C a r b o n a t e E q u i l i b r i u m . The method described de
pends on the carbonate equilibria, which according to Byck (1) are as follows:
[H + ] [ O H - ] = K „ = 0 .5 4 X lO “ 14 (20° C.) (6)
^ - ¡ H g g P = K i = 3 .1 8 X 1 0 - ’ (200 C.) (7)
= K* = 3-04 X 1 0 - ” (2 0 °
[nV jÜ 3 J C.) (8)
[H + ] = 2 [ C O , ~] + [ H C O r ] + [O H - 1 (9)
[HoCOj] = ncP (10)
nc — Bunsen coefficient = 0.0393
= 2fcife; ncP + h ncP + Iiw
[H+]' [H+J 1 [H+] (1 1) . For pressures of carbon dioxide over 1 X lO-4 atmosphere several of these constants may be neglected, and the equation becomes
[H+] = V k, ncP (12)
These equations apply only to pure water. When the salt of an indicator is present, more and more base is made avail
able for bicarbonate formation as the tension of carbon di
oxide increases. The situation is comparable to the two- acid problem of Michaelis (6) which he states is difficult to solve. The pH, observed with the glass electrode, of con
ductivity water in Pyrex glass in equilibrium with various carbon dioxide mixtures agrees well with the pH calculated by Equation 13.
Since some base is always present—e. g., from the indicator, impure distilled water, or alkali of soft glass—it is convenient to use as simple a system as possible to avoid laborious cal
culations. This is done by adding enough base so that the total HCO3“ concentration is high enough to make negligible both the additional IICO3“ from carbonic acid dissociation and the base freed through methyl red association. The buffer equation, 13, satisfies our needs, provided free bicarbon
ate is at least ten times as large as the two factors mentioned above.
pH = -log h - log[H2COj] + log IICOj (13) P h o t o m e t r y . In the photometric measurements, since the standard is completely transparent to the radiation passing through the filters, its transmission represents the incident light or 70. The light transmitted by the test solution cor
responds to I.
When Io/I is plotted against the carbon dioxide content of the gas in equilibrium with the test solution, a straight line results (Figure 2). This line has the equation
C02 (14)
The slope of the line and the range of carbon dioxide de
termination depend upon the concentration of methyl red and alkali, and upon the depth of the cuvette. The authors have used the method to determine carbon dioxide tensions be
tween 0.0 and 8.0 per cent. In practice Io/I is plotted against carbon dioxide, and a calibration is accomplished by drawing a straight line through C 02 = 0, 1 a/I = 1.0, and a point de
termined for a known carbon dioxide tension. Unknown carbon dioxide percentages can then be read directly from the graph from observed values of Io/I, or Equation 14 may be applied with the slope of the above line as k.
The change in concentration of associated methyl red (red form) with change in pH is measured photometrically as de
scribed above. The fundamental Beer-Lambert equation is , h .
log kd (1 — a)
where k = extinction coefficient 1 — a = red form of methyl redd = depth of cuvette
This equation applies rigidly only when monochromatic light is used. In this method a rather narrow range of wave length is obtained by passing the light through a Corning No.
397 and a W ratten No. 74 filter. Figure 3 shows the region of maximum absorption of both the yellowr and the red form of methyl red, and also the absorption of the two filters. It is clear from the figure that the requirement of the Beer-Lambert law is fairly well met. This equation was checked and found to hold by a dilution technique and also by calculating 1—a from the pH measured by the glass electrode.
Fig u r e 3. Ab s o r p t io n Sp e c t r a o f Me t h y l Re d a n d Fil t e r s
E. Eastm an Kodak Co. filter N o. 74«
C. Corning filter N o. 397 R. Red form of indicator Y. Yellow form of indicator
The molecular extinction coefficient (epsilon), in moles per liter and centimeters, was calculated from pH observations using Equation 16, the molar concentration of 1 — a, the depth of the cuvette, and Equation 15. It was found to be 0.51 X 105, which agrees very well with the value in the literature (8) for X = 530 m/x.
Since k, the extinction coefficient, and d, the depth, re
main constant and Zo and I are observed, I — a may be cal
culated. Its concentration varies with pH as follows:
pH = pK a methyl red ~ log 1
By combining Equations 13 and 16 it is clear that:
C02
(16)
(17)
(15)
This relation was tested experimentally in the following manner: The pH and color of an aqueous solution of methyl red were modified by bubbling known carbon dioxide mixtures Until equilibria had been reached. The pH was then deter
mined with the glass electrode and Io/I was observed. From the pH the value of log 1 g a was calculated with Equation 16, and in Figure 2 it is plotted against log C 02. The re
sulting straight line proves the validity of Equation 17 which gives the theoretical relationship between carbon dioxide ten
sion and the degree of dissociation of methyl red.
374 INDUSTRIAL AND ENGINEERING CHEMISTRY VOL. 11, NO. 7
Ex p l a n a t io n o f t h e Un e x p e c t e d Lin e a r it y b e t w e e n
C 0 2 a n d Jo/7. By combining the Beer-Lambert equation, 15, with the experimentally observed equation, 14, we get
CO: « 10'-“ (18)
This relation which is found experimentally is not to be ex
pected a priori.
Equations 17 and 18 would seem incompatible, since one is a hyperbolic and the other a logarithmic relation. However, between certain limits of alpha the two curves change at the same rate and both relations can hold. This is shown in Fig
ure 4, where these terms are plotted against alpha and against each other. It is seen that in the interval 1 > a > 0.45 the hyperbola, when plotted against the logarithm, yields a straight line. Therefore, it is necessary to work with carbon dioxide tensions low enough so that the pH will not fall below 5.1, the pKa of methyl red, in order that 1—a never exceeds 50 per cent of the total concentration of methyl red. For higher tensions of carbon dioxide more free base may be added.
T a b l e I. C a l c u l a t i o n o p P e r c e n t a g e E r r o r o r R a t i o A V hen 70 Is 1000 a n d 7 Is D e t e r m i n e d w i t h a n A c c u r a c y
o p O n e S c a l e D iv is io n
error in ratio is given in Equation 20 where k is the same as in Equation 14.
Jo Jo
I I - 1 ,
Jo Jo h _ i '
R atio I I I - l i 1
1.1111 900 0.00123 1.215
1.25 800 0.00156 0 .6 2
1.42857 700 0.00204 0 .4 76
1.66667 600 0.00277 0.4 1 5
2 .0 500 0.00399 0 .3 99
2 .5 400 0.00623 0 .4 16
3.33333 300 0.01107 0 .4 73
5 .0 200 0.02488 0 .6 22
10.0 100 0.09901 1.10
1 0 0 .0 10 9.09091 9 .1 0
x 100
Co m pa r iso n o f t h e Se n s it iv it y a n d Er r o r o f t h e pH
a n d Ph o t o m e t r ic Me t h o d s. Since a pH indicator is used in this method, it seems logical to consider whether or not the direct measurement of pH of dilute bicarbonate with which
•carbon dioxide is in equilibrium would be as suitable as the photometric method for determining carbon dioxide. The determination of carbon dioxide by pH has been used by Ivauko (5), and by Wilson, Orcutt, and Peterson (5). Such a
•determination could most conveniently be made with the glass electrode with a sensitivity of about 0.02 pH units. Equa
tion 13 would hold and so:
pH a log COi
dpH = 1/C02 X dC02 X 0.4343 wrn - ¿pH X CO;2 0.4343
^ -i.- -i dC02 <2pH X C02 Sensitivity = ^ 0 7 = 0.4343^< COi Sensitivity = 0.02/0.4343 = 2.3%
The error of the photometric method may be determined by calculating the effect on the ratio 70/7 resulting from the inaccuracy of the measurement of these two factors. The error in the ratio resulting from an error in reading the value can be calculated by Equation 19.
h _ 7 7 — error7, 7„
7 (19)
The authors have made calculations of error in the case
•where 7o equals 1000 and 7 is measured to within ± 1, as in these experiments. These calculations are given in Table I and show that the error resulting from an error in measure
ment of 7 is at a minimum when I is one half of 7o. The ab
solute error in measuring carbon dioxide resulting from this
k A -j = error in C02 (20) Dividing Equation 20 by Equation 14 and multiplying by 100 we obtain the error in terms of percentage of carbon di
oxide present. This error holds for all concentrations of carbon dioxide and dye for any depth of cuvette. It is also the percentage error of the ratio, and is therefore generally applicable to all cases of photometric measurement, provided
70 equals 1000 times the smallest significant scale division.
When the accuracy of measuring 7 and 70 is changed, the error in percentage of ratio changes.
■ F i g u r e 4
1 - L i n e a r i t y b e t w e e n C 0 2 a n d Io/ I plotted against a
O . 101_a plotted against a
--- - plotted against 10l_ct. This curve show s th at in the interval 1 > a > 0.45 a straight line results from plotting the hyperbola against the logarithm . This accounts for the conven
ient linear relationship betw een COj and Io/I.
Table I show's that the error is fairly constant for a wide range of ratios and has a minimal value of ±0.4 per cent when 70 equals 1000 and 7 equals 500.
Compared with the pH method, the photometric method is about six times as accurate with the authors’ setup.
S u m m ary
A method is described which permits the continuous photo
metric determination of carbon dioxide.
A method of making gas mixtures of known composition is described.
A mathematical relationship is derived that simplifies the determination of the concentration of acids by direct meas
urement of In/I.
A method of calculating the reliability of photometric measurements is developed and applied to the case of carbon dioxide.
Literature Cited
(1) Byck, H. T., Science, 75, 224 (1932).
(2) Fegler, J., and Modzelewski, T., Compt. rend. soc. biol., 116, 244, 24S (1934).
(3) Fenn, W. O., Am. J . Physiol., 84, 110 (1928).
(4) Higgins, H. L-, and Marriott, W . M„ J. Am . Chem. Soc., 39, 68 (1917).
JULY 15, 1939 375
(5) Kauko, Y., Angcw. Chem., 47, 164 (1934); 48, 539 (1935); Acta (9) Wilson, F. W ., Orcutt, F. S., and Peterson, W. H „ In d. En g.
Chem. Fennica, 5B, 54 (1932). Ch e m., Anal. Ed., 4, 357 (1932).
(6) Michaelis, L., “Die W asserstoffionen-Konzcntration”, 2nd ed., p.
40, Berlin, Julius Springer, 1922.
(7) Parker, G. H., J. Gen. Physiol., 7, 641 (1925). C o n t r i b u t i o n from the Physiology Laboratory, Stanford University.
(8) Thiel, A., Dasaler, H ., and Wulfken, F., Fortschr. Chem. Physik Work supported in part by a grant from the Rockefeller Foundation and by
physik. Chem., 18, 1 (1924). the National Youth Administration.