A General Program for pH-calculations in a Mixture of Acids and
Bases in Watersolutions.
By Stig Johansson Bräkne-Hoby, Sweden.
The Purpose. With the help of this program you will be able to caculate pH of a special
acid or base or a number of these substances in a watersolution. Both concentration pH
and activity pH may be calculated.
If there are other reactions than hydrolysis the calculations may not be valid.
The Mathematical Part of the solving consists of two sections.
The first section consists of a complete algebraic solving with regard to the conditions of
mass balance, charge balance and equilibriums for water, acids and bases. With the help
of these conditions you will be able to get [H3O+] as the only unknown factor, if the
values of concentrations and dissociation constants are known. [H3O+] will not be solved
explicitly but exists in an implicit form.
The second section of the calculation is performed with the help of a dator program,
which is based on Newton-Raphson’s method. This method is iterative in according with
the following formula, where [H3O+] is represented by X, and the derived function by
F’(X):
Xn+1 = Xn – F(Xn)/F’ (Xn)
The iteration proceeds until ABS(Xn+1 - Xn) < ABS(Xn+1/1e10). From the formula we
conclude that also the derivative F’ (X) must be calculated.
The categories of acids and bases, which have been used as examples, have been
taken from chemical handbooks for instance Handbook of Chemistry and Physics. It is
always possible to discuss which categories and how many of each sort of acid and base
will be represented. It is, however, very simple for the programmer to change and
construct programs with different constituents.
How large the program screen is depends on what sort of monitor is used. The
underlying basis of the program (ConcpH,ActpH) is adjusted to a 800*600 17¨monitor.
In using monitors with other performances it is possible to have more data on the screen
without scrolling.
Literature:
Handbook of Chemistry and Physics. 64th Edition. !983-84
G Christian: Analytical Chemistry. New York 1994.
A. Gordus: Analytical Chemistry. New York 1985.
G Hägg: Kemisk reaktionslära. Uppsala 1965.
A.E. Kehew: Applied Chemical Hydrogeology, New Jersey. 2001
E Lundqvist and A Sjöberg: Numeriska metoder. Uppsala 1970.
Acid-Base-functions in the pH-program.
In the functions, [H3O+] will be replaced by x and [OH-] by y.
Mathematical principle.
The final functions F(x) and F’(x) have been built by adding the partial functions fi(x)
respective fi’(x) , which represent the different sorts of substances where the function f1
represents the shared water and the other fi the special acids and bases. For each of the
acids and bases there is a complete solution, in which water is included, but in each of
their final functions the solitary terms of y and x have been taken away, as their watervalues must not be added to the shared water represented by f 1 , which participates in all
calculatioins.
Kw=1.00E-14 (pKw=14.00) is default but can be changed
1
Calculating ConcpH.
Pure water.
OH   H O 

H O  OH   K
1

3


3
y  x  f1 x   y  x  0
1
y  x  KW
2
2
KW
x0
x
K

f 1  x    W2  1  0
x
f1 x  
Strong 1-protic acid
C1
M
HA  H 2 O  H 3O   A 
HA
OH   A   H O 



3

y
yx
has
already
been
C1 
brought
x
and
 y  C1  x  0
f 2 x   C1  0

f 2 x   0
Strong 2-protic acid.
C1
M
H2 A
C2
M
NaHA
H 2 A  H 2O  H 3O   A
NaHA  Na  HA 
HA   H 2O  H 3O   A2 
C3
M
K 2 A  2 K   A2 
K2 A
OH  HA  2  A   Na  2  K  H O 
y  HA   2A   C
 2C
(1)
C  C  C  C  HA   A  (2)
K  HA   x  A  3


2



3

2
2

1
2
3
3
2
Tot

2
2
KW  x  y
y
4
x  CTot 2 K 2  CTot

 C2  2C3  x  0
x  K2
x  K2
f3 x  
x  CTot 2 K 2  CTot

 C2  2C3  0
x  K2
x  K2
f 3 x   
K 2  CTot
x  K 2 2
2
Weak 1-protic acid
HA  H 2 O  H 3 O   A 
C1
M
HA
C2
M
NaA
aq
NaA 
Na   A 
OH   A   Na   H O 
y  A  
C

x 1
C C C
 HA   A  2 
K  HA   x  A  3




3

2

1
2
Tot

4
KW  x  y
K  C Tot
 C2  x  0
Kx
K  C Tot
f 4 x  
 C2  0
Kx
K  C Tot
f 4 x   
0
K  x 2
y
Weak 2-protic acid.
H 2 A  H 2 O  H 3 O   HA 
C1
M
H2 A
C2
M
NaHA
C3
M
K2 A
aq
NaHA 
Na   HA 
aq
K 2 A 
2 K   A 2
OH   HA   2  A   Na   2  K   H O 
1
y  HA   2  A   C
 2C  x
2
C  C  C  C  H A  HA   A 
3
K  H A  x  HA 
4
K  HA   x  A 


2



3

2
2
3

1
2
3
Tot
2
2

1
2

2
2
KW  x  y
y
5
K 1CTot x
2 K 1 K 2 CTot

 C 2  2C 3  x  0
K1 K 2  K1 x  x 2 K1 K 2  K1 x  x 2
f 5 x  
K 1CTot  x  2 K 2 
 C 2  2C 3  0
K1 K 2  K1 x  x 2
f 5x   

K 1CTot K 1 K 2  x 2  4 K 2 x
K K
1
2
 K1 x  x

2 2
0
3
Weak 3-protic acid.
C1
M
H3 A
H 3 A  H 2 O  H 3O   H 2 A 
C2
M
LiH 2 A
aq
LiH 2 A 
Li   H 2 A 
C3
M
Na2 HA
aq
Na2 HA 
2 Na   HA 
C4
M
K3 A
aq
K 3 A 
3 K   A 3
OH   H A   2  HA   3  A   Li   2  Na   3  K   H O 
y  H A   2  HA   3  A   C  2C  3C  x 1
C  C  C  C  C  H A  H A   HA   A  2 
K H A  x  H A  3
K H A   x  HA  4 
5
K HA   x  A 


2
3



2

2
3
2
2

1
2

3
3
4
Tot
3
3
2
4
3
2

1
2
3
2

2
2
2
3
3
KW  x  y
y
6
K 1CTot x 2 2 K 1 K 2 CTot x 3K 1 K 2 K 3 CTot


 C 2  2C 3  3C 4  x  0
N
N
N
N  K1 K 2 K 3  K1 K 2 x  K1 x 2  x 3
f 6 x  
K 1CTot x 2 2 K 1 K 2 CTot x 3K 1 K 2 K 3 CTot


 C 2  2C 3  3C 4  0
N
N
N
f 6 x  
2 K 1CTot x K 1CTot x 2  N  2 K 1 K 2 CTot 2 K 1 K 2 CTot x  N  3K 1 K 2 K 3 CTot  N 




0
N
N
N2
N2
N2
N   K1 K 2  2 K1 x  3x 2
4
Weak 4-protic acid
C1
M
H4 A
H 4 A  H 2 O  H 3O   H 3 A 
C2
M
LiH 3 A
aq
LiH 3 A 
Li   H 3 A 
C3
M
Na 2 H 2 A
aq
Na 2 H 2 A 
2 Na   H 2 A 2 
C4
M
K 3 HA
aq
K 3 HA 
3K   HA 3
C5
M
Rb 4 A
aq
Rb 4 A 
4 Rb   A 4 
OH   H A   2  H A   3  HA   4  A   Li   2  Na   3  K   4  Rb   H O 
y  H A   2H A   3HA   4A   C  2C
 3C
 4C
 x 1
C  C  C  C  C  C  H A  H A   H A   HA   A  2 
K  H A  x  H A  3
K  H A   x  H A  4 
K  H A   x  HA  5
6
K  HA   x  A 


2
3

3
3
4




2
2
3
4
2
2
3

1

3
2
3
4
5
Tot
1
4
2
3
3
2
4
3
4
5
2
3
4
2

3

2
2
2
3
3
4
4
KW  x  y
7 
x 3 K 1CTot 2 x 2 K 1 K 2 CTot 3xK1 K 2 K 3 CTot 4 K 1 K 2 K 3 K 4 CTot



 C 2  2C 3  3C 4  4C 5  x  0
N
N
N
N
3
2
x K 1CTot 2 x K 1 K 2 CTot 3 xK1 K 2 K 3 CTot 4 K 1 K 2 K 3 K 4 CTot
f 7 x  



 C 2  2C 3  3C 4  4C 5  0
N
N
N
N
N  K1 K 2 K 3 K 4  K1 K 2 K 3 x  K1 K 2 x 2  K1 x 3  x 4
y
3K 1 x 2 CTot K 1 x 3 CTot N  4 K 1 K 2 xCTot 2 K 1 K 2 x 2 CTot N  3K 1 K 2 K 3 CTot 3K 1 K 2 K 3 xCTot N  4 K 1 K 2 K 3 K 4 CTot N 






0
N
N
N
N2
N2
N2
N2
N   K 1 K 2 K 3  2 K 1 K 2 x  3K 1 x 2  4 x 3
f 7  x  
5
Strong 1-protic base.
C
M

  
  H O 
OH   C
B  H 2 O  BH   OH 
B
OH   BH


y



3

C
x
y C  x  0
f 8  x   C  0
f 8 x   0
Strong 2-protic base.
B  H 2 O  BH   OH 
C1
M
B
C2
M
BHCl
C3
M
BH 2 Br2
aq
BH 2 Br2 
BH 2  2 Br 


2


2
3
2

2
3
2
2

1
2
3
Tot
2
2
2
KW  x  y
y  C 2  2C 3 
4
CTot K W  2 K b 2 x 
x0
K W  Kb2 x 
f 9  x   C 2  2C 3 
CTot K W  2 K b 2 x 
0
K W  K b 2 x 
CTot K b 2 K W
K W
1
2

b2
f 9 x   
B  0
aq
BHCl 
BH   Cl 
OH   Cl   Br   BH   2  BH  H O 
y C
 2C
 BH   2  BH   x
C  C  C  C  B   BH   BH 
3
K  BH   BH  y

dvs
 K b2 x
2
0
6
Weak 1-protic base.
B  H 2 O  BH   OH 
C1
M
B
C2
M
BHCl
aq
BHCl 
BH   Cl 
OH   Cl   BH   H O 
y  C  BH   x
C  C  C  B   BH 
K  B   BH  y




3
1

2
2

1
2
Tot
3

b
4
KW  x  y
y  C2 
K b CTot x
x0
K b x  KW
K b CTot x
0
K b x  KW
f10 x   C 2 
f10 x   
K b K W CTot
 K b x  K W 2
Weak 2-protic base.


B  H 2 O  BH  OH
C1
M
B
C2
M
BHCl
C3
M
BH 2 Br2
aq
BHCl 
BH   Cl 
aq
BH 2 Br2 
BH 2
2
 2 Br 
OH  Cl  Br   BH  2  BH  H O 



2


2
y
 C2

3


 BH   2  BH 2
 2C3
2
 

C1  C 2  C3  CTot  B   BH   BH 2
B K b1  BH   y
BH  K

b2

 BH 2
2
y  C 2  2C3 
2
1
2
3
 y
KW  x  y
 x

4
5
K b1CTot K W x
K b1 K b 2 x 2  K b1 K W x  K W
2

2 K b1 K b 2 CTot x 2
K b1 K b 2 x 2  K b1 K W x  K W
2
x0
K b1CTot K W x 2 K b1 K b 2 CTot x 2

0
N
N
2
2
K C 4 K b 2 x  K W  K b1 CTot x2 K b 2 x  K W 
f11 x    b1 Tot

0
2
N
N
f11 x   C 2  2C3 
N  K b1 K b 2 x 2  K b1 K W x  K W
2
7
The calculated partial functions will be added as below
F(x)=f1(x) + f2(x) + .... +f11(x) = 0 och F´(x)=f´1(x) + f´2(x) + ..... + f´11(x) = 0
After that the iterative expression xn+1 = xn - F(xn)/F´(xn) is used until ABS(xn+1-xn)
<ABS(xn/1e10).
[H3O+] = x2, where 1. 10-15 <= x2 =<1.
At last the final concentrations can be calculated according to the following.
Calculation of the final concentrations at the calculated pH.
Strong 1-protic acid.
C
M
HA  0
HA  H 2 O  H 3 O   A 
HA
A   C

C1
M
H2A
Strong 2-protic acid.
H 2 A  H 2O  H 3O   A 
C2
M
HA 
H 2 A  0
C3
M
A 2

   1
  H O  A  2
C1  C 2  C 3  C Tot  HA   A 2 

K 2 HA 
HA   KC

A   CK
x
x
2
Tot
2

2
3
Tot
2
K2
x
Weak 1-protic acid.
C1
M
HA
HA  H 2 O  H 3O   A 
C2
M
A
C1  C 2  CTot  HA   A 

 
K  HA   H 3O  A
A   Kx CK

Tot

 
1
2

HA  CTot  x
xK
Weak 2-protic acid.
C1
M
H2A
H 2 A  H 2 O  H 3 O   HA 
C2
M
HA 
HA   H 2 O  H 3 O   A 2 
C3
M
A 2
   
2
K H A  x  HA 
3
K HA   x  A 
C1  C 2  C 3  C Tot  H 2 A  HA   A 2 
1

1
2

2
2
A   x
K 1 K 2 C Tot
2
2
 K1 x  K1 K 2
HA   x
K 1C Tot x

2
 K1 x  K1 K 2
H 2 A 
C Tot x 2
x 2  K1 x  K1 K 2
8
Weak 3-protic acid.
M
H3A
C2
M
H 2 A
M
HA
C4
M
A 3
A   K K
H
2

A 
   x  HA
K HA   x  A 
K2 H2 A

2
2


  
3
HA   K K
K 3  K1 K 2 x  K1 x  x
3
1
K 1C Tot x 2
K 1 K 2 C Tot x
2
K 3  K1 K 2 x  K1 x 2  x 3
C Tot x 3
H 3 A 
K1 K 2 K 3  K1 K 2 x  K1 x 2  x 3
1
2
3
4
2
2
2
 
3
K 1 K 2 K 3 C Tot
3
1

K 1 H 3 A  x  H 2 A 
2
C3

C1  C 2  C 3  C 4  C Tot  H 3 A  H 2 A   HA 2   A 3
C1
K1 K 2 K 3  K1 K 2 x  K1 x 2  x 3
Weak 4-protic acid.
C1
M
H4A
C2
M
H 3 A
C3
M
H 2 A 2
C4
M
HA 3
C5
M
A 4
A   K K
4
1
2

K 1 H 4 A  x  H 3 A 
 
 
  


1
2
 
 3
  x  HA  4
  x  A  5
K 2 H 3 A   x  H 2 A 2

K 3 H 2 A 2

K 4 HA 3
K 3 K 4 C Tot
N
H 4 A  CTot x

C1  C 2  C 3  C 4  C Tot  H 4 A  H 3 A   H 2 A 2   HA 3  A 4 
HA   K K
3
1
2
K 3 C Tot x
N
3
4
H
2

A 2 
K 1 K 2 C Tot x 2
N
H
3

A 
K 1C Tot x 3
N
4
N  K1 K 2 K 3 K 4  K1 K 2 K 3 x  K1 K 2 x 2  K1 x 3  x 4
N
Strong 1-protic base.
C
M
B
B  H 2 O  BH   OH 
B  0
BH   C

9
Strong 2-protic base.
B  H 2 O  BH   OH 
C1
M
B
C2
M
BH 
C3
M
B H2
 

B  0
C1  C 2  C 3  C Tot  BH   BH 2
 

2
2
K b 2 BH   BH 2
 y
2
b2
C Tot
b2
y
2
1
2
BH   KC x KK
K b 2 C Tot x
K b2 x  KW


3
KW  x  y
BH   KK
2

Tot
W
b2
W
Weak 1-protic base.
B  H 2 O  BH   OH 
C1
M
B
C2
M
BH 

C1  C 2  C Tot  B   BH 

K b B   BH
 y


2
3
KW  x  y
BH   KK C y  KK xC Kx

b
Tot
b
b
B 
Tot
b
W
1
C Tot y
C Tot K W

K b  y K b x  KW
Weak 2-protic base.
B  H 2 O  BH   OH 
C1
M
B
C2
M
BH 
C3
M
BH 2

K b1 B   BH
2

K b 2 BH

2
 
BH   K

N
2
N  K b1 K b 2 x 2  K b1 K W x  K W
b1
C Tot K W x
N

 y
  BH  y
K W  xy
B  CTot K W

C1  C 2  C 3  C Tot  B   BH   BH 2
2
2
2

1
2
3
4
BH   K
2
2
b1
K b 2 C Tot x 2
N
Acid-Base-functions in the ActpH-program.
It is necessary to know the values of the ionic strength activity coefficients of the ions in
a solution if the ActpH is to be calculated
The ionic strength can according to all literature be calculated as below (2)
10


1
2
  Ci  Z i
2

Ci 
and
the
Zi 
conc
the
of
ch arg e
the
matter
of
the
i
matter
i are all the ions in the solution
The activity coefficient is calculated for each of the ions and is according to (3)
 log f i 

1  Bd i 
AZ i
2
 EDHE :

extended
Debye  Hückel
equation 

fi = the activity coefficient of the ion
Zi = the charge of the ion
di = hydrated diameter of the ion
 = the ionic strength of the solution
A and B are dependent on the temperature and their values can be received from tables,
for instance in (3). It will be seen that each of them appear as practically linear functions
between 0oC and 60oC. In the program, A and b have been input as functions:
A = (8.98e-4).T + 0.242
t= temp i oC
0<=t<=60
B = (1.61e-4) T + 0.28
.
EDHE is valid if  <= 0.1 but it can according to (2) be valid for
Davies modification
 log f i 
AZ i2 
1  Bd i 
 0.10Z i2  
The ionic activity can now be calculated
aZi = Zi.fi
<=0.6 if using the
Davies mod ification 
aZi = the ionic activity.
When calculating ConcpH the conc of the ions and the thermodynamic dissociation
constants from tables are used. These constants are valid at unlimited dilution and can
not be used together with conc of ions when calculating ActpH, rather, the concentration
dissociation constants must be calculated from ionic activity coefficients and used with
concentrations. This will give conc. [H+], which multiplied by its activity coefficient, gives
activity [H+] or {H+}.
In this program the activities of the ions have been calculated and with the help of these,
the thermodynamic dissociation constants have been modified in such a manner that the
concentrations can be used. After calculating of the ionic strength from the new values of
the concentrations the calculations of the ion activity coefficients have been done, the
ionic activities can be calculated by multiplying the concentrations of the ions by their
activity factors.
Ki = thermodynamic dissociation constant, and the corresponding concentration
dissociation constant = aKi or bKi for the corresponding dissociation constant concerning
acids and bases.
Kf = the effect of the activity factors on converting of K i to aKi and bKi .
After calculation of the participant aKi and ion conc values have been input, the NewtonRaphson method is used to calculate the ActpH in the same way as for ConcpH, but using
an iterative procedure.
11
i
The following will show how to calcualte aKi or bKi
Here is one example of rhe transfer
2-protic weak acid with the thermodynamic dissociation constants K 1 and K2

H 2 A  H 2 O  H 3 O  HA

H O  HA   H O  f  HA  f
K 
H A  f
H A 
H O  HA  f  f 
K 
H A  f
H O  HA  and K  f  f
aK 
f
H A 
H O  HA 
K
K  aK  K
aK 

K
H A 




H 3O 
3
3
HA
1
2
2

H2 A

H 3O 
3
HA
1
2

Input
H2 A

H 3O 
3
1
2
H2 A

1
1
1f
1
1f

HA  H 2 O  H 3 O  A
2

3
1

HA
1f
2
H O  A   H O  f  A  f
K 
HA  f
H A 
H O  A  f  f 
K 
HA  f
H O  A  and K  f  f
aK 
f
HA 
H O  A 
K
K  aK  K
aK 

K
H A 


2
2
H 3O 
3
3

2
A2 

HA

2
H 3O 
3
A2 

2
HA

Input
2
H 3O 
3

2
A2 
2f
HA2 

2
2
2
2f
2
3

2
2f
The program starts with calculating the conc pH. After that the ionic strength is
calculated. Now the activity coefficient for every participating ion can be determined and
the aKi-values then calculated. The new values (initial “Ed concentrations”) of every ion
concentration are used in the next loop of calculation, giving revised new values. This
continous until two following values of [H+] are sufficiently near each other. [H+] is then
multiplied by its activity factor to give {H+}.
The values you find of End concentrations after the calculation are just concentartions
and not activities.
I have taken methods and formulas from Gary Christian (2) and Adon Gordus (3).
To summarize the approach:
1. Calculate ConcpH using thermodynamic equilibrium constant, Ki, and the initial
concentrations
2. Calculate the ionic strength
3. Calculate activity coefficients
4. Calculate the concentration equilibrium constant, aKi (or bKi) from Ki and the
activity coefficients.
5. Use aKi (or bKi) and the initial concentrations [H+] (or [OH-]), and End
Concentrations. Use the End Concentrations to give a new ionic strength and
recalculate [H+] (or [OH-]) in an iterative fashion till [H+] (or [OH-]) is constant.
6. Multiply the final [H+] (or [OH-]) by its activity coefficient to get activity of H+ (or
OH-) and then ActpH.
12
2-protic strong acid with the thermodynamic dissociation constant K2
H 2 A  H 2 O  H 3 O   HA 
HA   H 2 O  H 3 O   A 2

  

 
H O  A  f  f

HA  f
H O  A 

and
HA 
K2 
H
3

K2


H 3 O   f H O   A 2   f A2 
O   A 2
3

HA 
HA   f HA

2
H 3O 
3

A2 

HA

Input
aK 2
2
K2 f 
3

K 2  aK 2  K 2 f
aK 2 

fH
3O

 f A2 
f HA

 

H 3O  A 2
K2

K2 f
HA 

1-protic weak acid with the thermodynamic dissociation constant K 1
HA  H 2 O  H 3 O   A 
  
 
H O  A  f
K 
HA  f
H O  A 
aK 
HA 
K1 
H
3


 
H 3 O   f H O   A   f A
O   A
3

HA
HA  f HA


3
H 3O 

 f A 
1

Input
3
1
K 1  aK 1  K 1 f
HA

and
aK 1 
K1 f 

f H O   f A
3
f HA
 
 
H 3O   A 
K1

K1 f
HA
13
2-protic weak acid with the thermodynamic dissociation constants K 1 and K2

H 2 A  H 2 O  H 3 O  HA

H O  HA   H O  f  HA  f
K 
H A  f
H A 
H O  HA  f  f 
K 
H A  f
H O  HA  and K  f  f
aK 
f
H A 
H O  HA 
K
K  aK  K
aK 

K
H A 




H 3O 
3
3
HA
1
2
2

H2 A

H 3O 
3
HA
1
2

Input
H2 A

H 3O 
3
1
1f
2
H2 A

1
1
1f
1
1f

HA  H 2 O  H 3 O  A
2

3
1

HA
2
H O  A   H O  f  A  f
K 
HA  f
H A 
H O  A  f  f 
K 
HA  f
H O  A  and K  f  f
aK 
f
HA 
H O  A 
K
K  aK  K
aK 

K
H A 


2
2
H 3O 
3
3

2
A2 

HA

2
H 3O 
3
A2 

2
HA

Input
2
H 3O 
3

2
A2 
2f
HA2 

2
2
2
2f
2
3

2
2f
14
3-protic weak acid with the thermodynamic dissociation constants K 1 K2 and K3

H 3 A  H 2 O  H 3O  H 2 A

H O  H A   H O  f  H A  f
K 
H A  f
H A 
H O  H A  f  f 
K 
H A  f
H O  H A  and K  f  f
aK 
f
H A 
H O  H A 
K
K  aK  K
aK 

K
H A 



3

H 3O 
3
2
2
H 2 A
1
3
3

H3 A

3
H 3O 
2
H 2 A
1
3
H3 A


3
Input
H 3O 
2
1
H 2 A
1f
3
H3 A

1
1
1f
2
1
1f


H 2 A  H 2 O  H 3 O  HA
2
3
H O  HA   H O  f  HA  f
K 
H A  f
H A 
H O  HA  f  f 
K 
H A  f
H O  HA  and K  f  f
aK 
f
H A 
H O  HA 
K
K  aK  K
aK 

K
H A 


2
2
H 3O 
3
3

2
HA2 
2

2
H 3O 
3
HA2 

2
H 2 A
2

2
H 3O 
3

2
H 2 A

2
2f

2
2f
HA

 H 2 O  H 3O  A
3
2
3
2
2
HA2 
2f
2
2
HA2 

2
Input

3
1
2
H O  A   H O  f  A  f
K 
HA  f
H A 
H O  A  f  f 
K 
HA  f
H O  A  and K  f  f
aK 
f
HA 
H O  A 
K
K  aK  K
aK 

K
H A 


3
3
H 3O 
3
3
2
3
A3 
2
HA2 

3
H 3O 
3
A3 
2
3
HA2 

Input
3
H 3O 
3
2
3
A3 
3f
HA2 

3
3
3
3f
3
3
2
3
3f
15
4-protic weak acid with the thermodynamic dissociation constants K 1 K2 K3 and K4

H 4 A  H 2 O  H 3O  H 3 A
H O H



K1
3
H 4 A
H O  H

K1 
Input
aK 1 
3
  H O  f

A
3
3



3
H 4 A
3
3

H 3 A  H 2 O  H 3O  H 2 A
2
A
3A

H O H A   H O  f
K 
H A 
H
H O  H A  f  f
K 
H A  f
H O  H A 
aK 
and
H A 


2
3
2
H 3O 
3
2

H 3O 
3
A

2
3

2
aK 2 
f H O  f H

3

fH




3
H 3O 
3
2
3
HA3 
2
H 2 A2 
2

3
H 3O 
HA3 
2
3
H 2 A2 
2

3
H 3O 
3
2
3
H 3 22 

3
3
3
3f
2
3

4
3
3
3
4
3
3
3f

H 3 A
3f
2
H 3 A  H 2 O  H 3O  A
2

H O   H 2 A 2
K2
 3
K2 f
H 3 A

3
3
3
4
3A
2A
H O HA   H O  f  HA  f
K 
H A 
H A  f
H O  HA  f  f
K 
H A  f
f
f
H O  HA 
aK 
and K 
H A 
f
H O  HA 
K
K  aK  K
aK 

K
H A 
H O A   H O  f  A  f
K 
HA 
HA  f
2

2
H 3 A
K2 f 
2
K 2  aK 2  K 2 f

2A
H 2 A2 
3
Input

H 3 A

 H 2 O  H 3 O  HA

 f
 H 2 A 2  f H


3
H2 A

f H4A
3A
2
2
2
3
3



f H O  f H
K1 f 
and
H 3O   H 3 A 
K1
aK 1 

H 4 A
K1 f
3
2


3
Input
3A
4
K 1  aK 1  K 1 f


4
H 4 A  f H A
H O  H

 H 3 A  f H
H 4 A f H A
A  f H O  f H
3
H 3O 
2
4
H 3O 
A4 
3
HA3 
H O  A  f
K 
HA  f
H O  A 
aK 
HA 

4
3
H 3O 
 f A4 
3
4
HA3 

Input
4
3
4
K 4  aK 4  K 4 f
3
and
aK 4 
K4 f 

f H O   f A4  
3
f HA3
 
 
H O   A 4
K4
 3
K4 f
HA 3
16
2-protic strong base with the themodynamic dissociation constant K 2
B  H 2  BH   OH 

BH  H 2 O  BH
2
BH  OH   BH  f  OH  f
 OH
K 
BH  f
BH 
BH  OH  f  f  BH  OH   f
K 
BH  f
BH 
BH  OH  and K  f
Input
bK 
f
BH 
K
BH  OH 
K  bK  K
bK 

K
BH 
2

2


BH 2 


2
OH 
BH 
2

BH 2 
2
OH 


2
BH 2 

f BH 
BH 
2

 f OH 
BH 2 

2
 f OH 
2f
2
BH 

2
2
2
2f

2
2f
1-protic weak base with the thermodynamic dissociation constant K 1
B  H 2 O  BH   OH 
K1 
BH  OH   BH  f



B
BH  OH  f

K1 
B f B
bK1 
Input

BH 
BH  OH 

BH 

B f B
 f OH 

BH  OH   f

K1  bK1  K1 f

BH 
 f OH 
fB
f BH   f OH 
Kf1 
and
bK1 

B

B

 OH   f OH 
fB

K1
BH   OH 

B
K1 f

2-protic weak base with the thermodynamic dissociation constants K1 och K2
B  H 2 O  BH   OH 
K1 
BH  OH   BH  f



B
BH  OH  f

K1 
B f B
bK 1 
Input

BH  OH 

BH  H 2 O  BH
2
 OH


B
B  f B
 f OH 

BH  OH   f


B
K1 f 
and
bK 1 

 OH   f OH 

K 1  bK 1  K 1 f

BH 
BH 

BH 
 f OH 
fB
f BH   f OH 
fB

K1
BH   OH 

B
K1 f

BH  OH   BH  f  OH  f
K 
BH  f
BH 
BH  OH   f  f
K 
f
BH 
BH  OH  and K  f  f
bK 
f
BH 
K
BH  OH 
K  bK  K
bK 

K
BH 
2
2


BH 2 


2
OH 
BH 
2

BH 2 
OH 

2
BH 
2
Input

BH 2 

2
OH 
2f
BH 
2

2
2
2
2f

2
2f
17