Ames Laboratory Technical Reports
Ames Laboratory
2-1964
Calculation of acid dissociation constants
Wayne Woodson Dunning
Iowa State University
Don S. Martin
Iowa State University
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IS-822
IOWA STATE UNIVERSITY
CALCULATION OF ACID DISSOCIATION
CONSTANTS
by
Wayne Woodson Dunning and DonS. Martin
RESEARCH AND
DEVELOPMENT
REPORT
U.S.A.E.C.
IS-822
Chemistry (UC -4)
TID 4500 April 1, 1964
UNITED STATES ATOMIC ENERGY COMMISSION
Research and Development Report
CALCULATION OF ACID DISSOCIATION
CONSTANTS
by
Wayne Woodson Dunning and DonS. Martin
February, 1964
Ames Laboratory
at
Iowa State University of Science and Technology
F. H. Spedding, Director
Contract W -7405 eng-82
ii
LS-822
This report is distributed according to the category Chemistry (UC -4)
as listed in TID-4500, April 1, 1964.
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A.
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iii
IS -822
TABLE OF CONTENTS
Page
I.
II.
III.
IV.
ABSTRACT • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
v
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .
1
A.
Experimental Background •••••••••••••••••••••••••••••
1
B.
Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
EXPERIMENTAL
11
A.
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
B.
Equipment ..•
11
MATHEMATICAL THEORY
12
A.
Titration of a Single Acid
12
B.
Mixture of Two Acids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
C.
Modified Titrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
D.
Calculation of Titration Curves
19
E.
Calculation of Acid Concentrations
20
F.
Determination of Errors
20
CALCULATIONS
A.
B.
v.
22
Calculation of the Dissociation Constants for
a Single Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Calculation of the Dissociation Constants for
a Two Acid Mixture ••••••••
25
.......... .
c.
Calculation of Intermediate Quantities ••••••
D.
Options and Considerations • • • • • • • • • • • • • • • • • • • • • • • • • • • 28
RESULTS
A.
. ............................................ .
27
30
Resolution of Successive Constants • • • • • • • • • • • • • • • • • • • • 30
iv
Page
VI.
. ............................ .
38
B.
Resolution of Mixtures
c.
Consistency .••••.••••
D.
Improper Calculations
E.
Experimental Titrations ••••••••••••••••••••••••••••• 41
DISCUSSION •••••
39
. ............................ .
40
..................................... . 43
VII.
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . 47
VIII.
LITERATURE CITED. • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 48
IX.
APPENDIX A .••
X.
APPENDIX B •••
. ..................................... .
50
53
v
IS-822
CALCULATION OF ACID DISSOCIATION CONSTANTS):<
Wayne Woodson Dunning and DonS. Martin
ABSTRACT
A computational method has been developed for the determination of
dissociation constants of acids (and bases). The method is applicable to
mono or polybasic acids and, in certain cases, to mixtures of acids. A
least-squares treatment is employed, and no simplifying assumptions
are made in any of the equations or factors involved. All the available
data are utilized, and provision is made for including or calculating
activity coefficients. Utilization of the method requires a medium to
large size digital computer.
The resolution of successive dissociation constants of small ratio
has been shown to present no difficulties attributable to such ratios;
ratios as small as three were resolved as readily as those much larger.
The accuracy attainable depends entirely upon the accuracy of the titration data. Tests have shown that titrations accurate to better than one
part in five thousand may still produce two percent errors in the resulting constants. Consistent or random titration errors are likely to cause
considerable variations in the dissociation constants. Experimental
work must be performed with the greatest care if constants of even two
significant figures are desired. Ordinary pH titrations may give uncertainties in even the first figure.
Mixtures of acids proved to be no more difficult to resolve than single acids, within certain severe limits. If there is a considerable separation in constants, and the data are extremely accurate, useful results
may be obtained. In most cases, overlap of constants and/ or experimental errors will prevent successful resolution.
Many previous methods of calculating dissociation constants have
treated polybasic acids as mixtures of monabasic acids. It has been
shown that if the ratio of successive constants is one hundred or more,
no appreciable error results. With the present method, however, such
assumptions are completely unnecessary.
Examination of the literature, together with experimental results
obtained in this work, indicates that many "accepted" values are either
very inaccurate, or at least given to more significant figures than justified.
~:~
This report is based on a Ph. D. thesis submitted by Wayne Woodson
Dunning, Febraury, 1964, to Iowa State University, Ames, Iowa.
1
I.
INTRODUCTION
The determination of the dissociation constants of
acids 1 -- though sometimes considered to be a very straightforward procedure -- entails in practice numerous experimental
and theoretical difficulties.
The variety of experimental
methods that have been used, the corrections applied to the
data, the numerous manners in which data for identical procedures have been treated, and the considerable disagreement
in values of the dissociation constants obtained are all indicative of the difficulties.
The research reported in this
work was undertaken primarily for the purpose of developing
computational methods that would furnish the best possible
values of dissociation constants from the data of any of
several different experimental procedures.
A.
,.
Experimental Background
Among the many experimental procedures used in the past,
one of the most common was the measurement of the conductance
of an aqueous solution of the acid.
This method was used
primarily on monobasic acids, since polybasic acids presented
As late as 1959, Dippy £!
great mathematical difficulties.
!!•
(1) expressed the opinion that there was still no wholly satisfactory method of calculating
second and higher thermo....
,'
1Although this work deals specifically with acids, the
treatment of bases is quite analogous, and the necessary modifications to the theoretical equations and the computer program are included.
th~
:'
·.... ·
2
dynamic dissociation constants of polybasic acids from conductivity data.
Conductance methods are now considered ob-
solete by some authorities (2).
Another common procedure, and one which still sees extensive use, is that of optical measurement.
either colorimetric or spectrophotometric.
This may be
The colorimetric
method enables one to obta.in a pH titration curve while avoiding some of the problems arising in standard electrometric determinations.
However, standards of known dissociation con-
stant are necessary for calibration of the indicators, and
can give rise to other difficulties.
thi~
Spectrophotometric meth-
ods are quite useful in many cases, though extensive computations, similar in principle to those employed in this work,
are often required.
1'-
. Among the other methods or measurements that have been
employed at one time or another for the determination of dissociation constants are:
a) the change in freezing point;
b) hydrolysis of salts (3); c) catalytic effect· on the r.ates
of sugar inversion (4); d) solubility of slightly soluble acids
in solutions of salts of other acids (5); and e) kinetic methods.
Perhaps the most common procedure, in both past and present use, for determining
dissoci~t~on
. ,.
constant'·· is that of
.
,
.
It has iong been recognized that
'
electrometric measurement.
the measurement of the change in pH during the neutralization
3
of a weak acid with a strong base can be employed to obtain an
accurate value for the concentration of the acid and its dissociation constant(s).
In an electrometric titration, the pH
values are measured at successive steps in the titration by
means of the changing potentials of suitable electrodes immersed in the solution.
According to usual practice, suc-
cessive portions of a solution of strong base are added to a
sample of pure acid and the pH values of the several mixtures
are determined. 1
The titration is performed in a vessel that
contains a hydrogen, glass, or quinhydrone electrode, and the
cell system is completed by a reference half-cell (calomel,
AgCl, etc.) whose electrolyte is brought into liquid-liquid
contact with the acid solution in the titration vessel.
The pH values derived from the e.m.f. measurements of
.
~
such a cell often involve considerable uncerta1nty, and in
unfavorable cases they may be in error by more than 0.05 pH
units (6).
A principle reason for this difficulty is the ne-
glect of the contribution of the potential of the liquid
junction or in the application of improper or inadequate cortections.
Such corrections are laborious and unsatisfactory,
and require a knowledge, often unavailable, of the mobilities
and activities of the ions of which the solutions are composed.
.
.
!Numerous variations are pos.ibl~, and u~qer certain conditions may be more practical. The · computational methods developed in this work are designed to accommodate some of the
variations.
4
It has been shown (7) that partial corrections may produce
a larger error than no corrections at all.
Therefore, for ac-
curate work, liquid junctions are to be avoided whenever possible.
For approximate work, however, cells with liquid junc-
tions are often useful.
A second limitation to the accuracy of these titrations
has its origin in the changing concentration of ionized solutes
in the vessel during the titration.
A solution of a weak acid
has a low ionic strength, whereas its salt is a strong electrolyte.
There is no unique titration curve for a weak acid;
it is well known that the curve is affected by the ionic strength
of the solution.
Several procedures are available for resolv-
ing this difficulty.
One can make
1)
a series of pH measure-
ments of buffered solutions at varying ionic strengths, with
extrapolation to zero ionic strength;
,..
2) a series of titrations
at varying ionic strength, with extrapolation to zero, or
3)
a titration with corrections for ionic strength at each point
by means of the Debye-Huckel equation
=
A
zi
2
.fP
1
1 + B ai -ijJ
where fi is the activity coefficient, Zi is the valence of
the ion i, ai is the average effective diameter of the ion in
Angstroms,
<?»
is an empirical coefficient dependent upon the
system under study, and A and B are coefficients whose values
vary with tbe temperature and dielectric constant of the
5
solvent.
The ionic strength,
p , is defined by Lewis and
Randall (8) as
2
)l =l.~m
a Lo i z i
where m.1 is the moial concentration of ion i. 1
2
The lack of a completely standardized scale of pH may
lead to further problems, particularly in the re-evaluation
of older data.
or
The simple hydrogen ion concentration
pcH - - log cH
3
pcH - - log mH
3a
is used in much of the older literature, and great care must
be taken when recalculating data from such work, if one wishes
to obtain meaningful answers.
The Sorensen scale is a conventional one2, defined in
terms of the potential of the celi Pt;H2,Soln. X/Salt bridge/ .
0.1 N Calomel electrode,
- 0.3376
4
= E 0.05916
It is a measure of neither concentration, nor activity of the
psH
1The standard state of unit activity coefficient at in-
finite dilution requires that activity on the scales of volume concentration (c) and molality (m) shall be related by
ac = amd 0 , where d 0 is the density of water. At 25° or less,
this difference is negligible and will be ignored in this work.
2A conventional scale is one in which the values of aH,
although not truly hydrogen ion activities, will nonetheless
be numbers which, inserted in equations involving ·aij, will .
furnish results consistent with th'Ose obtained·-.by r 1gorous
thermodynamic methods.
,·
·· ·,
·
6
hydrogen ion.
In spite of this fact that psH bears no simple
direct relationship to chemical equilibria, this scale has
been widely used, and extensive tables of psH values for
buffer mixtures are available (9).
The recognition that the e.m.f. of galvanic cells reveals
changes of activity rather than of concentration, brought
about the proposal by Sorensen and Linderstrom-Lang (10) of
a new pH unit
paH
= - log
aH
= - log
(fHcH)
5
where aH is the activity, and fH the activity coefficient
corresponding to the scale of concentration.
The fact that
the activity of a single ionic species is a concept lacking
unique physical definition does not preclude the establishment
of a reasonable scale of paH, but this scale must be a conventional one.
Guggenheim (11) and Hitchcock (12) have called attention
to the advantages of a unit of acidity defined as
6
Unlike paH, this quantity is physically defined at all ionic
strengths, and can be determined exactly from measurements
of cells without liquid junction comprising electrodes reversible to hydrogen and chloride ions.
Since pwH is therefore a quite useful concept, it is
,
fortunate that conversion can be made· between it and paH with
sufficient accuracy for most purposes, provided that the
7
ionic strength is not too high.
The common "PH meter" employing a glass electrode enjoys
widespread popularity, and it is worthwhile to note that the
development of pH standards allows one, without the necessity
of liquid junction corrections, to make determinations of what
is essentially paH, but only under certain specific conditions.
It is safe to say that no quantitative interpretation of
measured pH values should be attempted unless the medium can
be classified as a dilute aqueous solution of simple solutes.
This requirement excludes all non-aqueous media, suspensions, .
colloids, and aqueous solutions of ionic strengths greater
than 0.2.
From this point of view, the "ideal'' solutions are
those which match the standards of reference, namely aqueous
solutions of buffers and simple . salts with ionic strengths
between 0.01 and 0.1.
.
~
. .
Under these very restr1cted cond1t1ons,
the measured pH may be expected to approach an experimental
-log fHmH, where fH is defined in a conventional manner consistent with the assignment of the pH values of the standards
with which the instrument was adjusted.
For all practical
purposes, the value of fH in this dilute range is given by
equation
1
with ai values of 4 to 6.
B.
Computational Methods
The number of mathematical ,. t-teatments that have been ap.•
,
I
•'
·•
plied to electrometric or pH titration data may ' well be larger
8
than the number of experimental procedures devised for determining dissociation constants.
A thorough analysis of them is
not practicable in this work, but a brief discussion of some
of the methods will be given.
,.
A simple approximation -- one that has been in use for
more than fifty years (13) -- is that K
= [H+]
half neutralization point of a monobasic acid.
at the oneThis is, of
course, not strictly true, and has the further disadvantage
of making the value of the dissociation constant dependent
upon only one pH determination.
Auerbach and Smolczyk (14) proposed a set of theoretical equations involving some assumptions which were shown by
Britton (15) to be erroneous.
have been widely used.
Nonetheless, these equations
The final values of dissociation
/'-
constants obtained by this treatment are generally the averages of constants calculated from various combinations of the
titration points. 1
A logarithmic equation was derived by Cohn
~
but does not appear to have seen any further use.
!!·
(16)
It is ap-
parently limited to monobasic acids.
lThis method of "using all the data" is apparently still
popular. It is generally applied in a rather random fashion,
as there is no known formal procedure for its use. The process is essentially one of averi.-g..ing, and is not generally
considered to have the merit of the · ~east-squates method.
9
Several workers have developed equations by means of
which a polybasic acid is treated as a mixture of monobasic
acids.
The "titration constants" thus obtained are presum-
ably convertible to the true dissociation constants by means
of simple relationships.
This approach has apparently not
seen extensive use, despite its multiple development.
Speakman (17) devised an approach that appears to have
considerable merit, although it is in general limited to
dibasic acids.
The data, modified by the appropriate equa-
tions, are recorded graphically, and an essentially linear
plot is obtained.
The slope of the line gives K1, and the
intercept gives K1K2 •
Drawing the "best" straight line is
considered to give the best use of all the data.
Speakman
considers extrapolation to zero ionic strength to be inferior
""
to making proper activity corrections at finite ionic
strength.
The general treatment of polybasic acid titration curves
has been a matter of much study and many suggestions.
So
far as is known, this work presents the first completely
general treatment of such situations.
Previously, it was
common to develop specific data treatments for varying situations, depending largely upon the ratio(s) of the several
dissociation constants.
Where this ratio is over 500 to 1000,
the acid was generally treated
a mixture of.. monobasic acids.
.. ,.
As the ratio decreased, various approximations were necessary,
a~.·
,·
graphical treatment might be required, etc.
In at least one
10
study, equations were developed for five different cases, none
of the treatments being exact.
It is the intention of this
work to present a treatment that is applicable to mono- or
polybasic acids, or to mixtures of acids under certain condition~,
which makes no simplifying assumptions for any case,
and which utilizes all the data for any of several types of
titrations.
The method of least-squares is employed, and an
estimate of the errors in the individual constants is obtained.
·"
....
: f.
,•
·~
11
II.
EXPBRIMBNTAL
A. Materials
1.
Acids
Standard commercial reagents, in grades meeting A.
c. s.
specifications, were used without further purification.
2.
~
A stock solution of sodium hydroxide was prepared by the
dilution of a filtered, concentrated solution of the reagent
grade chemical.
It was standardized against potassium bi-
phthalate by potentiometric titration.
3.
Buffer
Primary standard grade potassium biphthalate was used to
standardize the pH meter for all titrations.
A 0.05 M solu-
tion has a pH of 4.01 at 25°.
4.
.I'
Water
Tap distilled water was redistilled from alkaline perman-
ganate solution for use in all preparations and titrations.
B.
Equipment
A Beckman model "G" pH meter was used in all pH determinations.
Shielded electrodes, model 1190-80, permitted pH
No electrode
determinations outside the shielded cabinet.
corrections were applied to any ireadings.
- .. , .
.
....
,.
·~
12
III.
A.
MATHEMATICAL THBORY
Titration of a Single Acid
The simplified chemical equation for the dissociation of ·
an acid, represented by the formula RnA, in water or other
suitable solvent is
Hn A or-
H+
+
J.J
A·"D-1
7
or, for the ith step of the reaction, where i varies from one
to n,
J.J
A-(i-1) -. H+
·"D+l-i
......-
8
The general equation for the thermodynamic dissociation constants of the acid is
9
.I'-
where the brackets represent ionic concentrations, and the
parentheses represent ionic activities.
The
~i
are the .ac-
tivity coefficients for the ions of charge -i; since uncharged
species are considered to be of unit activity,
~0
=1.
Equation 9 may be rearranged to
~i-1
[J-In-iA-i) = Ki Xi
[Hn+l-iA-(i-1))
(H+)
10
and proper substitution of successive terms will give
· = Xo
[ J.J-1·A -1)
."D
li
[ HnAl
(H+)l
1;!;_.
1
1
... . _~ • ·•
.' ·
....
11
13
Since the products of the dissociation constants will be
encountered so frequently, it is useful to define
12
and
k 0 = 1.
13
The total concentration of the acid, Ca, is the sum of
the concentrations of the unionized molecule and all the
ionized species, and may be expressed as
<1
n
+
E
i=l
k·
!)1
ri
(H
>•
1.4
For ionic balance in solution, it is necessary that the
following condition be met:
[~]
+ [Na+]
n
= [oH-1
+ [cl-] + E
i=l
15
where [Na+] is usually due to the base added, and [Cl-]
= 0.
If the salt of an acid is being titrated with a strong acid,
then [Na+] is equal to the salt concentration, · and [cl-] is
due to the acid added.
The equations may be simplified somewhat by making the
convenient definition
16
or, in terms including activities, activity coefficients, and
·-.•
the dissociation constant of wate'r 'o
•
N•
<r+>
H
+ [Na+] -
0
'
0
,
KW
(H+)
1 OH
..
.~
17
14
Combination of Equations 15 and 16 gives
18
and from Equations 11, 12 and 14, one finds that
Ca
)
.
19
Multiplication of this by (~)n/(H+)n gives
).
20
Rearrangement
N
t+)n
n
::
~
iki (H+)n-ica
~i
i=l
21
and the separation and recombination of terms gives
•
22
There is one such equation for each data point, resulting in a set of equations in n unknowns for a given titration.
There will most generally be more than n data points, and in
order that the best possible use may be made of all the experimental data for a
titration~
the method of least-squares,
or multiple regression, is employed in solving··
.. ·, the set of
,'
equations.
The particular treatment used in this work may be best
15
illustrated by simplifying the coefficients of Equation 22 to
give
23
The coefficients of the numerous individual equations are
multiplied and summed to form the following set of equations:
24
(~
•
•
•
•
AnY)
•
•
•
= (~ AnA1 )k1
•
+ ----------<~nAn)kn
•
This set may be solved for the ki by any of several
standard methods.
The dissociation constants,
~i'
are then
obtained from an equation analogous to Equation 12.
K.·
1
B.
=
ki
k·1- 1
2.5
•
Mixture of Two Acids
The addition of a second acid, HmB, to the system introduces certain complicatiqns.
Dissociation constants for
the second acid are Kj, and the equations which differ sig•
nificantly from those for a single acid are
k
J
• }.13J
•1
'J ' , 'r ·.....
....
,.
26
'p
27
and
16
N
=~~
ik·1
~i=l
G~
jkj
j=l
(jj+')m-j)
X·J
Cb
m
cjj+)m + 1: k J·(Ir')m-j
j=l
X·
28
~
This equation may be reduced to
n
N(H+)n+m = 1: k. ( i Ca-N )(H+)n+m-i
¥i
i=l 1
+
29•
+
n
m
iCa + J. Cb-N
· ·
~
~ k •k •(
)(H+)n+m-1-J
v
'I
i=l j=l 1 J
Oi Oj
~
•
~
A non-linear method must be employed in the solution of
,.
.
this system due to the fact that products of the var1ables are
present.
So far as is known, all methods for solving non-
linear systems of equations on digital computers are i te.rative
and have one of two disadvantages:
either reasonably good
initial estimates of the variables are required to assure convergence, or else the equations are expanded at each iteration
and the program may bog down of sheer complexity.
In this
work, the Newton-Raphson method of iteration, as employed by
Crosbie and Monahan (18), was
convenient.
cJ;l~_~en
as the mQ..ft sui table and
,.
The method of least-squares requires the minimization of
Q in the equation
17
30
where
31
and may be either a measured value, YM' by virtue of Equation
16, or a calculated value, Yc, from Equation 29 and the esti-
-
mates of the ki,j•
The appropriate corrections, (dk), in vector notation,
to the variables would result in
Q+
d9_
~
~(2)k)(dk)
= o.
32
From this, it follows that
l:( ~-~k)(dk)
= -Q
33
and
34
.1 '
Therefore
. 35
or
36
This can be most conveniently solved by multiplying by
(clYc/~k)
and expressing the entire equation in vector or
matrix notation,
~~D n;m(,.~(1))21 [dk)= ~~D n;m('Y,;j(l)":'Yc(l)><~~~l))l
~·1 a•l a J
[!~1 a•l
·
a ~
where ND is the number of points on the titration curve.
37
18
This system,
[ G][ dk]
= [R]
38
is solved for the vector [dk] by matrix inversion to yield improved values of the variables, ka
c.
= ka 0
+ d ka •
Modified Titrations
The principal variation that will be encountered is
probably the titration of a weak base with a strong acid.
In
addition, the free acid (or base) may occasionally be unobtainable, necessitating the titration of a sodium or potassium.
salt of the acid with strong acid.
Fortunately, modifications
to the equations already developed are either slight or unnecessary
and the changes in computer programming required to
handle all four situations are minor.
·"
1.
Salt of weak acid titrated with strong acid
The only changes needed are that [Na+] be · made equal to
the salt concentration, and [Cl-] is now .t he concentration of
the acid titrant.
2.
Weak base titrated with strong acid
For the titration of a base, M(OH)n, with acid, Equation
18 is changed to
N
=
n
1:
i=l
+i
i[M(OH)n_ 1. ]
· 1 ·,.·
•
·,
, ·
.. .
~
39
19
Equation 16 remains the same, except for a change in
sign, since N will now represent the total concentration of all
simple negative ions.
The value of (Na+] will generally be
zero, and [cl-] will be the concentration of the acid titrant.
3.
Salt of weak base titrated with strong base
This case is very similar to that of Section 1 above.
[Cl-) equals the salt concentration, and [Na+] is the concentration of the base titrant.
The quantity N has the same sign
as in Section 2.
D.
Calculation of Titration Curves
It is sometimes useful to calculate an exact titration
curve from known dissociation constants and concentrations.
An iterative process is necessary for curve calculation,
~
due to the continual corrections for volume change.
quired sodium ion concentration (or chloride ion,
The re-
depend~ng
upon the type of titration) at each desired pH on the titration curve is calculated from Equation 19 and 16.
The pre-
vious value of [Na+] is obtained from the expression
The difference, required [Na+) ~ Previous [Na•], is multiplied by the total solution volume
and, . divided··by
the con.. . .
,. ·,
centration of the base to obtain· the volume of base that must
be added.
20
This calculated amount is added to the solution, and
then a test is made to determine whether or not sufficient
accuracy has been achieved.
Gene rally, if the amount of base
added (or subtracted, since the quantity may have either sign)
is less than 0.001 ml, no further adjustments are made to the
particular point on the curve in question.
B.
Calculation of Acid Concentrations
The concentration of an acid can be readily obtained
from its titration curve and dissociation constants by means of
Equations 20 or 28 and the linear least-squares treatment.
For
a single acid, this is of doubtful utility since standard
analytical analysis of the titration curve will generally give
a value of completely satisfactory accuracy.
For a mixture of
acids, however, this calculation may be of some importance as
there may be no break in the curve.
F.
Determination of Errors
The method of least-squares can be reasonably depended
upon to do a good job of fitting a particular
given set of data.
equat~on
to a
It does not, however, necessarily indicate
how well the answers obtained satisfy the equation and data.
It is therefore highly advantageous
to, determine the standard
.
deviations of the final results; these will give indications
of the random errors in the data, or perhaps evidence that the
21
chosen equation was not the proper one.
In the latter case,
this would be due to improper choice of the initial parameters
specifying the number of acids or dissociation constants.
In all cases in this work, the same procedure is used to
Obtain the standard deviations of the particular results.
The
systems of linear equations are solved by the method for nonlinear equations on the last iteration.
The diagonal elements
of the inverse matrix G- 1 , from Equation 38, are transformed
into the standard deviations by the expression
G- 1 (L,L) x Q
ND-NC-1
where ND is the number of points on the titration curve, NC
is the number or variables in the problem, and Q is defined
as in Equation 30.
,1 '
. . r ..... •
,
. ...,.
(
...
22
IV.
CALCULATIONS
The development of the equations in Chapter III, and,
in particular, the reduction to practice of these equations,
actually constituted the major portion of this research.
For
this reason, the applications of the equations for the titration of a single acid, and for a mixture of acids, will be
described in some detail.
The applications of the equations
in the remaining sections do not differ sufficiently from the
above to warrant detailed descriptions.
Utilization of the procedures developed in this work
requires the availability of an electronic digital computer,
preferably of large size.
Much of the initial development and
testing was done on an International Business MachinesType 650
computer.
However, this machine is of
insuffici~nt
size to
contain the complete program, and final assembly and operation
was done on an IBM Type 704.
In general, the Fortran Auto-
matic Coding System was used to code the programming for both
machines.
A.
The program is listed in Appendix B.
Calculation of the Dissociation Constants for
a Single Acid
A simplified flow sheet for this section is given in
Figure 1.
1.
The individual steps ,· ·~.re explained ...below.
,
Read in the titration curve·, estimates of dissociation constants, control parameters, and any other
23
I.
INITIALIZATION
2. CLEAR ARRAY
SET LOOP
3. GO
r
TO
-...
CALC.
...
4. CALC. N ANDY
5. BUILD
~·
ARRAY
,
6. RAISE LOOP AND
TEST FOR END
NO
,1"-
YES
,,
7. SOLVE ARRAY
FORM SUM
REPLACE ANSWERS
TEST FOR ANSWERS
NO
YES
~·
' ,.
,
..•
. .
8. MODIFIED PROPER PARAMETERS
AND
ENTER TWO ACID SYSTEM
Figure 1.
Flow diagram for the calculation of dissociation constants for a single acid
24
necessary information.
The control parameters speci-
fy which of the numerous program options are to be
used, and control the introduction of additional
information necessary to satisfy those options.
The
products of the dissociation constants are formed by
means of Equation 12.
2.
The array specified by Equation 24 is set to zero
since it is to be built up by a summation process in
each iteration.
The "loop" will control the succes-
sive processing of each point on the titration curve.
as the array is built up.
3.
The "Gamma Calculator'' and the operations it performs
are described in detail in Section C.
4.
The quantity N is calculated by means of Equation 17,
and y
5.
= N(if')n
,..
as shown in Equations 22 and 23.
The elements of the array are formed by means of
Equations 22 and 23, and are added to the previous
values, as indicated by Equation 24.
6.
The
11
loop" is increased to process the next point, am .
a test is made to determine whether or not any points
remain.
7.
The array is solved by the conventional method of
triangularization.
Th~' . , .~nswers
..
.. ,
replace
. the previous
.·
values of ki, and the sum of the absolute values of
the fractional differences between the new results
25
and the previous results is compared with the desired
closeness of fit to determine the necessity for
another iteration.
8.
If the last answers obtained are satisfactory, the
control parameters for the two acid system (Section
B) are modified to allow one pass through that section in order to calculate the standard deviations.
B.
Calculation of Dissociation Constants for a Two
Acid Mixture
Figure 2, detailed below, shows the basic steps employed,
in this treatment.
1.
This step is essentially the same as Step 1 in Section A.
Equation 26 is used in addition to Equa-
tion 12 to form the products of the constants.
2.
The looping operation is the same as the one used in
Section A.
The array used in the single acid treat-
ment does not appear in this section.
3.
See Section c.
4.
From Equations 16, 29, and 31, Yc and YM are calculated.
These quantities are vectors, there being one
value of each for every point on the titration curve.
5.
The partial derivatives, aytfak, are calculated by
means of Equations 29
· a~
31.
These ...will form a
matrix whose dimensions are· determined by the total
number of dissociation constants and the number of
26
I. INITIALIZATION J
1
I...
I
2. SET LOOP
~
3. GO
TO
r
...
CALC.
,
4. CALC. Yc{L), YM(L)
1
5. CALC. VECTOR
8Yc{L) I 8k
~
6. RAISE LOOP AND
TEST FOR
END
NO
Jf..
,,!YES
7. FORM G AND R MATRICES
INVERT G , APPLY CORRECTIONS
:
8. FORM SUM
NO
REPLACE ANSWERS
TEST FOR END
, YES
9. CALC. STANDARD
Figure 2.
DEVIATIONS
I
Flow diagram for the calculation of dissociation constants for a mixture of two acids
27
points on the titration curve.
6.
Same as step 6 in Section A.
7.
The G and R matrices, given by Equations 37 and 38,
are formed and G is. inverted by a standard procedure.
The corrections, dka, are added to the dissociation
constants.
8.
The previous values of ki,j are replaced and the sum
of the absolute values of the fractional changes is
calculated to determine the necessity for any further
iterations.
9.
The standard deviations are calculated from the inverse matrix, G- 1 , by means of Equation 41.
c.
Calculation of Intermediate Quantities
,1 "
This section, previously referred to as the ''Gamma Calculator'', computes, in addition to the activity coefficients
for the various ions in solution, a large number of other
necessary factors.
Since the operations in this section are
performed in sequence. they will be described without reference
to a flow sheet.
1.
The activity coefficients for the hydrogen and hydroxide ions, and for all the ionic species derived
from the acids, are ca,.g.u lated by
me~ns
..
of the Debye-
~
Hucke! formula, Equation 1.· In the first calculation
for each point, the estimated ionic strength is used.
28
Successive calculations use the ionic strength previously determined at the end of this section.
2.
The concentrations of the acids are corrected for the
volume change, caused by the addition of base, by
means of the following equation.
Ccorr= Cinit
3.
X
Vinit
Vinit + Vbase
42
•
The sodium ion concentration is calculated from the
volume of base added, its normality, and the initial
solution volume.
[Na+] = Vbase x Nbase
Vbase + Vini t
4.
43
•
The concentrations of all the ionic species of the
acids are calculated from the current estimates of
the dissociation constants.
·" is first
Equation 14
employed to obtain the concentration of the unionized acid.
Equation 11, which relates the ionic
species to the unionized acid, is then utilized to
obtain the individual ion concentrations.
S.
The new value of ionic strength is calculated by
means of Equation 2.
D.
Options and Considerations
As previously mentioned,
se·ction~
c through F of Chapter
III do not warrant detailed description in this work.
They
29
do, however, add considerably to the complexity of the program
because of the many branches and multiple paths of similar
calculations that must be provided.
In addition to the options previously described, a number
of minor options have been incorporated in the final program
to increase its flexibility and utility.
They include the
following.
1.
Uydrogen ion concentration may be in the form of
pcH, paH, or pwH.
2.
Ionic strength may be calculated at each point,
specified for each point, or constant.
3.
Hydrogen ion activity coefficients may be calculated
or specified at each point.
4.
Total solution volume may be calculated or constant •
s.
The coefficients, A, B, and @in Equation 1 may be
.1'-
altered.
Also, the ion size parameter for hydrogen
ion, normally set at 5.0, may be changed.
The permissible decimal range of quantities in programs
coded by Fortran is lo- 38 to 10+ 38 •
This range is generally
sufficient, but in some problems may be greatly exceeded,
causing meaningless results.
Considerable attention was there-
fore given to providing means by which certain quantities
would
be automatically scaled in· it,be proper direction when nee.
~
'
..
~
essary, without changing the validity of any equations.
30
V.
RBSULTS
Much of the testing of the computational methods
devel~
oped herein could be done only with theoretical titration
curves.
For the most part, these curves were calculated from
various hypothetical dibasic acids.
The problem of the mono-
basic acid is considerably less complex in many respects, and
has been largely ignored since any method capable of handling
dibasic acids is almost certain to be more than sufficient
for monobasic acids.
Likewise, once the problem of the dibasic acid has been '
solved without resorting to graphical or other two dimensional
limitations, additional hydrogen ions give no particular complications, in theory.
In practice, the limited accuracy of ex-
perimental data will likely be the greatest
treatment of polybasic acids.
pro~em
in the
For these reasons, dibasic acids
are the principal ones considered in this work.
A.
Resolution of Successive Constants
One of the greatest problems in the calculation of acid
dissociation constants has been the differentiation of suecessive constants whose values differ only slightly.
Gener-
ally, if the ratio of the values of successive constants has
been less than about 500, previ'~-~s. c•lculatioiis have employed
na thematical artifices involving questionable assumptions.
It
31
has been calculated (14) that a break will appear in the
titration curve of a dibasic acid only if the ratio of constants is over sixteen.
Since there exist several acids whose
reported constants differ by even lesser amounts, it is important to test the differentiating ability, or "resolving
power" of any new computational method.
This has been done
using theoretical data, to test the ultimate limits, and with
modifications to simulate experimental data in order to determine the results likely to be obtained in practice.
A series of titration curves was calculated for dibasic
acids with ratios of dissociation constants varying from 1000
to 3.
The initial
parame~ers
used are given in Table 1.
Several of these curves have been plotted in Figure 3 to indicate the general curve shape obtained with various ratios of
.JO.
constants.
These calculated curves are precise in pH, and
within 0.001 ml of base at each point.
Activity corrections
were not made, since such corrections are somewhat time consuming, and ideally should have no effect on this part of the
investigation.
However, as a safeguard, several of the curves
with low ratios of constants were also calculated with activity corrections to see if there were any effects peculiar to
such corrections.
1.
Ultimate resolution
·-.•
•'
·.•
These calculated curves were then processed to determine
the dissociation constants, and the results are given in TableL
32
en
J-
z
::::>
:c
a..
2
I
EQUIVALENTS · OF . BASE -ADDED
.'
Figure 3.
/
.....·
....
Calculated titration ·curveJ 'for variot;{s hypothetical
dibasic acids with varying ratios of
I 2
Ki K
Conditions and results for the theoretical test of resolving power
upon titration curves of dibasic acids (Acid concentration
0.05 M,
base concentration
0.10 N, and initial solution volume
100.0 ml)
Table 1.
=
Initial constants
Ratio
Calculated constants
pK2
pel
=
=
pKl
PK2
Fractional
Standard deviations
.SK2
.I'Kl
3.00
3.52
4.00
4.52
6.00
6.00
6.00
6.00
1000
300
100
30
3.00
3.52
4.00
4.52
5.99
6.00
6.00
6.01
1
5
9
2
4.60
4.70
4.82
5.00
6.00
6.00
6.00
6. 00
25
20
15
10
4.60
4.70
4.82
5.00
6.00
6.00
6.00
6.01
1 X 10- 5
4 x 1o-5
2 x 1o-s
1 x 1o-s
1 x 10-6
4 x 1o-8
9 x 10-8
6.00
6.00
6.00
6.00
5
3
10
3
5.30
5.52
5.00
5.52
5.99
6.00
6.00
6.00
1
1
6
5
4
2
6
8
-~.'
'
5.30 .
..
5.52
5.ooa
5.52a
•.
awl ih activity correc-tions
)
X 10-5
x 1o-6
X 10-6
X 10-5
x lo-4
X 10-4
x 1o-6
x 1o-4
2
1
3
1
x lo-5
X 10-6
x 1o-1
x 10-1
1 x 1o-8
x
x
x
x
1o-8
1o-8
1o-9
1o-8
w
w
34
It is evident from these results that this method is inherently capable of excellent resolution.
It is instructive
to note, however, that the answers obtained are not always
exactly equal to the constants used to calculate the titration
curvea.
An error of 0.01 in pK is equivalent to an error of
about 2% in K, and in several cases such errors were obtained.
These errors are not reflected in the standard deviations,
since this latter quantity is merely a measure of how well the
answers fit the data, and not a measure of how accurate the
answers are.
The inclusion of corrections for. activity caused no
difficulties.
2.
Effects of errors
A consistent error in experimental data maf result in a
titration curve that is smooth and satisfactory in appearance
When
plot~ed,
but is nevertheless inaccurate.
An error -of up
to 0.05 units in the standardization of a pH meter might not
be unexpected under certain circumstances.
To determine the
effects of such an error, the pH values of the calculated
curves were all changed by the same amount, and the resulting
data processed to obtain the constants.
in Table 2.
/
The results are given
....·
The changes in pKl are quite .c:onsistent -·and not alarmingly large.
The changes in pK2 are considerable, and no posi-
Table 2.
Conditions and results for the test of the effect upon dissociation
constants of a consistent error in the titration curve
Initial constants
Calculated constants with
+0.05 pH unit error
Calculated constants with
-0.05 pH unit error
pKl
pK2
pKl
pK2
pKl
pK2
3.00
3.52
4.00
4.52
4.60
6.00
6.00
6.00
6.00
6.00
3.09
3.(>1
4.08
4.61
4.69
4.61
4.80
5.04
5.25
5.18
2.91
3.43
3.91
4.44
4.51
Neg.
4.70
4.82
6.00
6.00
4.77
4.90
5.65
5.58
"_a
5.00
5.30
5.52
5.oob
5.52b
6.00
6.00
6.00
6.00
6.00
5.08
5.37
. 5.60
5.08
5.60
5.47
5.74
5.63
5.64
5. 76
4.63
-- a
4.92
5.23
5.44
4.92
5.44
~his calculation was not completed
bwith activity corrections
"
"
,,"
Neg.
"
"
"
"
w
tJt
36
tive values were obtained when the pH values were lowered.
Activity corrections gave no change in pK1 , but did affect pK2 • The reason for this is not known.
Experimental titration curves are unlikely to exhibit
the precision demanded of the calculated curves, nor, it is
hoped, will they have a consistent error of disturbing magnitude.
gin.
~~at
will be found are random errors of multiple ori-
A precise determination of the effects of such errors is
more than likely impossible.
Since acids with suitably spaced
constants are not readily available (if at all), and experimental titrations are not sufficiently accurate, attempts to
determine the effects of random errors must be made with calculated data suitably modified to include such errors.
Ideally,
the errors should be applied to both pH and titrant volume
readings, but this would seem to be an unnecessary complication for what can at best be only an approximation.
the pH data were modified.
Thus, only
McComas and Rieman (19) estimated
the accuracy of pH measurements with a glass electrode as
t 0.03 units.
~ss
Although other electrodes may be subject to
error, it was considered reasonable to subject the pH
data to a standard deviation of 0.05 units.
A table of random
digits, in conjunction with the standard Gaussian Distribution
Curve, was used to calculate the error to be applied.
Table 3
gives the direction and magnitude of the error assigned for
each digit.
37
Table 3.
Errors in pH assigned by random digit table for
Sigma = 0.05
Digits
Error
+
0
1
2
3
4
5
6
7
8
9
0.006
0.020
0.034
0.072
0.083
All calculated curves were treated identically, and for
a 28 point curve, the average deviation applied was -0.01
units.
The results of the dissociation constant determina-
tions f rom these modified curves are given in Table 4.
Table 4.
Conditions and results for the test of the effect
upon dissociation constants of random errors in the
titration curve
Initial constants
PK2
pKl
Calculated constants
pK2
pKl
3.00
3.52
4.00
4.52
4.60
6.00
6.00
6.00
6.00
6.00
2.84
3.54
4.10
4.49
4.57
Neg.
6.33
4.30
5.45
5.49
4.70
4.82
5.00
5.30
5.52
5.ooa
5.52a
6.00
6.00
6.00
6.00
6.00
6.00
6.00
4.60
4. 72
4.88
5.29
5.51
4.89
5.51
Neg.
Neg.
Neg.
5.85
5.82
Neg.
5.92
awith activity corrections
38
Once again, it is pK2 that is predominantly affected,
both with and without activity corrections.
However, the
errors are consistent neither in direction nor magnitude.
B.
Resolution of Mixtures
The mathematical resolution of a mixture of acids is
considerably more difficult than the resolution of successive
constants of a single acid.
The first test was made upon a
calculated curve for a monobasic acid, pK
= 5,
treating this
curve as a mixture of monobasic acids in varying concentration
ratios, with the total concentration equal to that used in
calculating the curve.
Ideally, the computations should re-
sult in values of 1.0 x 10~ 5 for both constants.
The results
given in Table 5 indicate that this ideal was approached rather
closely.
Table 5.
Conditions and results for the mathematical treatment of a single monobasic acid as a mixture of two
monobasic acids
Molar concentrations
Acid 1
Acid 2
Dissociation constants x 105
Acid 1
Acid 2
0.0500
0.0495
o. 0490
0.0475
0.0450
0.0000
0.0005
0.0010
0.0025
0.0050
1.000
1.000
.999
.999
1.00
1.02
1.05a
1.02a
.995
0.0425
0.0400
0.0350
0.0300
0.0250
0.0075
0.0100
0.0150
0.0200
0.0250
.998
.999
1.00
.991
1.00
1.01
1.00
.995
1.01
.997
aBetter values were obtained in these runs
39
As mentioned in Chapter IV, this part of the program is
reiterative.
Such
methods~y
occasionally oscillate around
the correct answer, or perhaps even diverge, rather than converge to a final solution.
was encountered.
In the runs in Table 5, oscillation
Ten iterations were permitted in each run,
and the answers were taken from the · iteration which gave the
smallest least-squares parameter.
This was usually the best
answer, although there were exceptions in two cases.
A calculated curve for what might be considered a typical
acid mixture was then processed; the results are given in
Table 6.
No difficulties were encountered in this calculation.
Table 6.
Conditions and results for the resolution of a mixture of a monobasic and a dibasic acid (Concentration of each acid
0.01 M)
=
Acid 1
Acid 2
Dissociation
given
6.40 X 10-5
2.70 X l0-6
1.77 X lo-4
Kl
K2
c.
constants
found
6.41 X lo-5
2.69 X l0-6
1.77 X l0-4
Consistency
Experimental consistency was examined by titrating aliquots of a tartaric acid solution with sodium hydroxide, using
a glass electrode.
The pH meter was standardized against 0.05
M potassium biphthalate at the beginning of each titration.
40
There was no significant drift.
The constants obtained from
these titrations, with activity corrections applied, are given
in Table 7.
Table 7.
Conditions and results for repetitive titrations of
tartaric acid (Initial volume = 100.0 ml, acid concentration
0.1031 M, base concentration
0.05 N)
=
=
Dissociation constants
K1 X 104
K2 X 105
8.64
8.64
8.81
9.33
8.34
10.07
10.05
Standard deviations
K1 X 106
~ X 1010
4.34
6.24
4.35
4.28
4.19
5.34
12.5
10.7
4.33
8.8
3.76
5.32
2.41
3.00
7.8
10.5
5.2
6.5
4.31
4.12
4.17
Although 12 to 14 pH measurements were made in each titration, and all reasonable care was taken, there are considerable variations in the results.
Here, unlike the tests with
calculated data, it is K1 that shows the greatest variation,
with values that bracket the commonly accepted 9.6 x 10·4 •
The variations in K2 are much smaller than those in K1 , but
the values differ somewhat from the accepted one of 2.9 x lo-5.
Once again, the standard deviation is obviously not a valid
indication of the accuracy of the computed constants.
D.
Improper Calculations
The errors introduced by treating a dibasic acid as a
mixture of monobasic acids were investigated by processing the
41
single acid calculated data as mixtures.
From the results
given in Table 8, it can be seen that this procedure will produce very good results when the ratio of constants is 100 or
more, and reasonably good results are obtained at even lower
ratios.
Table 8.
Below a ratio of 10, results were unsatisfactory.
Conditions and results for the test of the effect
upon dissociation constants of the treatment of a
dibasic acid as a mixture of two monobasic acids
Initial constants
pKl
pK2
Calculated constants
pKl
pK2
3.00
3.52
4.00
4.52
4.60
6.00
6.00
6.00
6.00
6.00
3.00
3.52
4.00
4.54
4.62
5.99
6.00
5.99
5.99
5.99
4.70
4.81
5.00
5.30
5.52
6.00
6.00
6.00
6.00
6.00
4. 72
4.86
5.05
Neg.
5.39
5.97
5.97
5.96
3.72
Neg.
B.
Experimental Titrations
In addition to the titrations of tartaric acid previously
mentioned, titrations were made on acetic acid, succinic acid,
and on mixtures of tartaric and acetic, succinic and acetic,
and tartaric and succinic acids.
The last three would provide
a severe experimental test of the resolving power of the method
for mixtures.
Unfortunately, of these three, only the last
produced any positive results.
Prom Table 9, it can be seen
that the single acids approached the literature values fairly
42
closely, and the values for tartaric acid in the mixture are
very close to the commonly accepted values.
However, negative
constants were obtained for succinic acid in this mixture.
Table 9.
Results of experimental titration& of various single
acids and mixtures
Acid
Constant
Experimental
value
1.7S
X
1o·S
5.96 x 10-S
6.6
X
l0-5
2.18 x 10·6 .
9.81 X 10•4
2.8
x 1o·6
9.6
X
10-4
2.9
X
lo-5
Acetic
K
1.66
Succinic
Kl
K2
Tartaric +
Succinic
Tar. K1
Tar. K2
10-s
Literature
value
2.88
X
X
Sue. Kl
Neg.
Sue. K2
Neg.
10• 5
Data for the titrations giving successful results are
tabulated in Appendix A.
•
43
VI.
DISCUSSION
The computational procedures developed in this work have
been shown, it is believed, to be fundamentally sound and inherently extremely accurate.
However, the accuracy that can
be attained in practice, by any method, may be far poorer than
generally expected.
Successive dissociation constants of small ratio, a
major obstacle in most former methods, have been shown to present no difficulties attributable to such ratios.
The thor-
oughly satisfactory manner in which closely spaced constants
are resolved is not, however, accompanied by equally satisfactory accuracy.
It is this latter fact which casts some sus-
picion upon all constants previously determined by titration
rethocis .
It should be emphasized that the matter of accuracy
is la rgely an experimental problem, and neither necessarily
nor probably a fundamental limitation of the method.
Since
the ''titrations" used for testing were calculated by reiterative methods, some tolerance had to be allowed.
This was in
the volume of base, which was within 0.001 ml at every point
in a titration requiring 100.0 ml.
Such accuracy in practice
would be attainable only with exceptional care, even in weight
titrations.
Although, as shown, this may lead to a 2% error
in dissociation constants, it is not unusual to find literature
values expressed to three significant figures when even the
44
second is probably uncertain.
Other experimental methods may
involve less error, but they can not be judged in this work.
Consistent pH errors produce effects somewhat greater
than anticipated.
For a monobasic acid, it might be expected
that the change in pK would essentially equal the change in pH.
Such a test was not made, but it now appears that the effect
might not be quite so simple.
The dibasic acid tests demon-
strate that a simple displacement of the titration curve gives
a not so simple displacements of pKs.
An analysis of the math-
ematical niceties involved in these displacements would have
little practical value.
conclusions can be drawn.
However, two simple and constructive
First, the magnitude of errors
to be expected, if the work if performed with care, has been
overestimated.
Second, the effect of such errors as may be
present can be considerable.
These conclusions are also con-
sistent with the experimental titrations done in this work.
When considering the results of introducing random errors
into the titration curves, it is well to remember that the
curves do not retain their original shape, nor are they altered
identically in two dimensions.
With this in mind, one may
conclude that the rather random results indicate excessive
pessimism in the estimation of likely experimental error; this
is then essentially the same result as obtained for consistent
errors.
Mixtures of acids proved to be no more difficult to re-
45
solve than single acids, within limits.
Certainly the re-
sults given in Tables 5 and 6 are as good as one could reasonably expect.
However, more severe examples, such as mixtures
of dibasic acids with overlapping constants, proved to be insoluble.
Inaccuracies in the data are the most reasonable ex-
planation for this failure.
It is possible that titration data
accurate to more significant figures would enable satisfactory
resolution to be attained, but there would be no practical application.
The almost total failure of the experimental titra-
tions of mixtures to be properly resolved is not surprising.
The solitary exception cannot be explained.
In general, ti-
tration of acid mixtures does not appear to be a fruitful undertaking, unless there is considerable separation of constants.
The multiple titrations of tartaric acid give an indication of the results likely to be obtained by ordinary analytical procedures.
These titrations were not the ultimate in re-
finement nor accuracy, but were typical of ordinary laboratory
practice.
Better results would be desired, but the range of
pK1 , 3.00 - 3.08, is not unreasonable in view of the previous
tests, and pK2 , 4.36 - 4.39, covers an almost remarkably small
range.
The "accepted" values, to which these results were compared in section
v,
are not to be considered as ultimate
stand:: . rds for comparison.
They are perhaps the most widely
accepted, but a cursory search of published constants will re-
46
veal both higher and lower values than those obtained in this
work.
For this reason, consistency is considered to be a more
important criterion than conformance to any other particular
work.
On that basis, and in view of the effects of errors,
these titrations may be considered quite satisfactory.
The errors that resulted by assuming a dibasic acid to
be a mixture of monobasic acids were not as extensive as anticipated.
If all other factors are equal, the ratio of constants
may apparently be as small as 100 without introducing appreciable errors.
There is no merit in such an assumption if com-
puting methods and facilities of the type used in this work are
available, (in fact the mixture involves more work), but it
does indicate that the limits of 500 to 1000 used by others are
rather conservative.
And that is, perhaps, the only assumption
in previous calculations that is conservative.
47
VII.
SUMMARY
The primary goal of this work has been attained; dissociation constants have been calculated without use of
simplifying assumptions, and with full consideration of all
available data.
Errors in the data have a considerable ef-
fect on the answers obtained, and some calculations are simply
not possible due to experimental inaccuracies.
Many literature values must now be accepted with reservation for several reasons.
These include unjustified or un-
necessary simplifications in calculations, and the use of
different pH scales in older work.
Extremely accurate titrations are required to obtain dissociation constants of two or more significant figures, particularly for polybasic acids.
If good data are available,
however, the mathematical means to make full use of them are
now at hand.
48
VIII.
LITERATURE CITED
l•
~· ~.,
1.
Dippy, J., S. Hughes, and A. Rozanski,
1959, 2492.
2.
Jones, R. and D. Stock,
3.
Kolthoff, I., Chern. Weekb1ad, 16, 1154, (1918).
[Original n~vai1ab1e for-examination; abstracted
in Chern. Abstr. 14, 13 (1920).]
4.
Paul, T.,
~·
physik. Chern., 110, 417 (1924).
5.
Dhar, N.,
z.
anorg. allgem.
6.
Bates, R., W. Hamer, G. Manov, and S. Acree,
Bur. ~., ~' 183 (1942).
7.
Hamer, W. and S. Acree,
605 (1936).
B.
Lewis, G. and M. Randall. Thermodynamics and the Free
Energy of Chemical Substitution. New York, N.Y.,
McGraw-Hill Co. 1923.
9.
Britton, H. Hydrogen Ions. 3rd ed., Vol. 1.
N.Y., D. Van Nostrand Co. 1943.
10.
Sorensen, S. and K. Linderstrom-Lang, Compt.
Lab. Carlsberg, 15, No. 6 (1924).
11.
Guggenheim, E.,
12.
Hitchcock, D.,
13.
Mori, Y., ~· QQ!!. Sci. Eng., 1, 205. [Original not
available for examination; abstracted in Chem. Abstr.,
!, 2529 (1907).]
14.
Auerbach, F. and E. Smo1czyk,
65 (1924).
15.
Britton, H.,
16.
Cohnt E. 1 F. Heyroth, and M. Menkin,
~' 6V6 (1928).
l•
J·
~.
§2£., 1960, 102.
Chern.,~'
l·~·
li!!!·
323 (1926).
~·
l•
~·
li!!!·
!1•
Std.,
New York,
~·
I!!Y·
J. Phys. ~., ~' 1758 (1930).
J. ~· ~· §2£., ~' 855 (1936).
~-
Soc.,
~·
~'
physik.
~.,
110,
1896 (1925).
l•
~· ~- ~.,
49
l·
Chern. Soc., 12iQ, 855.
17.
Speakman, J.,
18.
Crosbie, E. and J. Monahan. A General Least Squares
Fitting Procedure. (Typewritten) Argonne, Illinois,
Argonne National Laboratory. 1959.
19.
McComas, W. and
(1942).
20.
The Rand Corporation. A Million Random Digits with
100,000 Normal Deviates. Glencoe, Illinois, The Free
Press. 1955.
w.
Rieman, l•
~· ~·
22£•,
~'
2948
50
X.
Table 10.
Data for titrations of tartaric acid
Acid concentration
0.01031 M; base concentration
0.0501 N; initial solution volume
50.00 ml.
=
=
Ml of base
APPENDIX A
1
2
=
Solution pH
4
3
5
6
7
4.00
2.98
2.98
2.97
2.96
2.99
2.94
2.94
8.50
3.45
3.45
3.45
3.44
3.46
3.43
3.43
12.50
3.94
3.94
3.93
3.93
3.94
3.91
3.90
16.50
4.47
4.46
4.47
4.47
4.49
4.45
4.45
19.00
5.04
5.02
5.05
5.03
5.05
5.01
5.01
20.00
5.66
5.67
5.70
5.67
5.70
5.67
5.68
20.20
5.99
5.98
6.07
6.00
6.06
5.99
6.00
6.34
6.43
6.30
6.48
6.32
6.38
7.00
7.11
7.01
7.20
7.01
7.00
7.55
7.58
7.50
7.87
7.70
7.55
8.30
8.55
8.40
8.30
20.30
20.40
6.95
20.45
20.50
20.52
8.17
8.39
8.56
51
Table 11.
Data for titration of acetic acid
Acid concentration
0.02174 M; base concentration
0.0501 N; initial solution volume
50.00 ml.
=
=
Table 12.
=
Ml of base
Solution pH
1.00
3.00
6.00
10.00
14.00
3.56
3.99
4.32
4.67
5.00
18.00
20.00
21.00
21.30
21.50
5.45
5.85
6.34
6.70
7.26
Data for titration of succ1n1c acid
Acid concentration = 0.01051 M; base concentration
0.0501 N; initial solution volume
50.00 ml.
=
=
Ml of base
Solution pH
13.50
3.63
4.05 .
4.46
4.87
5.28
16.50
18.50
19.60
20.20
20.60
5.70
6.08
6.41
6. 73
7.17
2.00
4.50
7.50
-lo. so
52
Table 13.
Ml of base
Data for titration of tartaric and succ1n1c acid
mixture (Acid concentrations = 0.01031 M and
0.010437 M; base concentration = 0.0501 N; initial solution volume = 100.00 ml.)
·
Solution pH
Ml of base
Solution pH
1.00
2.00
3.00
4.00
5.00
2.77
2.84
2.92
3.00
3.09
26.00
27.00
28.00
29.00
30.00
4.63
4.70
4.80
4.89
4.99
6.00
7.00
8.00
9.00
10.00
3.15
3.23
3.31
3.40
3.48
31.00
32.00
33.00
34.00
35.00
5.09
5.20
5.30
5.40
5.53
11.00
12.00
13.00
14.00
15.00
3.55
3.62
3.70
3.77
3. 86
36.00
37.00
38.00
39.00
39.50
5.65
5.80
5.95
6.18
6.31
16.00
17.00
18.00
19.00
20.00
3.92
4.00
4.05
4.12
4.19
40.00
40.20
40.40
40.60
40.80
6.50
6.60
6. 71
6.84
7.05
21.00
22.00
23.00
24.00
25.00
4.26
4.32
4.40
4.48
4.55
41.00
41.10
7.35
7.55
53
XI.
c
c
c
c
c
c
c
c
c
c
c
c
APPENDIX B
WAYNE DUNNING, IOWA STATE UNIVERSITY
THIS PROGqAM CALCULATES DISSOCIATION CONSTANTS AND THEIR STANDARD
DEVIATIONS FOR A SINGLE ACID OR A MIXTURE OF TWO ACIDS. PRINCIPAL
INPUT DATA CONSIST OF A PH TITRATION CURVE AND KNOWN ACID CONCENTRATIONS. IN ADDITION, CONCENTRATIONS MAY BE CALCULATED FROM A
TITRATION CURVE AND KNOWN CONSTANTS, OR A THEORETICAL TITRATION
CURVE MAY · BE COMPUTED. OPTIONS INCLUDE THE TITRATION OF A BASE,
OR OF A SALT OF A WEAK ACID OR BASE.
WWOllOO
WW01026
WWD1002
WWD
DIMENSION JUNK11401r JUNK2140r21r JUNK3(401
00000 DIMENSION SIZIONibl, CKPROOiblr CKINDiblr HAIONI81 0 HYDACTI41 0
WW01003
000001 OHACTI41, CMUI511, PHI511r ACTCFHI511o GAMMAI61, TNTHL(511o
WWD10C4
OOOJ02 FACTORibl, ACTSCLI71, RESULTI61, ARI50,61, Ell41, FCI50lr FHI501oWWD1005
000003 DFI501t G(40,401, Rl40,11r Ul6,6l
WW01006
COMMON JUNKl, JUNK2, JUNK3
00000 COMMON SIZIONo CKPROD, CKINO, HAlON, HYDACT 1 OHACT 1 CHU 1 PH 1
WWD1007
000001 ACTCFH, GAMMA, TNTML, FAC.TOR, ACTSCL, RESULT, AR, Eo FC, FH, OF, WWD1008
000002 G, R, U, AC!Dl, ACID2, CAC!Dl, CACID2, NACIOS, NCONl, NCON2,
WWD1009
000003 NCONT, OENOM1, DENOM2, VNUMl, VNUM2, SCFTR, FN, Yr FB, MSCALE,
WWD1010
000004 SUM, Q, 01, NDPTS, CMLTNT, ROOCL, VOLTL, CNAo VOLC, CLION
WWD1011
C
BEGINNING OF PROGRAM. THIS SECTION IS PRIMARILY FOR PROPER
C
INITIALIZATION OF THE PROGRAM. THE NECESSARY DATA IS READ IN, AND
C
THE VALUES OF CERTAIN CONSTANTS ARE SET.
SET ICN SIZE PARAMETERS, DISSOCIATION CONSTANTS, AND INDIVIDUAL
WWD1101
C
WWD1102
C
ION CONCENTRATIONS TO ZERO
WWD1103
00401 00 403 I X 1, 6
WWD1104
00000 SIZIONIII 2 O.
RESULTIII oa O.
WWD1105
00000 CKPRODI I I " O.
WWD1106
00403 CKIND([ I = 0.
WWD1107
00000 00 405 I 2 1 0 8
WWD1108
00405 HAIONIII
O.
WWD1109
c
WW01110
ZERO POWER OF HYDROGEN AND HYDROXIDE ACTIVITY EQUALS ONE
c
WWD1111
00000 HYDAC Til I "' 1.
WWD1112
00000 OHAC Tl 11 = 1.
2
c
c
c
UNLESS OTHERWISE SPECIFIED, HYDROGEN ION SIZE IS 5
HIONSZ 2 5.
WWD 1113
C
C
00000
00000
UNLESS OTHERWISt SPECIFIED, TEMPERATURE IS 25 DEGREES, AND DEBYE
HUCKEL COEFFICIENTS ARE
DBHFA 2 0.5092
DOHFB 2 0.32B6
WWD1115
WWDlll6
WWD1117
WWD1118
C
UNLESS
BETADH
WWDlll9
c
c
c
OTHERWIS~
= O.
SPECIFIED, BETA IN D-H FORMULA IS ZERO
WRITE HEADING, READ AND WRITE OPENING STATEMENT
00000 WRITE OUTPUT TAPE 9, 8
00000 READ 10
00000 WRITE OUTPUT TAPE 9, 10
PUNCH 10
WWD1121
WW01122
WWD1123
WWD1124
C
READ CONTROL PARAMETERS
00000 READ 12 0 NTYP~1, NTVPE2 0 NTVPE3, NTVPE4, NTVPE5 0 NTVPE6o NTVPE7o
000001 NTVPE8, NTYPE9
WWD1125
WWD1126
WWIH127
c
c
C
WRITE TYPE OF CALCULATION TO BE DONE IN THIS PARTICULAR RUN
54
1413, 411, 409), NTYPE1
OUTPUT TAPE 9, lit
415
OUTPUT TAPE 9, 16
415
OUTPUT TAPE 9, 18
001t07
001t09
00000
OOit 11
00000
001t13
GO TO
WRITE
GO TO
WRITE
GO TO
WRITE
001t15
001t17
001t19
00000
001t21
READ ACID CONCENTRATIONS, AND WRITE WITH APPROPRIATE COMMENT
OEPENOING . UPON WHETHER CONCENTRATION IS EXACT, OR AN INITIAL
ESTIMATE. IF THERE IS ONLY ONE ACID, THE CONCENTRATION OF THE
SECOND IS ZERO.
READ 20, ACI01, ACI02
GO TU 1421, 421, 4191, NTYPE1
WRITE OUTPUT TAPE 9, 22, ACIOl, ACI02
GO TO 423
WRITE OUTPUT TAPE 9, 21t, ACI01, ACI02
c
c
c
c
c
c
c
c
c
c
c
c
001t23
c
c
00000
c
c
c
c
c
WWD1129
WWil1130
WWOll:H
WW01132
WW01133
WWIH l31t
WW01135
WW01137
WWilll38
WW01139
WWOllltO
WW01141
WWD1142
READ INITIAL VOLUME OF SOLUTION IN ML., NORMALITY OF TITRANT,
IONIC STRENGTH IAN ESTIMATE UNLESS VALUE IS CONSTANT), MOLAR CONCENTRATION OF ANY ADDEO MONOVALENT SALT, DISSOCIATION CONSTANT OF
WATER, AND DESIRED CLOSENESS OF FIT OF RESULTS IN PERCENT
WW01147
READ 26, SOLNV, 8ASEN, GMU, CKN03, WCON, ERROR
WRITE THE ABOVE QUANTITIES
WRITE OUTPUT TAPE 9, 28, SOLNV, BASEN, GMU, CKN03, WCON, ERROR
WW0111t9
WW01150
READ THE INDIVIDUAL DISSOCIATION CONSTANTS OR THE ESTIMATED VALUES
OF THEM. THE FIRST FOUR ARE FOR THE FIRST ACID, THE LAST TWO FOR
THE SECOND ACID. ANY CONSTANT WHICH DOES NOT EXIST MUST HAVE A
VALUE OF ZERO.
WW01152
00000 READ 30, ICKINOIII , I • 1, 61
c
c
001t25
001t27
00000
001t29
c
c
c
c
00431
00433
00000
00434
00000
c
c
c
WRITE
GO TO
WRITE
GO TO
WRITE
INDIVIDUAL DISSOCIATION CONSTANTS WITH COMMENTS
1429,427,4271, NTYPE1
OUTPUT TAPE 9, 32, ICKINOIII, I • lt 61
431
OUTPUT TAPE 9, 31t, ICKINDIII,
• lt 61
IF ACTIVITY CORRECTIONS ARE NOT TO BE MADE, WRITE THIS FACT, SET
ALL VALUES OF IONIC STRENGTH TO ZERO, AND DO NOT REAO IN THE ION
SIZE PARAMETERS.
GO TO 1435, 433, 4351, NTYPE2
WRITE OUTPUT TAPE 9, 36
DO 434 I = lt 50
CMUIIl,. O.
GO TO 437
WWD1154
WWD1155
WWD1156
WW1Hl57
WW0115B
WW01161
WW01162
WWD1163
WW01164
READ ANO WRITE THE ION SIZE PARAMETERS. VALUES FOR NONEXISTENT
IONS SHOULD BE ZERO.
00435 READ 38, ISIZIONIIlo I • 1, 61
00000 WRitE OUTPUT TAPE 9, 40, ISIZIONIII, I • lt 6)
WWDll66
WW01167
IF IONIC STRENGTH IS SPECIFIED AS CONSTANT, WRITE ITS VALUE AND
SET ALL STORAGE VALUES TO IT.
GO TO 1440, 439, 4431, NTYPE3
WRITE OUTPUT TAPE 9, 42o GMU
DO 441 I • 1, 50
CMUIII • GMU
WWD1168
WW01172
WW01173
WWDll 74
c
c
c
00437
00439
00440
OOH 1
c
c
IF VOLUME IS SPECIFIED AS CONSTANT, WRITE THIS FACT AND THE VALUE
55
00443 GO TO (447, 445), NTYPE4
00445 WRITE OUTPUT TAPE 9 I 441 SOL NV
c
c
00447
c
00449
00000
c
00451
00000
c
00453
c
c
WRITE THE TYPE OF PH READING SPECIFIED BY CONTROL PARAMETER 5
GO TO 1453 1 451, 449) 1 NTYPE5
W~ITE THAT PWH IS USED
WRITE OUTPUT TAPE 9, 46
GO TO 455
WRITE THAT PCH IS USED
W~ITE OUTPUT TAPE 9, 48
GO TO 455
WRITE THAT PAH IS USED
WRITE OUTPUT TAPE 9, 50
WW01175
WWI)ll77
WWD1178
WWD1180
WWD1181
WWD1183
WWDll84
WWDll86
READ AND WRITE ION SIZE PARAMETER FOR HYDROGEN IF VALUE IS NOT 5.0
WWD1187
00455 GO TO 1459, 457), NTYPE6
WWDll90
00457 READ 52, HIONSZ
WWD1191
00000 WR!Tt OUTPUT TAPt 9, 54, HIONSZ
c
c
c
READ AND WRITE NEW D-H COEFFICIENTS A AND B IF TEMPERATURE IS NOT
25 DEGREES
WW01192
00459 GO TO 1463 1 4611 1 NTYPE7
WWD1194
00461 READ 56, DBHFA, ODHFB
WWD1195
00000 WRITE OUTPUT TAPE 9, 58 1 DBHFA, DBHFD
c
c
READ AND WRITE BETA IN D-H FORMULA IF BETA IS TO BE USED
00463 GO TO 1467, 465), NTYPEB
00465 READ 60, BETADii
00000 WRITE OUTPUT TAPE 9 1 62 1 DETADH
c
c
00467
c
00469
00000
c
00471
00000
c
00473
00000
c
00475
c
c
WRITE
GO TO
WRITE
WRITE
GO TO
WRITE
WRITE
GO TO
WRITE
WRITE
GO TO
WRITE
WRITE
THE TYPE OF TITRATION SPECIFIED BY CONTROL PARAMETER 9
1475 1 473 1 471, 469) 1 NTYPE9
SALT OF BASE TITRATED WITH BASE
OUTPUT TAPE 9, 64
477
BASE TITRATED WITH ACID
OUTPUT TAPE 9, 66
477
SALT OF ACID TITRATED WITH ACID
OUTPUT TAPE 9, 68
477
ACID TITRATED WITH BASE
OUTPUT TAPE 9, 70
COUNT NUMBER OF ACIDS
00477 NACIDS = 1
004H IF IACID2l 901, 483, 481
00481 NACIDS = 2
c
c
WWDll96
WWD1198
WWD1199
WWD1200
WWD1203
WW01204
WWD1206
WW01207
WWD1209
WW IH210
WWD1212
WWD1213
WWD1214
WWD1215
WWD1216
WWD1217
COUNT NUMBER OF DISSOCIATION CONSTANTS FOR FIRST ACID
NCON1 = 0
DO 489 I " 1, 4
IF ICKINDI!ll 902, 489, 487
NCON1 = NCON1 + 1
CONTINUE
NCON2 = 0
WWbl218
WWD1219
WWD1220
WWD1221
WWD1222
WWD1223
WWD1224
COUNT DISSOCIATION CONSTANTS FOR SECOND ACID, IF ANY
GO TO 14923, 49151, NACIDS
AN INITIALIZATION FOR NON-LINEAR LEAST SQUARES, IF THERE ARE
TWO ACIDS
04915 000 " 1000.
WWD1225
WW01226
004tl3
00000
00485
00487
00489
o4n 1
c
c
04913
c
c
WWD1228
56
c
00000
00000
04917
04919
04921
c
c
MAXIMUM NUMBER OF ITERATIONS ALLOWED PER PASS IN N-L LEAST SQS.
NIO • 5
DO 4921 I • 5, 6
IF ICKINOII l l 913, 4921, 4919
NCUN2 • NCON2 + 1
CONTINUE
TOTAL NUMOER OF CONSTANTS
04923 NCONT • NCON1 + NCON2
c
c
00000
c
c
c
00000
04925
00000
04927
04929
04931
00000
c
00000
c
c
04933
c
c
c
c
00101
c
c
ABSOLUTE ERROR
ERROR • ERROR I 100.
OBTAIN PRODUCTS OF DISSOCIATION CONSTANTS
RESET OVERFLOW TRIGGER
IF ACCUMULATOR OVERFLOW 4925, 4925
CKPRODI1l • CKINDI1l
DO 4927 I • 2, NCON1
CKPROOIIl : CKPROOII- 11 • CKINOIII
GO TO 14933, 4931), NACIDS
CKPROOI5l • CKIND15l
CKPROOI6l • CKINDI5l • CKIN016l
ERROR STOP IF A PRODUCT OF CONSTANTS IS OUT OF RANGE
IF ACCUMULATOR OVERFLOW 909, 4933
TRANSFER TO PROPER SECTION OF PROGRAM DEPENDING ON TYPE OF
CALCULATION OEING DONE
GO TO 1101, 201, 3011, NTYPE1
SUBMASTER PROGRAM TO DETERMINE DISSOCIATION CONSTANTS
GO TO READ IN TITRATION CURVE
GO TO 501
RETURN FROM CURVE READ IN. DETERMINE MATRIX SIZE FOR SINGLE ACID
00103 NROWS = NCON1
00000 NCOLMS = NROWS + 1
c
c
c
c
00000
00000
c
c
00000
c
c
00000
WWD122<,~
wwrH230
WWD123l
WW01232
WWD1233
WWD1234
wwD1235
WWD1236
WWD1237
WWD1238
WWD1239
wWD1240
WWD1241
WWD1242
WWD1H3
WWD1244
WWD1245
WWD1246
WWD1247
WWD1248
WW01249
WWD1250
WWD139B
WWD1399
WW01400
WW01401
WWD1402
WWD1403
WWD1404
WWD1405
WWD1406
SET ITERATION COUNTER AND SCALE CHANGE CHECKER TO ZERO
CHECKER PREVENTS AN ATTEMPT TO SCALE IN BOTH DIRECTIONS
SCALING IS NECESSARY TO KEEP DECIMAL POINT WITHIN ALLOWABLE RANGE
WWD1407
MSCALE • 0
WWD1408
MCOUNT = 0
MCLLD = 0
INITIAL SCALING FACTOR EQUALS ZERO
WWD1410
SCFTR = 0.
WW01412
SET NONLINEAR ERROR FACTOR
WWD1413
ERQO = .0001 • FLOATFINOPTSI
GET SCALING QUANTITY
00000 IF ACCUMULATOR OVERFLOW 105, 105
00105 ACTSCLI1l = 10. •• ISCFTR • FLOATFINCONTll
ERROR STOP IF SCALING QUANTITY OUT OF RANGE
c
00000 I~ ACCUMULATOR OVERFLOW 906, 107
WWD1414
WW01415
SET NCNLINEAR ITERATION COUNTER TO ZERO
00107 LLD = 0
CLEAR RESULTS AND ARRAY FOR SINGLE ACID
c
00109 GO TO 1111, 1151, NACIOS
00111 00 113 I • 1, NROWS
ocooo RESULTIIl • O.
OOOuO DO 113 J • 1, NCOLMS
00113 AR I I , J l " 0.
WWD1418
WWD1419
WWD1420
WWD1421
WWD1422
WWD1423
WW01424
WWD1425
c
c
WWD1417
57
c
c
c
00115
c
INITIALIZE LOOP AND GO TO ACTIVITY CORRECTION SECTION (GAMMA
CALCULATOR)
K • 1
OPTIONAL OUTPUT
IF I SENSE SWITCH 11 116, 801
116 WRITE OUTPUT TAPE 9, 61
00000 GO TO 801
WWD1427
RETURN FROM GAMMA CALCULATOR AND BRANCH ON TITRATION TYPE
00117 GO TO 1125, 125, 119, 1191, NTYPE9
WWD1428
WWD1429
WWD11t30
WWD1431
OBTAIN SCALED HYDROXIDE ACTIVITIES
00119 DO 123 I = 1, NCONT
00123 ACTSCLII + 11 • ACTSCLIII • OHACTI21 ·
TOTAL NF.GATIVE ION CONCENTRATION
c
00000 FN = -HYDACTI21 I GHYORO - CNA + COH + CLION
00000 GO TO 131
WWD1432
WW01433
WW0l434
WWD1435
WW01436
WW01437
OBTAIN SCALED HYDROGEN ACTIVITIES
00125 DO 129 I = 1, NCONT
00129 ACTSCLII + 11 • ACTSCLIII • HYDACTI21
TOTAL POSITIVE ION CONCENTRATION
c
00000 FN = HYDACTI2l I GHYDRO + CNA - COH - CLION
WWD1438
WW()1439
WWD1440
WWD1441
WWD1442
c
c
c
c
c
c
c
c
00131
c
00000
c
c
00133
c
c
c
00135
c
00137
c
SCALEC FN TIMES PROPER POWER OF ACTIVITY
Y • FN • ACTSCLINCONT + 11
IF OUT OF RANGE, SCALE UP
IF ACCUMULATOR OVERFLOW 161, 133
TRANSFER OUT TO CALCULATE PROPER MATRIX ELEMENTS FOR THIS POINT
GO TO 1601, 10011, NACIDS
RETURN FROM OBTAINING ELEMENTS, SET LOOP FOR NEXT POINT
SINGLE ACID
K •
K +
1
WWD1444
WWD1445
WWD1446
WWD1447
WW01448
WW01449
TEST FOR LAST POINT USED. IF FINISHED, SOLVE MATRIX.
IF IK - NOPTSl BOlt 801, 701
WW01451
RETURN FROM MATRIX SOLVE. OBTAIN DIFFERENCE FROM OLD ANSWERS
00139 SUM : 0.
00141 DO 143 I = 1, NCONl
00000 SUM= SUM+ ABSFIIRESULTIII - CKPRODIIII I RESULTIIII
REPLACE OLD ANSWERS WITH NEW
c
00143 CKPRODIII = RESULTIII
00000 CKIND11l : RESULTI11
00145 IF I NCON1 - 1 I 904, 151, 147
00147 DO 149 I = 2, NCONl
00149 CKINDIII "RESULTIII I RESULTII- 11
WWD1452
WWD1453
WWD1454
WWD1455
WWD1456
WWD1457
WW01458
WWD1459
WWD1460
WWD1461
c
c
c
WRITE ANSWERS AND DIFFERENCE FROM PREVIOUS SET
WWD1462
WWD1463
00151 WRITE OUTPUT TAPE 9, 72, ICKINDIII, I • 1, 4), SUM
PUNCH 3~, ICKINDillo I • 1, 4), SUM
TEST FOR END. IF ANSWERS SATISFACTORY, OBTAIN STANDARD DEVIATIONS
c
WWD1469
00000 MCOUNT • MCOUNT + 1
153 IF !SUM- ERROR • FLOATFINROWSII 6001, 6001, 155
c
c
ANSWERS NOT GOOD ENOUGH. REPEAT IF NOT TOO MANY ITERATIONS
00155 MSCALE = 0
00157 IF IMCOUNT - 101 107, 6001, 6001
c
c
SCALE UP
WWD1467
WWD1468
WWD1470
WWD1471
WW01472
58
00161
00163
00000
00000
c
c
IF !MSCALE) 905, 163, 163
MSCALE • 1
SCFTR • SCFTR + 1.
Gn TO 105
SCALE DOWN
00171 IF !MSCALEl 173, 173, 905
00173 MSCALE • -1
00000 SCFTR ,. SCFTR - 1.
O:JrrJ O Gn TO 105
c
c
c
c
TWO ACID, NON-LINEAR LEAST SQUARE SOLUTION
SET MEASURED VALUt
0100 1 FM!K) "' Y
CALCULATE ELEMENTS FOR SIXTH CONSTANT, IF ANY
c
100 3 IF !CKPRODI6)) 1005, 1009, 1C05
01005 DO lu07 I • 1, NCONl
00000 Z = I
00000 L = NCONT - 1 - I
01007 Ell+l0l=!IZ•CACI01+2.•CACI02-FNl•ACTSCLillliiGAMMAIIl•GAMMAI6ll
00000 El6l ,. 112. • CAC!D2- FN) • ACTSCLINCONT- lll I GAMMA(6)
00000 GO TO lOll
CALCULATE ELEMENTS FOR FIFTH CONSTANT, IF ANY
c
1009 IF ICKPRODI5)) 1011, 1015, lOll
01011 00 1013 I • lo NCONl
00000 l = I
OOOuO L • NCONT - I
01013 Ell+6l = IIZ•CAC!Ul + CACID2- FNI•ACTSCLill IIIGAMMAIII•GAMMA(5))
00000 El5) = IICACID2- FNl • ACTSCLINCONTll I GAMMAI5l
CALCULATE ELEMENTS FOR FIRST ACID
c
01015 DO 1017 I "' 1, NCONl
00000 l ,. I
00000 L • NCONT + 1 - I
01017 Elll = I l l • CACIDl- FNl • ACTSCLIL)l I GAMMAIIl
c
C
OBTAIN DIFFERENTIAL ELEMENTS OF MATRIX
01019 IF ACCUMULATOR OVERFLOW 1021, 1021
01021 AR(K, ll = Elll + E(7) • CKPRODI5l + Ellll • CKPRODI6)
00000 AR'!K, 21 ,. El2l + E!Bl • CKPROD!5l + Ell2l • CKPROD(6)
00000 AR(K, 31 • El3l + El9) • CKPRODI5) + El13) • CKPR00(6)
00000 AR!K, 4) a El4l + EllOl • CKPRODI5l + Ell4) • CKPRODI6l
00000 AR!K, NCONl+ll = El5l + El7l • CKPROD!ll + El8l • CKPRODI2l
000001 + El9l • CKPROD!3) + EllOl • CKPROD!It)
00000 AR(K, NCON1+2) a E!6) + Ellll • CKPROD!1) + Ell2) • CKPRODI2)
000001 + Ell3l • CKPROD!3l + Elllt) • CKPROD!Itl
CALCULATE FC
C
01023 FC!Kl • CKPROD!ll•Eill+CKPRODI21•EI2)+CKPRODI3l•EI3)+CKPRODI1tl•
010231 El4l+CKPROD!51•EI51+CKPROOI6l•EI6)+CKPR001ll•EI71•CKPROD(5)+
010232 CKPRODI2l•EIBI•CKPRODI5l+CKPRODI3l•EI9l•CKPRODI5l+CKPROD!Itl•
010233 EllOI•CKPR00!5l+CKPROD!ll•EI1ll•CKPRODI6l+CKPRODI2l•EI12)•
010234 CKPRCDI6l+CKPRODI3l•Eil3l•CKPRODI6)+CKPROD!Itl•EI11tl•CKPPOOI6)
00000 IF ACCUMULATOR OVERFLOW 161, 1025
C
SET LOOP FOR NEXT POINT
01025 K " K + 1
TEST FOR LAST POINT USED
C
01027 IF IK - NDPTS) 801, 801 1 1380
c
c
c
MODIFIED AN E 208 PROGRAM
01380 DO 1430 J • 1, NUPTS
Ollt30 DFIJl • FMIJl - FC!Jl
WWD1473
WW01474
WW01475
WWD1476
WWD11t77
WWD1478
WW01479
WW0141l0
WW014B1
W10Dl4A2
WW0141!3
WWD1484
WWD14B5
WW01486
WWD1407
WW01488
WWD1490
WWD1491
WW01492
WW01493
WW0llt91t
WWOl495
WWD1496
WWD149B
WW01499
WW01500
WW01501
WWD1502
WWD1503
WWD1504
WWD1505
WWD1506
WW01507
WWD1508
WWD1509
WWD1510
WW015ll
WWD1512
WWD1513
WW01514
WWD1515
WWD1516
WW01517
WWD1518
WW01519
WWD1520
WW01521
WWD1522
WW01523
WWD1524
WWD1525
WW01526
WWD1527
WW0152B
WWD1529
WWD1530
WWD1532
WW01533
WWOl534
.,
..
59
c
TESTING SWITCH FOR OPTIONAL OUTPUT
01450 IF !SENSE SWITCH 31 1460, 1490
01460 no 1480 J = 1, ~OPTS
01480 WRITE OUTPUT TAPE 9, 74, IARIJ,KI,K • 1, NCONTit DFIJI
c
c
CALCULATE
DO 1520 l = 1, NCUNT
Rllt 11 "O.
DO 1520 J a 1, NDPTS
R(L, 11 a Rllo 11 + ARIJ, ll • DFIJI
DO 1570 l = 1, NCONT
DO 1570 K " 1, NCONT
Gilt Kl "' O.
no 1570 J "' 1, NDPTS
GIL,KI = GIL, Kl + ARIJ, ll • ARIJ, Kl
IF ACCUMULATOR OVERFLOW 161, 1580
TEST SWITCH FOR R AND G OPTIONAL OUTPUT
c
01580 IF !SENSE SWITCH 41 1590, 1600
01590 WRITE OUTPUT TAP~ 9, 74, IIGIL,KI, K•1,NCONTI, Rlloll, L•1oNCONTI
01490
01500
01510
01520
01530
01540
01550
01560
01570
00000
c
c
c
01600
01610
01615
01617
00000
00000
00000
01619
00000
c
c
_IF G Y =R, THEN MATINV RETURNS WITH R REPLACED
BY Y AND G REPLACED BY G INVERSE
f"M
"
1
CALL MATINV IG, NCONT, R, MM, DETERMI
GO TO 11620, 911, 16171, NTYPE1
CKINDill ,. ACIDl
CKINDI21 " ACID2
Q 1 .. 0.
DO 1619 J = 1, NDPTS
01 "' 01 + IDFIJI •• 2.1
GO TO 1760
WW01536
WW01537
wwrH538
WWD1539
WWD1540
WWD1541
WWD1542
WWD1543
WWD1544
WWD1545
WWD1546
WWD1547
wwrH548
WWD1549
WWD1551
WWD1552
WWD1553
WWD155't
WWD1555
WWD1556
WWD1557
WWD1558
WWD1559
WWD1560
WWD1561
WWD1562
WWD1563
CHECK THE FIT
01620 01 .. o.
01625 DO 1630 J = 1, NDPTS
01630 01 = Ql + IDFIJI •• 2.1
IF ACCUMULATOR OVERFLOW 161, 1635
01635 Q "' A~SFIOOD- 011
01640 QOD = Q1
01645 LLD = LLD + 1
MCLLD • MCLLD + 1
IF FIT IS NOT GOOD ENOUGH, REPEAT IF NO. OF ITERATIONS IS
c
NOT TOO LARGE. CKPRODIJI CORRECTED IN ANY CASE
c
1650 DO 1653 J • 1, NCONl
1653 CKPRODIJI • CKPRODIJI + RIJ,ll
IF INCON21 911, 1660, 1655
1655 no 1659 J • 1, NCON2
1657 JJ • J + NCON1
1659 CKPRODIJ + 41 • CKPRODIJ + 41 + R(JJ, 11
01660 IF IQ - ERQDI 1680, 1680, 1665
01665 IF ILLO- NIDI 1670, 1680, l760
C
TEST SWITCH FOR Y AND G INVERSE, OPTIONAL OUTPUT
01670 IF !SENSE SWITCH 41 1675, 1677
01675 WRITE OUTPUT TAPE 9, 74, II GIL,KJ, K • lt NCONTI, Rllt lie
1 l • 1, NCONTI, ICKPRODIL21t L2 • lt 61, 01
01677 GO TO 107
WWD1564
WWD1565
WWD1566
WWD1567
REPLACE DISSOCIATION CONSTANTS AND TEST FOR CLOSENESS,
RECORRECT FOR ACTIVITY AND REITERATE IF NECESSARY
01680 SUM • O.
01685 RESULTI11 • CKPRODI1l
01690 IF INCON1 - 21 1705, 1695, 1695
WWD1582
WW01583
WW01584
WWD1585
WWD1586
c
c
c
WWD1568
WWD1569
WW01570
WWD1571
WWD1572
WWD1575
WWD1576
WWD1578
WWD1579
WW01581
60
01135
01740
01745
01750
DO 1700 I a 2, NCON1
RESULTIII • CKPRODIII I CKPRODII- 11
IF INCON2- 11 1720, 1715, 1110
RESULTI61 • CKPRODI61 I CKPROOI51
RESULTI51 = CKPR00(51
DO 1130 I = 1,
6
IF (CKINOIIll 914,1130,1725
SUM= SUM+ ABSFIIRESULTIII- CKINDIIII I CKINOIIII
CKINOIII = RESULTIII
CONTINUE
WRITE OUTPUT TAPE 9, 97, ICKINOIIlo I • 1, 61, SUM
PUNCH 30, ICKINDIII, I • 1, 61
IF ISUM- ERROR • FLOATFINCONTII 1760, 1760, l11t0
MSCALE "' 0
MCOUNT = MCOUNT + 1
IF IMCOUNT- 101 107, 1760, 1760
01760
01765
01770
01775
01780
CALCULATE
DO 1770 J
DO 1770 K
UIJ, Kl =
DO 1780 L
U(L, Ll
01695
01700
01705
1710
1715
01720
0172 5
1727
1130
00000
c
c
c
WW01592
WW01593
WW0159
THE CORRELATION MATRIX
= 1, NCONT
= 1, NCONT
GIJ , Kl I SCRTFI GIJ, Jl • G(K, Kl )
= 1, NCONT
SCRTFI GIL, Ll • Q1 I FLOATFINDPTS - NCONT - 11 l
C
WRITE FINAL RESULTS
01785 WRITE OUTPUT TAPE 9, 76,
01790 WRITE OUTPUT TAPE 9, 78
01795 WRITE OUTPUT TAPE 9, 80,
01800 WRITE OUTPUT TAPE 9, 82
01810 00 1820 L a 1, NCONT
01820 WRITE OUTPUT TAPE 9, 84,
01830 WRITE OUTPUT TAPE 9, 86
01840 WRITE OUTPUT TAPE 9, 88,
018401 FMIJI, FCIJl, OFIJI, J
00000 GO TO 401
c
c
c
c
WWD1587
WWD1588
WWD1589
Q1, MCLLD, ACTSCLI11
IJ, CKINDIJI, U(J,J), J • 1,
6)
(U(L,Kl, K = 1, NCONTI
ITNTMLIJI, PHIJI, CMU(J), ACTCFHIJI,
NDPTSI
= 1,
SUBMASTER PROGRAM FOR CALCULATION OF TITRATION CURVE
READ LIMITS OF PH AND STEPPING INTERVAL
00201 READ 90, PHLOWR, PHUPPR, PHSTEP
WRITE OUTPUT TAPE 9, 91
c
INITIAL CONDITIONS
00000 K = 1
00000 CMUI1 I • GMU
00000 TNTML I 1 l = 0.
00000 PH I 11 • PHLOWR
c
c
SCALING FACTOR
00203 ACTSCLI11 = l.OE30
IF TWO ACIDS, MOVE CONSTANTS OVER TO MAKE THEM CONSECUTIVE
c
00205 GO TO 1801, 2071, NACIDS
00207 DO 209 I a 1, NCON2
00000 M • NCON 1 + I
00209 CKPROOIMI • CKPRODII + ltl
c
c
00000
c
c
00211
c
c
WW01596
WW01597
WW01598
WWD1599
WW01600
WWD1601
WW01602
WW01603
WWD1601t
WWfH605
WWD1606
WWD1609
WWD1610
WWD16ll
WWD1612
WW01613
WW01611t
WWD1615
WWD1616
WWD1617
WWD1618
WWD1620
WW01621
WWD1622
WWD1623
WWD1621t
WWD1625
WWD1626
WWD1627
WWD1628
WWD1629
WWD1631
WWD1632
WW01633
GO TO ACTIVITY CORRECTION SECTION
GO TO 801
WWD1631t
RETURN AND TRANSFER ACCORDING TO TYPE OF TITRATION
GO TO 1217, 217, 213, 2131, NTYPE9
WWD1635
TITRATIONS OF BASE
WW01636
61
00 213 DO 215 I ., 1, 4
00215 ACTSCLII + 1 I • ACTSCLIII • OHACTI21
00000 GO TO 221
WW01631
WW01638
WW01639
TITRATIONS OF ACID
00217 DO 21c; I = 1, 4
00219 AC TSCLI I + 11 • ACTSCLI I I • HYOACTI21
WW0l640
WWDl641
WWD1642
WWD1643
c
c
c
c
00221
00000
00223
00225
00227
00000
00000
00229
c
c
00231
c
c
OBTAI~ ELEMENTS OF EQUATION
DO 223 I • 1r NCONl
WWDl645
WWD1646
WWD1647
EIII = IHAIONI11 • Z • ACTSCLI111 I CGAMMAIII • ACTSCLIJ + 111
WW01648
GO TO 1231, 2271, NACIDS
WWD1649
DO 229 I = 1, NCON2
WWOl650
Z • I
WW0l65l
L = NCON1 + I
EILI = IHAIONI61 • Z • ACTSCLI111 I IGAMMAII + 41 • ACTSCLII + 111WW01652
Z •
I
TRANSFER ACCORDING TO TYPE OF TITRATION
GO TO 1245, 241, 237, 2331, NTYPE9
WWD1653
00233
00000
00235
00000
00000
TYPE 4 TITRATION, SALT OF BASE WITH BASE
RQDNA = -HYOACTI21 I GHYDRO + COH + CLION
DO 235 I = 1r NCONT
RQONA = RQONA- IEIII • CKPROOIIII
CMLTNT = IRQONA - CNAI • IVOLTL I BASENI
GO TO 249
00237
00000
00239
00000
00000
TYPE 3 TITRATION, BASE WITH ACID
TYPE 3 TITRATION
RQOCL • HYOACTI21 I GHYORO - COH
DO 239 I = lr NCONT
RQDCL • RQDCL + IEIII • CKPRODIIII
CMLTNT • IRQOCL - CLIONI • IVOLTL I BASENI
GO TO 249
WW01660
WWD1661
WWDl662
WWD1663
WWD1664
WWD1665
00241
00000
00243
00000
00000
TYPE 2 TITRATION, SALT OF ACID WITH ACID
TYPE 2 TITRATION
RQDCL • HYOACTI21 I GHYDRO + CNA - COH
DO 243 I • 1r NCONT
RQDCL • RQDCL- IEIII • CKPRODIIII
CMLTNT = IRQDCL- CLIONI • IVOLTL I BASENI
GO TO 249
WWD1666
WWD1667
WWD1668
WW01669
WW01670
WWD161l
TYPE 1 TITRATION, ACID WITH BASE
TYPE 1 TITRATION
RQONA • COH - HYDACTI21 I GHYDRO
DO 247 I = lr NCONT
RQDNA • RQDNA + IEIII • CKPROOIIII
CMLTNT • IRQDNA- CNAI • IVOLTL I BASENI
CONTINUF
WWD1672
WW01613
WW01674
WWD1675
WWD1675A
WWD1675B
c
c
c
c
c
c
c
c
c
00245
00000
00247
c
c
249
ADD TITRANT AND TEST FOR LIMIT
00251 TNTML Ill • TNTML Ill + CML TNT
IF IICMLTNT •• 2.1- 1.0E-61 255, 255, 253
OPTIONAL OUTPUT
c
253 IF !SENSE SWITCH 21 254, 803
254 wRITE OUTPUT TAPE 9, 92, TNTMLillr PHil), CMUillt GHYDRO
GO TO 803
00255 WRITE OUTPUT TAPE 9, 92, TNTMLillt PHillt CMUillt GHYDRO
c
c
DO NEXT POINT, If ANY LEFT
WW01655
WW01656
WWD1657
WWD1658
· WWD1659
WW01676
WWD1677
WW01679
WWD1680
62
00257 PH(ll • PH(ll + PHSTEP
00259 IF IPH(1l- PHUPPRl 801, 801, 261
261 PUNCH 92, TNTMLillr PHill, CMUill, GHYDRO
GO TO 401
c
c
c
c
SUBMASTfR PROGRAM TO CALCULATE CONCENTRATIONS OF ACIDS
WW0168l
WW01761t
GO TO READ IN TITRATION CURVE
00301 GO TO 501
WWD1766
WWD1767
WWD1768
RETURN FROM READ IN.
00303 NROWS • NACIDS
00000 NCOLMS • NROWS + 1
WW01769
WWD1770
WW01771
c
c
DETERMINE MATRIX SIZE
c
c
SET ITERATION COUNTER AND INITIAL SUM
00000 MCOUNT ,. 0
00000 SUM • 1.
c
c
00305
00000
00000
00000
00307
c
c
CLEAR RESULTS AND ARRAY
00 307 I • 1, NROWS
RESULTIIl • O.
E I II •
o.
DO 307 J • 1, NCOLMS
ARII, Jl • O.
INITIALIZE LOOP
00309 K • 1
00000 GO TO 801
RETURN FROM GAMMA CALCULATOR AND BRANCH ON TITRATION TYPE
c
00311 GO TO 1315, 315, 313, 313), NTYPE9
00313 FN • -1.
00000 GO TO 317
00315 FN • 1.
00317 Y • FN • (HYDACTI211GHYDRO + CNA - COH- CLIONI
c
c
OBTAIN ELEMENTS OF MATRIX
VNUM1 • O.
DO 321 I • 1, NCON1
WWD1772
WWD1773
WWD1774
WWD1775
WWD1776
WWD1777
WWD1778
WWD1779
WW01780
WWD1781
WWD1782
WWD1783
WWD1784
WWD1785
WWD1786
WWD1787
WWD1788
WWD1789
VNUM1 • VNUM1 + l • FACTOR! I l
Elll • VNUMl • VOLC I DENOMl
GO TO I 328, 3251, NACIDS
ELEMENTS FOR SECOND ACID
VNUM2 "' 0.
DO 327 I • 1, NCON2
l = I
VNUM2 = VNUM2 + l • FACTORII + 4)
El2l = VNUM2 • VOLC I DENOM2
WWD1791
WWD1792
WWD1793
WWD1791t
WWD1795
WWD1796
WWD1797
WWD1798
WWD1799
WW01800
WW01801
WWD1802
SOLVE FOR ANSWERS OR ERRORS
IF ISUM- 1.0E-6l 345, 345, 329
WW01803
WWD1801t
BUILD MATRIX
DO 333 L • 1, NROWS
ARIL, NCOLMSl "' ARIL, NCOLMSI + Elll • Y
DO 33 3 M = 1, L
ARIL, Ml : AR(L, Ml + EILI • EIMl
K "' K + 1
IF IK - NOPTSl 801, 801, 701
WWD1805
WWD1806
WW01807
WWD1808
WWD1809
WWD1810
WWD18ll
RETURN FROM MATRIX SOLVE
00337 SUM • IIACIDl - RESULT( 111 I AClDll •• 2.
WWD1812
WWD1813
00319
00000
00000
00321
00000
00323
c
0032 5
00000
00000
00327
00000
c
c
00328
c
c
00329
00000
00331
00333
00000
00335
c
c
l
.. I
63
00000
00339
00341
00000
00343
c
c
ACID1 • RESULTI11
GO TO 1343, 341lr NACIDS
SUM= SUM+ IIACID2 - RESULT(211 I ACID21 •• 2.
ACID2 a RESULTI21
WRITE OUTPUT TAPE 9, 99, ACIDlt ACID2r SUM
MCOUNT s MCOUNT + 1
IF IMCOUNT - 101 305, 344, 344
344 SUM • -1.0E+10
GO TO 309
DETERMINE STANDARD DEVIATIONS. OBTAIN DIFFERENTIAL ELEMENT$
00345 FMIKI • Y
00000 ARIK, 11 "'El11
00000 AR(K, 21 • El21
00 00 0 FCIKI • Elll • CACID1 + El21 • CACID2
00000 K " K + 1
TEST FOR LAST POINT USEO
c
00347 IF 1K - NbPTSI 801, 801 1 349
c
c
SET NUMBE~ OF UNKNOWNS FOR ERROR TREATMENT
NCONT • NACIOS
NID • 0
ERQD • 0.
COD " o.
PUNCH 99, ACID1r ACI02, SUM
00000 GO TO 1380
00349
00000
00000
00000
c
c
c
00501
00503
00505
00507
00509
00000
00511
00000
00513
00000
00515
c
c
00517
c
c
WW01814
WWD1815
WW01il16
WW01817
WW01818
WWD1821
WW01822
WW01823
WW01i24
WW01825
WW01826
WW01827
WWD1828
WW01829
WW01830
WW01831
WWD1832
GO TO 1505, 505, 5071, NTYPE2
GO TO 1515, 515, 5131, NTYPE3
GO TO 1511, 511, 5091, NTYPE3
READ 93, TNTMLIII, PHIIJ, CMUIIlr ACTCFHIII
GO TO 517
READ 94, TNTMLIII, PHIII, ACTCFHIII
GO TO 517
READ 96, TNTMLIII, PHIII, CMUIII
GO TO 517
READ ~8, TNTMLIII, PHIII
WWD1833
WW01683
WWD1684
WWD1685
WWD1686
WWD1687
WWD1688
WWD1689
WWD1690
WW01691
WW01692
WWD1693
WWD1694
WWD1695
WWD1696
TEST FOR END AT PH OF ZERO
IF IPHIIII 907, 521, 519
WWD1697
WWDl698
READ IN TITRATION CURVE
I •
1
READ NEXT POINT
00519 I • I + 1
00000 GO TO 503
WWD1699
WW01700
WWD1701
COUNT NUMBER OF POINTS
00521 NDPTS • I - 1
00523 GO TO 1103, 401, 3031, NTYPEl
WWD1702 ·
WWD17D3
c
c
c
c
c
c
c
FORMATION OF MATRIX
OBTAIN ELEMENTS OF EQUATION
0060 1 DO 605 I • 1, NCON1
00 000 l • I
00000 L • NCON1 + 1 - I
00603 Elll • llll • CACID11- FNI I GAMMA(II I • ACTSCL(LI
IF OVERFLOW, SCALE UP
00000 IF ACCUMULATOR OVERFLOW 161, 605
c
WWD1705
WWfH706
WWD1707
WWD1708
WWD1709
WWD1710
WWD1711
WW0l712
WWD17U
64
00605 CONTINUE
c
c
BUILD MATRIX
00607 DO 615 L = 1, NROWS
00000 ARIL, NCOLMSI :a ARIL, NCOLMSI + Elll • Y
IF OVERFLOW, SCALE UP
c
00000 IF ACCUMULATOR OVERFLOW 161, 609
00609 00 615 M • 1, L
00000 ARIL, Ml :a ARIL, Ml + EILI • EIMI
00000 IF ACCUMULATOR OVERFLOW 613, 615
DETER~INE WHETHER TO SCALE UP OR DOWN
c
00613 IF IAHSFIEILII- 1.01 161, 161, 171
00615 CONTINUE
00617 GO TO 135
c
c
c
c
SOLVt: MATRIX
RESET TRIGGERS
00701 IF ACCUMULATOR OVERFLOW 703, 703
00703 J '" NROWS - 1
00705 !FIJI 912, 707, 711
IF ONLY ONE ROW, SOLVE DIRECTLY
c
00707 RESULTill • ARI1 0 21 I ARil, 11
00709 GO TO 1139, 912, 3371, NTYPEl
c
c
FILL OUT MATRIX
00711 DO 715 K • 2, NROWS
00000 M • K - 1
00713 DO 715 L = 1, M
00715 ARIL, Kl :a ARIK, Ll
OPTIONAL OUTPUT
c
IF ISENSE SWITCH 51 716, 717
716 WRITE OUTPUT TAPE 9, 51t IIARIK,LI, L • 1, 51t K • 1, NROWSI
c
c
717
00000
00719
00000
00723
00000
00000
0072'5
00727
c
728
00729
00000
00000
00000
00731
00733
00000
00000
00000
00735
00737
00739
c
c
c
SOLVE OY TRIANGULARIZATION
00 727 I = 1, J
M• I + 1
DO 727 K :a M, NROwS
FB = -ARII, II I ARIK, II
00 727 L = I, NCOLMS
ARIK, Ll " FB • ARIK, L1 + ARII, Ll
IF ACCUMULATOR OVERFLOW 725, 727
IF IFB - 1.01 161, 161, 171
CONTINUE
OPTIONAL OUTPUT
IF ISENSE SWITCH 51 728, 729
WRITE OUTPUT TAPE 9, 55, IIARIK,L), L • 1, 51, K • lt NROWSI
DO 737 I :a 1, NROWS
SUM ,. 0.
K • I - 1
M • NCOL MS - I
IF IKI 912, 737, 733
DO 735 L • 1, K
N = NCOLMS - L
SUM • SUM + RESULTINI • ARIM, Nl
IF ACCUMULATOR OVERFLOW 161, 735
CONTINUE
RESULTIMI :a I ARIM, NCOLMSI -SUM
I ARIM, Ml
GO TO 1139, 912, 3371, NTYPEl
ACTIVITY CORRECTION SECTION.
THIS SECTION CALCULATES ACTIVITY
COEFFICIENTS, VOLUME CORRECTIONS, IONIC CONCENTRATIONS, IONIC
WWD1714
WWD1715
WW01716
WWD1717
WW01718
WWD1719
WW01720
WW01721
WWD1722
WWD1723
WW01724
WW01725
WW01726
WWD1727
WWD1728
WW01729
WW01730
WW0173l
WW01732
WW01733
WW01734
WWD1735
WWD1736
WWD1737
WW01738
WWD1139
WW01740
WW017't1
WW01 742
WWD1741t
WW01 745
WW01746
WWOl 747
WWD1 748
WWOl 749
WW01750
WW01751
WW01752
WWD1753
WWD1754
WWD1755
WWD1756
WW01757
WWD1758
WW01759
WWD1760
WWD1761
WWD1762
WWD1763
WW01251
65
c
c
c
c
c
00801
c
c
STRENGTH, ETC.
THE SUBSCRIPT IKI IS SET OUTSIDE THIS SECTION FOR THE POINT UNDER
CONSIDERATION.
OOTAIN HYDROGEN ION ACTIVITY OR CONCENTRATION AT THIS POINT
HYDACTI21 z 10. •• 1-PHIKII
RESET OVERFLOft TRIGGER
00000 IF ACCUMULATOR OVERFLOW 803, 803
c
c
c
00803
c
c
00805
00000
00000
00000
00807
00000
c
c
00809
00000
00000
00000
c
c
c
00811
00813
00000
00815
00000
00817
c
c
00819
00000
00821
c
c
00823
00825
00000
00827
c
c
c
00829
00831
00000
c
c
c
833
WWDl259
TRANSFER DEPENDING UPON WHETHER OR NOT ACTIVITY CORRECTIONS ARE TO
OE MADE.
GO TO 1809, 805, 809), NTYPE2
WWD1260
NO CORRECTIONS. ALL ACTIVITIES EQUAL ONE
GHYDRO : 1.
ACTCFI-IKI z 1.
GOXIDE : 1.
DO 8J7 I : 1, 6
GAMMA I II z 1.
GO TO 835
WWD1261
WWD1262
WWD1263
WWD1264
WWD1265
WWD1266
WWIH267
WWD1268
WW01269
WWD1270
WWD127l
WWD1272
WWD1273
WW01274
ACTIVITY COEFFICIENTS FOR HYDROGEN AND HYDROXIDE IONS
WWD1275
IF HYDROGEN ACTIVITY READ IN, DO NOT CALCULATE
WW01276
GO TO 1815, 903, 8131, NTYPE2
WW01277
GHYDRO : hCTCFHIKI
WWD1278
GO TO 817
WWD1279
GHYORO = 10. •• IASRMU I 11. + 8SRMU • HIONSZI - BETAMUI
WWD1280
ACTCF~IKI z GHYORO
WW0128l
GOXIDE : 10. •• IASRMU I 11. + BSRMU • 3.51 - BETAMUI
WWD1282
WWD1283
ACTIVITY COEFFICIENTS FOR IONS OF FIRST ACID
WWD1284
00 821 I : 1, NCON1
WW01285
l = I • I
GAMMAIII : 10. •• IASRMU • l I 11. + 8SRHU • SlZION(Ill - BETAHUI WWD1286
WWD1287
WWD1288
ACTIVITY COEFFICIENTS FOR IONS OF SECOND ACID
WW01289
GO TO 1829, 8251, NACIOS
WWD1290
DO 827 I "' 1, NCON2
WW01291
Z " I • 1
GAMMAII+41 • lO.••IASRMU•Z I 11. + BSRMU • SIZ10NII+411- BETAMUI WWD1292
ACTIVITY CORRECTIONS MADE. 0-H FACTORS COMPUTED
SRMU = SQRTFICMUIKII
ASRMU = -DBHFA • SRMU
8SRMU = DOHFB • SRMU
8ETAML = OETADH • CMUIKI
CORRECT VALUE OF HYDROGEN ACTIVITY FOR CALCULATING USE DEPENDING
UPON TYPE OF PH MEASUREMENT MADE.
GO TO (835, 833, 8311, NTYPE5
HYDACTI21 • HYDACTI21 I GHYDRO
GO TO 835
HYDACTI21 • HYDACTI21 • GHYDRO
VOLUME CORRECTION SECTION
TEST FOR CONSTANT VOLUME
00835 GO TO 1839, 8371, NTYPE4
00837 VOLTL • SOLNV
00000 GO TO 841
c
c
WWD1254
WWD1255
WW01256
WWD1257
CALCULATE TOTAL VOLUME
WWD1293
WWD1294
WWD1295
WWD1294
WWD1297
WWD1298
WWD1299
WWD1300
WWOUOl
WWD1302
WWDU03
66
00839 VOLTL • SOLNV + TNTMLIKI
c
c
00841
c
c
HYDROXIDE ION CONCENTRATION
COH = WCON I IHYDACTI21 • GOXIDEI
WW01304
WWD1305
WWD1306
VOLUME CORRECTION FACTOR
00000 VOLC • SOLNV I VOLTL
WWDl307
WW01308
CONCENTRATIONS OF ACIDS, CORRECTED
00000 CAC!Dl = A~ID1 • VOLC
00000 CACID2 • ACID2 • VOLC
WW0l309
WWDlllO
WWD13ll
c
c
c
c
00843
c
c
BRANCH ACCORDING TO TYPE OF TITRATION
GO TO (867, 865 0 847, 8451, NTYPE9
SALT OF BASE TITRATED WITH BASE
00845 CNA = TNTMLIKI • BASEN I VOLTL
OOOJO CLION = CACID2 + CACID1
oo rJ£) 0 GO TO 849
(
UASE TITRATED WITH ACID
00 - h 7 CNA = O.
000 00 CLION a TNTMLIKI • BASEN I VOLTL
(
POWERS OF HYDROXIDE ION ACTIVITY
00849 OHACTI21 • WCON I HYOACTI21
00000 OHACTI3l • OHACTI21 • OHACTI21
00000 0HACTI4l = OHACTI31 • OHACTI21
ERROR STOP IF PH OUT OF RANGE
c
00000 IF ACCUMULATOR OVERFLOW 910, 850
(
c
c
00850
00851
00000
00000
00000
c
00000
00852
0085 3
c
00000
c
00000
00855
00857
c
00859
00000
00000
00861
00000
00000
00863
00000
c
c
ION CONCENTRATION CALCULATION SECTION
IF QUOTIENT OVERFLOW 851, 851
OENOM1 = 1.
DO 853 I a 1, NCON1
FACTORIII = CKPRODIII I IGAMMAIII • OHACTIII-I
FACTORIIl • FACTORitl I OHACTI21
IF RESULT OUT OF RANGE, SET VALUE TO ZERO
IF QUOTIENT OVERFLOW 852, 853
FACTOR(()= O.
DENOM1: OENOM1 + FACTORIII
CONCENTRATION OF UNIONIZED BASE NUMBER ONE
HAlON( 11 " CACID1 I DENOM1
CONCE~TRATION OF IONIC SPECIES
DO 855 I • 1o NCON1
HAIONII + 11 • HAIONI11 • FACTORIII
GO TO 1885, 8591, NACIDS
REPEAT ABOVE FOR SECOND BASE
DENOM2 = 1.
DO 861 I .. 1, NCON2
FACTORII+41 • CKPRODII + 41 I IGAMMAII + 41 • OHACTII + 111
DENOM2" OENOM2 + FACTORII + 41
HAIONI61 • CACID2 I OENOM2
DO 863 I = 1, NCON2
HAIONII + 61 • HAIONI61 • FACTOR(l + 41
GO TO 885
TITRATION OF ACID SALT WITH ACID
00865 CNA " CACID1 + CACID2
00000 CLION a TNTMLIKI • BASEN I VOLTL
00000 GO TO 869
ACID TITRATED WITH BASE
c
00867 CNA • TNTMLIKI • BASEN I VOLTL
WW01312
WWD1313
WWDl311t
WWOl315
WWDl316
WWD1317
WWDl3l8
WWD1319
WWD1320
WWD1321
WWD1322
WWD1323
WWD1324
WWD1325
WWD1326
WW01327
WWD1328
WWD1329
WW01330
WWD1331
WWD1332
WWD1333
WWD1334
WWD1335
WWD1336
WWD1331
WW01338
WW01339
WWD1340
WW0134l
WW01342
WW01343
WWDl344
WWD1345
WW01346
WW01347
WWD134B
WW01349
WWD1350
WW01351
WW01352
WWDl353
WWD1354
WW01355
WW01356
WWD1357
WW01358
67
00000 CLION a O.
POWERS OF HYDROGEN ION ACTIVITY
00669 HYOACTI31 • HYOACT!21 • HYOACTI21
00000 HYOACTI41 a HYDACTI31 • HYDACTI21
ERROR STOP IF PH OUT OF RANGE
c
00000 IF ACCUMULATOR OVERFLOW 910, 870
ION CCNCENTRATION CALCULATION SECTION
c
00670 IF QUOTIENT OVERFLOW 871, 871
00671 OENOM1 a 1.
00000 DO 873 1 ~ 1t NCON1
00000 FACTORIII • CKPRODIII I IGAMMAIII • HYOACTIIII
00000 FACTORIII • FACTORIII I HYDACTI21
IF RESULT OUT OF RANGE, SET VALUE TO ZERO
c
00000 IF QUOTIENT OVERFLOW 872, 873
00872 FACTORIII • O.
00873 OENOM1 • DENOM1 + FACTORIII
CONCENTRATION OF UNIONIZED ACID NUMBER ONE
c
00000 HAIONI11 • CACID1 I OENOM1
c
WWD1359
WW01360
WW01361
WWDl362
WWD1363
WW01364
WWD1365
WW01366
WWOU67
WWDU68
WW01369
CONCENTRATION OF IONIC SPECIES
00000 DO 675 I = 1, NCONl
00675 HAIONil + 11 z HAIONill • FACTORIII
00671 GO TO 1885, 8791, NACIDS
REPEAT ABOVE FOR SECOND ACID
c
00879 OENOM2 • l.
00000 00 681 I a 1, NCON2
00000 FACTORII + 41 • CKPROOII + 41 I IGAMMAII + 4) • HYOACTII + lll
00861 OENOM2: DENOM2 + FACTORII + 4)
00000 HAIONI6l z CACI02 I DENOM2
00000 DO 663 I = 1, NCON2
00883 HAIONII + 61 • HAIONI61 • FACTORII + It)
WWD1370
WW0137l
WWD1372
WW01373
WWD1371t
WWD1375
WWD1376
WWD1377
WWD1378
WWD1379
WWD1380
WW0l381
WWD1382
WWD1383
WWD1384
WWD1385
WWD1386
WWD1387
TRANSFER OUT IF NO ACTIVITY CORRECTIONS
00865 GO TO 1887, 891, 887), NTYPE2
TRANSFER OUT IF IONIC STRENGTH CONSTANT OR GIVEN AT EACH POINT
c
00887 GO TO 1889, 891, 8911, NTYP£3
WWD1388
WW01389
WWD1390
WWD1391
C
COMPUTE
00889 CMUIKI a
006691 + 4. •
006692 + 4. •
WWD1392
WWD1393
WW01391t
WWD1395
c
c
c
c
c
c
c
c
c
c
c
c
TRANSFER OUT TO PROPER SECTION Of MAIN PROGRAM. OUTPUT OPTIONAL.
891 IF !SENSE SWITCH ll 893, 895
893 WRITE OUTPUT TAPE 9, 63, Kt HYOACTI2lt GHYORO, CMUIKit VOLCt
l IHAIONIKS), KS • 1, 81
895 GO TO 1117, 211, 311), NTYPE1
06001
00000
00000
00000
00000
c
c
c
NEW IONIC STRENGTH
.5 • I HYDACTI211GHYORO + CNA + COH + CLION + HA10NI21
HAIONI31 + 9. • HAIONI41 + 16. • HA10NI51 + HAIONI71
HAIONI81 I + ICKN03 • VOLCI
WWD1396
WW01831o
ERROR CALCULATION FOR DISSOCIATION CONSTANTS
WWD1835
MUST SPECIFY TWO ACIDS TO ENTER NON-LINEAR TREATMENT AND CALCULATE
STAND~RD DEVIATIONS
WWD1836
WWD1837
NACIDS = 2
WWD1838
NIO " 0
WWDl839
ERQD • 0.
WWD181t0
COD • O.
WW0181tl
GO TO 107
ERROR STOPS
00901 WRITE OUTPUT TAPE 9, 9010
GO TO lt83
WWD1842
WW0181t3
WW01844
68
00902 WRITE OUTPUT TAPE 9, 9020
GO TO 487
00903 WRITE OUTPUT TAPE 9, 9030
PUNCH 9030
PRINT 9080
STOP 3333'3
00904 WRITE OUTPUT TAPE 9, 9040
WRITE OUTPUT TAPE 9, 97, ICKINDII2l, 12 • 1 , 6 l , SUM
00000 GO TO 401
00905 WRITE OUTPUT TAPE 9, 9050
00000 GO TO 401
00906 WRITE OUTPUT TAPE 9, 9060
00000 GO TO 401
00907 WRITE OUTPUT TAPE 9, 9070
GO TO 503
00909 WRITE OUTPUT TAPE 9, 9090
NTYPE1 • 2
GO TO 501
910 GO TO 1920, 920, 9301, NTYPE6
920 WRITE OUTPUT TAPE 9, 9010
NTYPE6 • 3
930 GO TO 1135, 401, 940), NTYPE1
940 K • K + 1
IF ISUM- l.OE-61 347, 347, 335
00911 WRITE OUTPUT TAPE 9, 9110
PUNCH 9110
PRINT 9080
STOP 33333
00912 WRITE OUTPUT TAPE 9, 9120
00000 GO TO 401
913 WRITE OUTPUT TAPE 9, 9020
GO TO 4919
914 WRITE OUTPUT TAPE 9, 9020
GO TO 1760
c
WW01846
WW018411
WWD1850
WW01851
WWD1852
WWDlUJ
WW011154
WWOU55
WWD1856
WWDl860
WW01864
WW01866
WWD1867
00008 FORMAT I 92H1DUNNING DISSOCIATION CONSTANT CALCULATOR, TITRATION CWWD1027
WWD1028
000081URVE CRAWER, ANO CONCENTRATION FINDER
WWD1029
0001D FORMAT 172HO
WWD1030
000101
WWD1031
00012 FORMAT I9I21
WWD1032
00014 FORMAT 144HOTHIS PROGRAM CALCULATES ACID CONCENTRATIONS)
WWD1033
00016 FORMAT 142HOTHIS PROGRAM CALCULATES A TITRATION CURVEI
WWD1034
00018 FORMAT 147HOTHIS PROGRAM CALCULATES DISSOCIATION CONSTANTS)
WW01035
00020 FORMAT IF8.5, F10.5)
00022 FORMAT 148HOORIGINAL ESTIMATES OF ACID CONCENTRATIONS ARE
F9.6, WWD1036
000221 8H
AND F9.6, 8H
MOLAR)
WW01038
00024 FORMAT 135HOORIGINAL ACID CONCENTRATIONS ARE
F9.6,
000241 8H
AND F9.6, 8H
MOLARI
WW01040
00026 FORMAT IF1D.5, 3F1D.7, E9.2, F6.11
WWD1041
00028 FORMAT 125H INITIAL SOLUTION VOLUME F8.3, 19HML, TITRANT CONC.
000281F7.4, 23HN, EST. IONIC STRENGTH F6.3, 18HMOLAR, ADDED SALT F6.~, WW01042
WW01043
0002826HMOLAR,/ 32H DISSOCIATION CONSTANT OF WATER E9.2,
WW01041t
00028328H, DESIRED CLOSENESS OF FIT F5.1, 16H PERCENT AVERAGE)
WW01045
00030 FORMAT I6F.10.31
00032 FORMAT 147HOINITIAL VALUES OF DISSOCIATION CONSTANTS ARE 6Ell.~l WW01046
WW01047
00034 FORMAT 150HOINITIAL ESTIMATES OF DISSOCIATION CONSTANTS ARE
WW01048
000341 6Ell.3l
WW01049
00036 FORMAT 133HOND ACTIVITY CORRECTIONS ARE MADE l
WWDlO!SO
00038 FORMAT IF4.1, 5F5.1l
40 FORMAT 125HOIDN SIZE PARAMETERS ARE 6F5.ll
WW01052
00042 FORMAT I32HOIONIC STRENGTH IS CONSTANT AT
F6.3, 6H MOLAR)
WW01053
00044 FORMAT I25HOVOLUME IS CONSTANT AT F8.3, 3H MLI
69
WWD1054
FORMAT !12HOPWH IS USEDI
WWD1055
FORMAT 112HOPCH IS USEOI
WWD1056
FORMAT ll2HOPAH IS USEDI
I
FORMAT !50HOTHE FILLED OUT MATRIX, AFTER INSTRUCTION 715, IS
1 I 5E20.8 II
WW01057
00052 FORMAT IF4.11
WWD1058
00054 FORMAT !23H HYDROGEN ION SIZE IS F4.ll
55 FORMAT (54HOTHE TRIANGULARIZED MATRIX, AFTER INSTRUCTION 727, IS I
1 I 5E20.8 II
WWD1059
00056 FORMAT IF8.5, Fl0.51
WWD1060
00058 FORMAT !26H 0-H COEFFICIENTS ARE A• F8.5, 6H, 8• F8.51
WWD1061
00060 FORMAl !FB.51
VOLC HAIONill HAlO
61 FORMAT !ll9HO K
HYOACT
GHYDRO ION.ST.
1N!21 HAION(31 HAION(4) HAION!51 HAION!61 HAIONI71 HAIONIBI I
WWD1062
00062 FORMAT !14H D-H BETA IS
F8.51
63 FORMAT !lH 13, E11.2, F8.4, F9.5, F7.3, 8F10.61
WWD1063
00064 FORMAT (32HOSALT OF BASE TITRATED WITH BASEl
WWD1064
00066 FORMAT (24HOBASE TITRATED WITH AClDI
WWD1065
00068 FORMAT !32HOSALT OF ACID TITRATED WITH ACIDI
70 FORMAT !24HOACID TITRATED WITH BASEl
WWD1067
4Ello3t
00072 FORMAT (29HODISSOCIATION CONSTANTS ARE
WW01068
000721 ZOH
SUM OF SQUARES • F8.51
WWD1069
00074 FORMAT !1HO 8El4.71
76 FORMAT (4H001zE16.8, 36H TOTAL NO. OF NON-LINEAR ITERATIONS• 14,
1 18H SCALING FACTORz EB.ll
WWD1071
STANDARD DEVIATION
00078 FORMAT !42HO J FINAL CKINDIJI
WWD1072
00080 FORMAT !lH 13, E13.4, E22.81
WWD1073
00082 FORMAT 126HOTHE CORRELATION MATRIX lSI
WW01074
00084 FORMAT llH 10E11.31
00086 FORMAT !83HOTITRANT ML PH IONIC STRENGTH H ACT COEF. FMIMEAS.IWWD1075
000861
FC!CALC.I DIFFERENCE
I
WW01077
00088 FORMAT!1H F8.3, F7.2, F11.6, F14.4, El4.3, 2El2.31
WW01078
00090 FORMAT !F6.2, F10.2, F9.21
H ACT COEF.IWWD1079
91 FORMAT !54HCTlTRANT ML
PH
IONIC STRENGTH
92 FORMAT !1H F8.4, F12.3, Fl2.6, Fl9.41
WWD1081
00093 FORMAT !F7.3, F10.3, Fl2.6, F8.41
WWD1082
00094 FORMAT IF7.3, Fl0.3, F10.41
WWD1083
00096 FORMAT !F7.3, F10.3, Fl2.6 I
WWD1084
6El1.3,
00097 FORMAT !29HOOlSSOCIATION CONSTANTS ARE
WWD1085
000971 20H
SUM OF SQUARES •
FB.5 I
WWD1086
00098 FORMAT !F7.3, F10.3)
99 FORMAT !9HOACID1 = F8.5, 13H M., ACID2 • F8.5,
1 22H M., SUM OF SQUARES • E11.41
09010 FORMAT 173HOACID NUMBER TWO HAS NEGATIVE CONCENTRATION. ASSUME ONL
1Y ONE ACID PRESENT)
WWD1089
09020 FORMAT !30HOA CKIND OR CKPROD IS NEGATIVE I
WWD1090
09030 FORMAT (22HOWRONG TRANSFER IN 800 I
WW01091
09040 FORMAT !34HOFIRST ACID HAS NO DISC. CONSTANTS!
WWD1092
09050 FORMAT !38HOATTEMPTED TO SCALE IN BOTH DIRECTIONS
WWD1093
09060 FORMAT !35HOOVERFLOW IN GETTING SCALING FACTOR )
WWD1094
09070 FORMAT 126HOA NEGATIVE PH ENCOUNTERED I
9080 FORMAT 145H IMPOSSIBLE ERROR STOP. DISCONTINUE PROGRAM.)
WWD1096
09090 FORMAT !20HOCKPROD OUT OF RANGE I
WWD1097
09100 FORMAT 128HOPH OUT OF RANGE, 849 OR 869
WWD1098
09110 FORMAT 123HOWRONG TRANSFER IN 1600 I
WWD1099
09120 FORMAT 122HOWRONG TRANSFER IN 700 I
00046
00048
00050
51