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Final Answers
© 2000-2022 Gérard P. Michon, Ph.D.

Wet  Chemistry

 Michon
 

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 Self-Ionization of Water

Liquid Water  &  Aqueous Solutions


(2011-07-17)   Self-Ionization of Pure Water
Equilibrium between water molecules, hydronium and hydroxide ions.

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 still working on this one...

pKw  (pure water)
Temperature
(°C)
H2OD2O
100°C 12.24   
50°C 13.26214.103
45°C 13.396 
40°C 13.53514.385
37°C 13.62   
35°C 13.680 
30°C 13.83014.699
25°C   13.9965   14.869  
  24.87°C  14.000 
20°C 14.16715.049
18°C 14.23   
15°C 14.346 
10°C 14.53515.439
5°C 14.734 
0°C 14.943 

According to that table, the  pH  of pure water is  7.000  only at  24.9°C.

Dissociation constants of water and heacy water as a function of temperature
 
Wikipedia :   Self-ionization of Water   |   Chemical equilibrium   |   Equilibrium constant
Activity   |   Activity coefficient   |   Dissociation constant   |   Acid dissociation constant
Solvation of hydronium ion  (dissolution and dilution of all hydrated proton species)


(2011-07-27)   Arrhenius Acids   (Svante Arrhenius, 1884)
Acidity is the symptom of protonation and deprotonation equilibria.

In 1884, Svante Arrhenius (1857-1927; Nobel 1903) attributed the acidity of water solutions to ...

Hydrochloric acid  (HCl)  is a strong monoacid :

Hydrochloric gas dissolved in water is essentially  completely  ionized:

HCl (g)  +  H2O   ®   H3O+  +  Cl -

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 still working on this one...

 van 't Hoff coat-of-arms
(2011-07-27)   Van 't Hoff's Law   (Van t'Hoff, 1886)
Osmotic pressure is to a solute what pressure is to an ideal gas.

In 1877, osmotic pressure was first measured by the botanist Wilhelm Pfeffer (1845-1920).

Chemical affinity.

In 1886, Jacobus H. van 't Hoff (1852-1911) found that osmotic pressure ...

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 still working on this one...


(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:

(997.0479 g/L) / (18.015268 g/mol)   =   55.34460 mol/L
       (1000 g/kg) / (18.015268 g/mol)   =   55.50847 mol/kg

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+).

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 still working on this one...


(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 i  ai / (1 + h/KAi )   =   h  +  å j  bj / (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 i  ai KAi / h   =   h  +  å j  bj 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 j  bj KBj  
  )   -   ½ log ( Kw )    
 
vinculum
Kw  +  å i  ai 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 M0.1 M0.01 M 1 mM0.1 mM0.01 mM 1 mM 0.1 mM
pH 2.382.883.383.884.384.88 5.385.88

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 nM0.1 nM0.01 nM1 pM
pH 6.3686.7816.9656.9966.9996

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.

Practical pH Theory and Use   |   pH (Wikipedia)


(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).

The simplest  carboxylic  example is  methylsuccinic acid  (CAS 498-21-5)  or  2-methylbutanedioic acid  C5H8O4  which can be synthesized as follows:

 Barium hydroxide-catalyzed methylsuccinic acid preparation

Also known as  pyrotartaric acid,  it is  listed  as having a pKa of  4.27.

Let  HAH  denote such an acid where  HA-  and  AH -  are different ions.

HAH  +  H2O « H3O+  +  AH - [ AH - ]  =   [HAH]  k1 / h
HAH  +  H2O « H3O+  +  HA- [ HA- ]  =   [HAH]  k'1 / h
AH -  +  H2O « H3O+  +  A- - [ A- - ]  =   [AH - ]  k2 / h
HA-  +  H2O « H3O+  +  A- - [ A- - ]  =   [HA- ]  k'2 / h
 

Let's introduce  K1   =   k1 + k'1   and   K2   =   kk2 / K1   =   k'k'2 / K1

As advertised, those new capitalized constants correspond to a viewpoint where both singly-ionized species are lumped together:

( [AH - ] + [HA- ] )       =     [HAH]  K1 / h
[A- - ]    =  ( [AH - ] + [HA- ] )     K2 / h

The total concentration  (a)  of the acid is then given by:

a   =   [HAH]  +  ( [AH - ] + [HA- ] )  +  [A- - ]
=   [HAH]  (1 + K1 / h + K1 K2 / h2 )

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:

( [AH - ] + [HA- ] )  +  2 [A- - ]
=   a  (K1 / h  +  2 K1 K2 / h2 ) / (1 + K1 / h + K1 K2 / h2 )

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 :

KA (x)   =   1  +  K1 x  +  K1 K2 x2  +  K1 K2 K3 x3  +  K1 K2 K3 K4 x4  +  ...

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:

KB (x)   =   1  +  K1 x  +  K1 K2 x2  +  K1 K2 K3 x3  +  K1 K2 K3 K4 x4  +  ...

For strong bases  (very large values of K1 and/or complete first ionizations):

  • KB (x)   =   x     for any strong monoprotic base, like  NaOH
  • KB (x)   =   x  +  ??? x2     for  barium hydroxide  Ba(OH)2

The concentration of the charges carried by all the cations of a base is:

b   (h/KW ) K'B (h/KW /  KB (h/KW )

Equating the charge of anions to the of cations and rearranging, we obtain the algebraic equation that is satisfied by  h = [H3O+ ]  at equilibrium:

h = 10-pH  for polyprotic acids and bases
   KW / h2   =    KW  +  å j  b j  K'Bj (h/KW /  KBj (h/KW )    
Vinculum
KW  +  å i  a i  K'Ai (1/h)  /  KAi (1/h)

 Come back later, we're
 still working on this one...

Citric acid  C3H4OH(COOH)3   (192.1235 g/mol).

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.

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(2011-08-01)   Ampholytes
Water is not the only compound that can act as either an acid or a base.

Carbonate ion

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 still working on this one...


(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). 

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(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  ( H2CO)  and bicarbonate ions  ( HCO3- ). 

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Wikipedia :   Buffer solution   |   Common-ion effect


(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:

 The two tautomeric forms of bisulfite ions.

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:

5 HSO3-  +  2 MnO4-  +  6 H +   ®   3 H2O  +  2 Mn++  +  5 HSO4-

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:

  1. 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...
     
  2. 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.

Wikipedia :   Bisulfite   |   Sulfite


(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.

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 still working on this one...

Wikipedia :   Carbonate hardness   |   Hard water   |   Water softening
Aquarium chemistry

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