electric charges carried by the
anions
of that particular acid is given by the following expression, involving the
derivative of the above polynomial:
a (1/h) K'A (1/h) / KA (1/h)
Note that this crucial formula is unchanged if we scale KA
arbitrarily. In particular, we obtain exactly the same result with:
KA (x) = 1/K1 + x +
K2 x2 +
K2 K3 x3 +
K2 K3 K4 x4 + ...
In the case of a strong acid where K1
is extremely large, the first term is negligible
(except for vanishing values of x corresponding to extremely acidic
solutions where the dissociation of the acid may no longer be considered perfect).
Thus:
KA (x) = x for any strong monoprotic acid, like HCl
KA (x) = x +
1.31
10-2
x2 for H2SO4
[pK2 = 1.883(2) ]
Similarly, for a weak polyprotic base, we introduce:
Sulfuric acid (H2SO4 )
is an example of a diprotic acid which is
a strong acid with respect to its first ionization and a
(relatively) weak acid with respect to the second ionization.
(2011-08-01) Ampholytes
Water is not the only compound that can act as either an acid or a base.
Carbonate ion
(2011-07-25) Neutralization of a Strong Acid and a Strong Base :
Simple arithmetic implies a sharp change in pH during titration.
In dilute aqueous solutions,
both NaOH and HCl are fully dissociated into ions
(this property is, by definition, what makes a simple acid or base a
strong one).
(2011-07-26) Buffer Solutions
Solutes that stabilize pH
Blood maintains its pH between 7.35 and 7.45
with a buffer of
carbonic acid ( H2CO3 ) and
bicarbonate ions ( HCO3- ).
(2011-08-09) Bisulfite and Sulfite Ions. Sulfurous Acid.
Unlike sulfuric acid, sulfurous acid only exists in an aqueous solution.
Dissolving sulfur dioxide in an alkaline solution
yields bisulfite ions:
SO2 + OH -
«
HSO3-
Those bisulfite ions are formally identical to the ions that would result
from the "first" dissociation of a fictitious acid
H2SO3 dubbed sulfurous acid,
which actually doesn't exist at all in its virtual un-ionized state:
H2SO3 + H2O
®
H3O+ +
HSO3-
The "second" dissociation of that virtual diprotic acid is quite real and produces
sulfite ions (pK2 = 6.97).
HSO3- + H2O
«
H3O+ +
SO3- -
The bisulfite ion
HSO3-
exists in two tautomeric forms:
Corrector (1964) :
Sulfurous acid is mostly used as a reducing agent based on the
redox properties of bisulfite,
which are normally expressed backward :
HSO4-
+ 2 H+ + 2 e-
«
H2O +
HSO3-
This can match the following half-reaction of oxidizing by permanganate :
MnO4-
+ 8 H +
+ 5 e-
«
Mn++ + 4 H2O
(+1.507 V)
Thus, we obtain the following balanced reaction, in an acidic solution:
That reaction, the discoloration and neutralization of permanganate
by bisulfite, is one-half of an invention that revolutionized
accounting around 1964, by forsaking the need to take a complex accounting
action for every simple mistake. The new product called
Corrector allowed such mistakes to be corrected by
erasing ink chemically in a clever two-step process:
A "red" solution of deeply-colored permanganate is applied
which attacks the ink by oxidation.
The ink disappears in seconds and the solution can be removed with blotting paper
(buvard). This leaves no trace of the original
ink but the spot is stained brown...
The "clear" solution of bisulfite
(sulfurous acid, with the distinctive smell of sulfur dioxide)
is then applied to that stain and discolors it instantly
via the above reaction. The clear liquid is then removed
with blotting paper and the spot is allowed to dry for a few seconds.
Some residue from the second solution is no big problem because
it's colorless and won't attack new ink. However, it can make it
difficult to apply Corrector again on the same spot
if you make another mistake...
Since the availability of Corrector (1964)
the rulings of
all accounting books have been printed with a pigment that resists the
action of Corrector.
Corrector was first sold in pairs of squarish glass bottles
with built-in applicators in the caps: Red cap and brown bottle for
the permanganate, white cap and clear bottle for the bisulfite.
As sold, the sulfurous solution was ready to use but the "red" bottle
contained just distilled water; you had to add the potassium permanganate
crystals supplied in a tiny plastic vial.
(2011-07-30) KH (alkalinity carbonate hardness )
GH (non-carbonate hardness)
Natural buffering. The concentration of carbonate and bicarbonates stabilizes pH.
Benefits of hard water include the presence of dietary calcium and magnesium ions
(Ca++, Mg++) and the reduced solubility of toxic metals like copper or lead.
