UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN No. 30 FEBRUARY, 1909
ON THE RATE OF FORMATION OF CARBON MONOXIDE
IN GAS PRODUCERS
BY J. K. CLEMENT, PHYSICIST, U. S. G. S., TECHNOLOGIC BRANCH, ASSISTED BY
L. H. ADAMS, JUNIOR CHEMIST, U. S. G. S., TECHNOLOGIC BRANCH;
APPENDIX BY C. N. HASKINS, ASSISTANT PROFESSOR OF
MATHEMATICS, UNIVERSITY OF ILLINOIS.
I. INTRODUCTORY STATEMENT.
The rapid advance in the use of producer gas in recent years
has given rise to a demand for a more accurate knowledge of the
processes taking place in the fuel bed of the producer and the
effect on these processes of certain variations in the conditions of
operation. The primary function of the gas producer is to trans
form solid fuel into a more readily combustible gaseous fuel
This transformation, which is relatively slow, consists of the fol
lowing processes:
1. The distillation of the volatile hydrocarbons from the
freshly fired fuel at relatively low temperatures.
2. The combustion of fuel by combination with the oxygen
of the air.
3. The formation of producer gas proper in accordance with
the equations:
I. CO, + C = 2 CO,
II. H20 + C = CO + H2.
The first of these reactions, the formation of carbon monox
ide, is the one with which the present investigation deals. The
problem proposed is in broad terms to determine the factors that
govern the production of CO in the gas producer; and the effect
of the temperature and of the time of contact of the gas and
carbon on the percentage of CO in the producer gas. The question
ILLINOIS ENGINEERING EXPERIMENT STATION
of the effect of temperature was brought forth by certain experi
ments made by one of the writers at the Fuel Testing Plant of
the U. S. Geological Survey at the Jamestown Exposition, in
which it was found that the temperature in the fuel bed of the
gas producer varies greatly from one portion of the bed to
another. In order, therefore, to ascertain the conditions of tem
perature most favorable to the efficient operation of the producer,
it becomes necessary to determine the temperature requisite for
the formation of carbon monoxide and hydrogen in accordance
with the reactions quoted in the preceding paragraph.
A study of the conditions for the reduction of CO2 by carbon
seems desirable from another consideration. A small amount of
CO is invariably contained in the flue gases of boiler furnaces.
It was hoped, therefore, that the investigation might furnish an
explanation of the formation of CO in boiler furnaces and per
haps suggest a means of preventing such formation.
The investigations herein described were made in, and with
the facilities of, the Physical Laboratory of the University of
Illinois.
II. FUNDAMENTAL EQUATIONS.
According to the law of chemical mass action, a chemical
reaction, as for example the reaction expressed by the equation
C + CO2 = 2 CO,
proceeds in one direction until equilibrium is established and
then stops; and when the system is in equilibrium there is for a
given temperature a certain constant relation between the
amounts of the components entering into the reaction. Thus, in
the system under consideration, let
[CO] = the concentration of CO in gram molecules' per liter,
[CO,] = the concentration of CO2 in gram molecules per liter;
then for equilibrium, the relation
1A gram molecule of a substance Is a weight of the substance In grams
numerically equal to the molecular weight. Thus a gram molecule of CO is
28 grams, one of CO, is 44 grams, etc.
THE FORMATION OF CO IN GAS PRODUCERS
[CO = constant = K (1)
[C021
must be satisfied.
The relation (1) may be thrown into another form as follows:
Let 100 x = per cent of CO in the gas by volume;
100 (1  x) = per cent of CO2 in the gas by volume;
p = pressure in atmospheres;
T = absolute temperature;
R = absolute gas constant = 0.0821 for the sys
tem of units here employed;
n = number of gram molecules of the gas unuder
consideration;
v = volume of gas in liters.
The characteristic equation of gases is
pv = nRT
from which l e p
Now xn and (1  x)n. are respectively the numbers of
gram molecules of CO and CO, in the gas; hence the concentra
tions of CO and CO, are respectively
[CO]  _ xp_
v RT
001  (1  x)n (1  x)p
[C02] 
v HT
Placing these values in (1), the resulting equation is
1 ) Y (2)
1 x R 1' 
If the pressure and temp rature are kept constant, the factor
i s a.constant, and (2) may be written
Xe KRT
I _  In "
ILLINOIS ENGINEERING EXPERIMENT STATION
where K' is a new constant.
When CO2 gas is maintained in contact with carbon at con
stant temperature and pressure the two will react rapidly at first
and then more slowly until the amount of CO formed is 100 x per
cent of the total, where x is given by equation (3).
The relation expressed by (1) may be considered as a special
case of a more general law. According to' the theory of the
kinetics of reactions now generally accepted in reversible reac
tions, two reactions take place simultaneously, one from left to
right and one from right to left. In the reection
CO2 + C = 2 CO
the velocity of the reaction from left to right, that is, the rate of
formation of CO, is at any instant proportional to the number of
CO2 molecules in the unit volume; thus denoting the v.locity by v,
v = k,[CO2].
Similarly the velocity of the reaction from right to left, that is,
the rate of formation of CO2, is proportional to the square of the
number of CO molecules in a unit volume. Hence
v' = k, [CO]2
The increase in the number of CO molecules per unit volume in
the time dt is the difference of the two velocities r and r'; that is,
d(I [] = v  [(;02] = k1 1(dO]M (4)
As [C(()] the number of gram molecules cf CO in the unit vo lume
increases and [(CO,] the number (f gramii molecules of CO., de
creases, the velocity dt will beceme smaller and smaller
until finally [CO] and [CO,] become constant, that is, the system
attains equilibrium. In this case we have, therefore,
d [  =
dt
ck,1 [(CO] = k, [(1CO]2.
whence
THE FORMATION OF CO IN GAS PRODUCERS
That is, the number of CO2 molecules formed in a given time is
equal to the number that are decomposed to form CO. The last
equation may be written
[CO] ki
[CO _  .K (6)
[GO0] k *
K is called the equilibrium constant.
Equation (4) giving the rate of formation of CO may be
modified as follows:
Let 100 a = per cent of CO2 by volume at the beginning of
the reaction, that is, when t = 0,
100 x = per cent of CO by volume after the time t has
elapsed.
At the beginning of the reaction, that is when t = 0, there is
no CO, hence x = 0. If now n is the number of gram molecules
of the gas when t = 0, then na is the number of gram molecules
of CO2, and the concentration of the CO2 is therefore
[CO21 = nn a
But since pr = n R T, this relation may be written
[CO2] = ap = M gram molecules per liter.
At the time t suppose m gram molecules per liter of CO to have
been formed. This involves the disappearance of 2 gram mole
m
cules per liter of CO2, leaving M  2 . The number of gram
molecules of gas is now
n (1a) + [ M  + m ] v
=n(1a) + ( M+ ) 
ILLINOIS ENGINEERING EXPERIMENT STATION
= n1 + m a)
2M
m RT
= n( 1 + )'
and the volume of the gas is therefore increased from v to
v ( 1 + R ). The concentrations at the time t are there
fore
[CO] = m
m RT (7)
2 p
ifO m
[CO] = . (8)
m RI (8)
+ 2
2 p
But since 100 x per cent of the gas is CO by volume the concentra
tion of the CO is also given by the relation
[CO] = . (9)
Combining the two expressions for [CO] given by (7) and (9),
we obtain.
2x _
m 2x RT'
and introducing this expression for m in (8), the result after
slight reduction is
[COs] = (a  a 1 x) P . (10)
Introducing in equation (4) the expressions for the concentrations
given by (9) and (10), the result is
dE00]  k a±1 2 r 1]
But from (9)
d[CO] _ p dx
dt RT dt '
THE FORMATION OF CO IN GAS PRODUCERS
whence
dX , a + 1 x) p X.
dt  (a 2  2 RB1
Replacing the constant k2 R by a single symbol k'2, the final
equation for the reaction velocity is
dx a12(
dt = k, ( a 2 )  k' x. (12)
The integration of the differential equation (12) offers no
great difficulty.
The determination of the constants kc and k2 seems to have
been made hitherto by assuming the value of the ratio k. This
method is applicable when the equilibrium conditions are readily
realized. As, however, this is not the case in the present reac
tion it has been necessary to devise a method for the determina
tion of k, and k'2 from two or more pairs of simultaneous obser
vations of x and t. It turns out that the method is applicable not
only to the reaction in question, but also to the most general in
complete reactions of the second order.
The great difficulty of realizing, with certainty, the condition
of the equilibrium in reactions like the one under consideration,
makes it highly desirable, therefore, to obtain a general solution of
the differential equation, without introducing a particular numer
ical value for the ratio Y. We are indebted to Prof. C. N.
Haskins for the following solution, the developments of which will
be found in the appendix:
4 a r tank at
a + 1 1 + r tank at
In equation (13) a and y are determined by the relation
k  4a
a + 1'
ILLINOIS ENGINEERING EXPERIMENT STATION
a (a (1 ).
F2  4 a
When the initial percentage of CO2 is 100, then a = 1,
dx
 k, ( 1  x)  k'2x,
and 2 r tank at (14)
1 + r tank 'it
In the case of the gas producer the value of a is about 0.21.
III. BOUDOUARD'S EXPERIMENTS.
An elaborate series of determinations of the amount of CO
formed at different temperatures has been made by 0. Boudouard.1
Boudouard's observations were made at 6500, 800' and 9250 C.
In his experiments at 6500 and 800' glass tubes containing char
coal, coke, retort carbon or lamp black were filled with CO,, heat
ed to the temperature of the experiment and then sealed. The
tubes were maintained at constant temperature until equilibrium
was reached, that is, when further heating at the same tempera
ture produced no increase in the percentage of CO present. At
6500 the heating was continued for twelve hours before equi
librium was attained. At 8000 equilibrium was reached in one
hour in the tubes containing charcoal and in two and onehalf
hours in those containing lamp black. With coke and retort car
bon the process was not complete at the end of nine hours. In
the experiment at 9250 the carbon was heated in a porcelain tube,
through which was passed a stream of CO, gas. The average time
of contact between CO2 and carbon calculated from the data given
20. Boudouard, Comptes Rendus de I'Academie des Sciences,
Vol. 128, page 824, 154; 1899.
Vol. 131, page 1204, 1900.
Vol. 130, page 132, 1900.
Bulletin Hoc. Chim., Paris, Vol. 21, 1901.
Bulletin Soc. Chim., Paris, Vol. 25, 1901.
Ann. de Chimie et de Physique, Vol. 354, 1901.
See also Haber, Thermodynamics of Technical Gas Reactions, 1908, p. 311.
THE FORMATION OF CO IN GAS PRODUCERS
in Boudouard's account of his experiments was approximately
30 seconds.
A summary of Boudouard's results is given in the following
table:
TEMPERATURE PER CENT OF PER CENT OF
C.0 C02 CO
650. 61. 39.
800. 7. 93.
925. 4. 96.
These values have been made the basis of computation by many
writers on the chemistry of combustion and of the water gas
reaction and especially in treatises on the gas producer.
In the first references to Boudouard's work which came to the
attention of the writers, no notice was taken of the remarkably
low reaction velocity of the formation of CO from CO2 and carbon,
and of the great length of time required to obtain the percentages
of CO that are given in the preceding table. In at least one case
the values shown above were offered as representing the quality
of gas that should be obtained in the gas producer at the tempera
tures given. The writers were therefore led to regard Boudouard's
figures as defining the relative proportions of CO2 and CO that
should be formed in a gas producer at various temperatures of the
fuel bed.
The experiments which form the subject of this paper had
originally as their object the confirmation of Boudouard's results
as well as a continuation of them at higher temperatures. Pre
liminary experiments made by Dr. C. S. Hudson demonstrated
that the amount of CO formed at a given temperature depends
largely on the time of contact or in other words on the rate of
flow of the gas through the fuel bed. Apparently Boudouard's
results represent limiting values, which can be obtained only
with a very low gas velocity. In order to ascertain the condi
tions for the formation of CO in producer furnaces, it is neces
sary, therefore, to determine the rate of formation of CO from
CO2 and carbon at various temperatures; that is, to determine the
ILLINOIF ENGINEERING EXPERIMENT STATION
amount of CO formed with different rates of flow of the gas
through the fuel bed.
IV. METHOD OF EXPERIMENT.
The arrangement of the apparatus is shown in Fig. 1, 2, and
3. A porcelain tube of 1.5 cm inside diameter, 60 cm long, and
glazed on the outside was filled with charcoal, coal, or coke and
heated in an electric furnace. The furnace, which was designed
FIG. 1
especially for this investigation, is shown in detail in Fig. 3. It
has been operated continuously for a period of six months at tem
peratures of from 800' to 12000, and even 13000 C. The temper
ature inside the porcelain tube could be maintained at any value
up to 13000 C with no fluctuations greater than 10 or 2'. The
.. . 4' . . 
FIG. 2
heating coil consists of a coil of No. 13 nickel wire, wound with
eight turns per inch on an electrical porcelain insulating tube of
38 mm inside diameter and 35 cm long. At either end of the coil
THE FORMATION OF CO IN GAS PRODUCERS 11
ILLINOIS ENGINEERING EXPERIMENT STATION
the number of turns was increased slightly to compensate for the
cooling effect of the end at the furnace. As a protection against
corrosion, the coil was painted over with a thin layer of magnesite
cementa material that is capable of withstanding very high tem
peratures. The heating tube was then mounted inside of and con
centric with several terra cotta pipes and the spaces between
the pipes were filled with light calcined magnesia. The cost of
the material used in the furnace and the amount of labor required
were very small. A temperature of 10000 C could be maintained
by the expenditure of 600 watts.
The temperature inside the porcelain tube was measured by
means of a platinum platinumrhodium thermocouple (see Fig.
2) and a Siemens & Halske millivoltmeter. As the thermoelec
tric height of the couple fell slightly with use at high temperature
due probably to the reducing action of CO gas on the insulating
tubes and the consequent contamination of the coupleit was
found necessary to calibrate the couple from time to time. This
was accomplished by determination of the melting points of zinc,
silver and copper. The error of individual temperature observa
tions does not exceed 50 below 1100° and 10° to 150 between 11000
and 1300°.
The carbon with which the porcelain tube was filled was
crushed to pieces of a uniform sizeabout 5 mm on a side. Only
the central portion of the tube (see Fig. 2) contained carbon, the
remainder of the space being occupied by pieces of broken porce
lain, which served at one end to heat the gas entering the tube,
and at the other end, by reducing the size of the passage way, to
increase the velocity of the gas through the region of falling tem
perature. Through the porcelain tube was passed a stream of
CO2 gas. In the earlier experiments CO2 was prepared from mar
ble and hydrochloric acid. Later, CO2 was taken from a tank of
liquid carbon dioxide.
The velocity of the gas over the carbon was determined by
the dimensions of the tube, the weight and density of the carbon,
THE FORMATION OF CO IN GAS PRODUCERS
and the temperature and the volume of gas passed through the
tube per minute.1
The analyses were made by the Hempel method, both CO,
and CO being absorbed. The amount of gas remaining in the
burette after the absorption in cuprous chloride was seldom great
er than two per cent.
V. EXPERIMENTS WITH CHARCOAL.
With the apparatus described in the preceding pages, experi
ments were conducted at temperatures ranging from 700° to 13000
C. The experiments with charcoal extended over a period of sev
eral months. The results are contained in tables 16.
TABLE 1
RATE OF FORMATION OF CO FROM C02 AND CHARCOAL
AT A TEMPERATURE OF 8000 C.
k = 0.01968
k2 = 3.031
Time of contact 1 % CO,100 t CO/100
in seconds " Observed Calculated
t
oc 0 ..... 0.535
188.6 0.0053 0.503 0.534
115.9 0.0086 0.504 0.527
57.18 0.0175 0.518 0.508
45.70 0.0219 0.522 0.468
24.20 0.0413 0.375 0.345
15.50 0.0645 0.283 0.252
12.32 0.0810 0.'45 0.209
2.686 0.354 0.063 0.051
1.550 0.645 0.03io 0.030
'The increase in the volume of the gas in its passage through the reaction
tube, due to the formation of two CO molecules in place of every molecule of
CO, which disappears, makes it difficult to determine accurately the time of con
tact and consequently the velocity of the gas.
The values of t, the time of contact, given in the following tables are based
on the volume of gas leaving the tube and are therefore somewhat too low.
Since the major portion of the expansion takes place within a short distance
from the entrance to the tube, the error here introduced is probably not appre
ciable.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 2
RATE OF FORMATION OF CO FROM C02 AND CHARCOAL
AT A TEMPERATURE OF 850° C.
ki = 0.07174
k2 = 3.238
TABLE 3
RATE OF FORMATION OF CO FROM C02 AND CHARCOAL
AT A TEMPERATURE OF 9000 C.
ki = 0.1540
k2 = 2.599
Time ,f contact 1 % CO/100 % CO/100
in seconds  Observed Calculated
t
0 ..... 0.873
64.29 0.0156 0.873 0.873
41.18 0.0226 0.867 0.872
10.008 0.0999 0.708 0.739
4.257 0.234 0.498 0.472
2.840 0.352 0.311 0.351
2.172 0.461 0.344 0.284
Time of contact
in seconds 1 00CO/100 CO/100
t t Observed Calculated
a 0 ..... 0.742
123 0 0.0082 0.743 0.742
54.18 0.0184 0.702 0.741
24.43 0.0410 0.572 0.694
13.,,3 0.0756 0.526 0 564
.268 0.1070 0.297 0.463
4.630 0.216 0,297 0.281
3 6,4 0.271 0.224 0.231
:1. 254 0.307 0.225 0.207
THE FORMATION OF CO IN GAS PRODUCERS
TABLE 4
RATE OF FORMATION OF CO FROM C02 AND CHARCOAL
AT A TEMPERATURE OF 925' C.
ki =.0.2175
k2 = 2.298
TABLE 5
RATE OF FORMATION OF CO FROM CO2 AND CHARCOAL
AT A TEMPERATURE OF 10000 C.
ki = 0.6404
ks = 4.708
Time of contact 1 % CO/100 % CO/100
in seconds Observed Calculated
t
oo 0 ...... 0.942
70.0 0.0143 0.949 0.942
18.60 0.0538 0.943 0.941
8.245 0.1195 0.903 0.938
3.675 0.272 0.797 0.869
2.296 0.436 0.795 0.752
Time of contact 1 %CO/100 % CO/100
in seconds 7 Observed Calculated
t
oo 0 .... . 0.914
118.8 0.0084 0.947 0.914
81.2 0.0123 0.933 0.914
12.37 0.0807 0.848 0.875
5.80 0.1725 0.718 0.697
4.277 0.234 0.642 0.595
2.272 0.440 0.375 0.387
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 6
RATE OF FORMATION OF CO FROM CO2 AND CHARCOAL
AT A TEMAERATURE OF 1100° C.
kl = 1.495
k2 = 5.275
Time of contact 1 * CO/100 % CO/100
in seconds t Observed Calculated
t
oo 0 ..... 0.972
36.48 0.0274 0.987 0.972
10.43 0.0958 0.983 0.972
4.968 0.2010 0.981 0.971
3.640 0.2745 0.973 0.968
1.921 0.521 0.946 0.955
The first and second columns of each table give the time of
1
contact t and the reciprocal of the time of contact , which is
equal to the velocity of the gas divided by the length of the char
coal column; thus
t ~l*
The third column contains the percentages of CO observed, and the
values in the last column were calculated by means of equa
tion (13), viz:
4 a y tank at
a + 1 1 + r tank at
The method of computing a and y of this equation is described
in the appendix. The constants ki and k'2 are determined by the
relations,
k  4ar
, a a + 1)
k2 a 4a+
THE FORMATION OF CO IN GAS PRODUCERS
Values of kj, k2, k'2, a, and y are given in table 7.
TABLE 7
CONSTANTS USED IN COMPUTATION OF X
Temp.
Co
800.
850.
900.
92.5
1000
1100
a
(a = 1)
0.0276
0.0612
0.09998
0.1297
0.3617
0.7921
7
(a =1)
0.3568
.5853
.7711
.8388
.8853
.9437
k
2
0.03373
0.03443
0.02646
0.02291
0.04416
0.04588
k2
3.031
3.238
2.599
2.298
4.708
5.275
0.01968
0.07174 '
0.1540
0.2175
0.6404
1.4950
K= k
0.006493
0.02216
0.05925
0.09465
0.13603
0.28341
The calculated and observed values of x, the per cent of CO,
agree within two or three per cent.
A comparison of the results in tables 1 to 6 shows in the
first place that with increasing temperature there is a rapid in
crease in the percentage of CO obtained with any given rate of
100
90
80
0 70
S60
0 50
S40
30
20
jo
1.4
§ ~ _F
.10 .20 .30 .40 50 .60 .70 .80 .90
1
RECIPROCAL OF TnuiE OF CONTACT, 
FIG. 4.
ILLINOIS ENGINEERING EXPERIMENT STATION
flow of the gas; in the second place that with increasing gas ve
locity at low temperatures the percentage of CO formed falls off
very rapidly, at higher temperatures very slowly. These varia
tions are illustrated by the curves in Fig. 4 in which the per
1 v
centage of CO is plotted as a function of t . When 1, the
length of the charcoal column, is equal to one, i. e., is equal to
the unit of length, then the numbers along the abscissa give the
velocity of the gas in terms of the same unit of length, and per
second. For example, the length of the charcoal column in the
experiments here recorded, was approximately 20 cm. The velocity
corresponding to the point t = 1 at the extreme right of Fig. 4,
is therefore 20 cm. per second.
The general shape of all the curves in Fig. 4 is the same.
The percentage of CO is greatest at zero velocity. With
1
increasing values of t each curve falls off, slowly at first
then more rapidly, passing a point of inflection and finally becom
ing nearly horizontal. The intersections of the curves with the
CO axis give the percentage of CO corresponding to the condition
of equilibrium.
That a considerable amount of time is required to reach equi
librium in the reaction under consideration, is further illustrated
in Fig. 5, in which the percentage of CO is plotted as a function of
t, the time of contact. (One small division = 1 second). At 800'
for example, the percentage of CO reaches a practically constant
value at the end of 50 sec.; at 1000' in 6 sec.
The curves in Fig. 4 were plotted from values of x (= per
cent CO) calculated from equation (13). The observed values are
indicated by the small circles. By means of equation (13) it is
possible to calculate the per cent of CO corresponding to any giv
en gas velocity providing a and y or ki and k2 are known.
THE FORMATION OF CO IN GAS PRODUCERS
100
90
80
0
0 70
o 60
50
a 40
30
20
10
100
90
80
70
60
50
40
30
20
10
10 20 30 40 50 60 70 80 90
TIME OF CONTACT (CHARCOAL)
FIG. 5.
VI. VARIATIONS OF K AND k1 WITH TEMPERATURE.
The values of ki and K given in table 7 exhibit a systematic
variation with temperature. If equations can be found that will
express ki and K as functions of the temperature it will then be
possible to calculate the per cent of CO for any time of contact
and any desired temperature. Such equations have been deduced
by Van't Hoff from purely thermodynamical considerations. They
are the following.1
d (In K) _ Q
dT R T2'
and
d ( Inn ) A
dT(ln + B. (16)
In these equations, Q is the latent heat of reaction at the abso
lute temperature T, A is a function of Q but is selected arbitrar
ily, and B is an arbitrary function of the temperature. By inte
gration the latter equation becomes
'The symbol In in the following equations stands for natural logarithm.
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ILLINOIS ENGINEERING EXPERIMENT STATION
A
In k =  + BT C, (17)
where C is an integration constant. The values of A, B, and C, in
equation (5) have been determined from the simultaneous values
of ki and T of tables 16. Table 8 contains the values of k,, ob
tained at various temperatures as well as the values of k1 calcu
lated from equation (17). The agreement is remarkable good.
TABLE 8
VARIATION OF ki WITH TEMPERATURE (CHARCOAL)
In = 5010  0.0203 T+ 65.376
Temp. Absolute Temp. ki (obs) ki (calc)
Deg. C. T
800 1073 0.020 0.021
850 1123 0.073 0.064
900 1173 0.154 0.159
925 1198 0.217 0.237
1000 1273 0.640 0.629
1100 1373 1.490 1.53
In order to integrate the equation
d(ln K) _ _ Q
d T RI'2
it is first necessary to determine the heat Q as a function of the
temperature. It has been shown by Kirchoff that the increase of
Q per degree rise in temperature is equal to the difference of the
molecular heats of the factors and of the products of the reaction.
Following this law, and taking the specific heat of a factor or
product as a linear function of the temperature, which is very
nearly true for gases, the relation between Q and T is given by
the equation'
Q = Qo + Ci T+ c2 T2 (18)
In this equation Qo denotes the heat of reaction for T = 0, and
c, and c2 are obtained as follows: Assuming that the mean specific
heat of each gas is given by an expression of the form
'Haber, Thermodynamics of Technical Gas Reactions, p. 49, eq. (7a).
THE FORMATION OF CO IN GAS PRODUCERS
c== a+ bT,
then cl is the difference between the sum of the a's of the factors
and the sum of the a's of the products; likewise c2 is the sum of
the b's of the factors less the sum of the b's for the products. Sub
stituting the value of Q given by (18) in (15) the result is
d(In K) 1 Qo c+
dt R T T
whence by integration
In K = (  c In T c, T + C. (19)
The constants in this equation, (19), may be determined by
either of two different methods: by experimental determinations
of Q0 c,, and c2, or from four or more simultaneous observations of
K obs and T. The first method was adopted in this instance,
the following being the values of the quantities in question:
Q0o=  40166
c, =  2.055
c2 = 0.003104
C = 8.604
In the determination of c, and c2, Langen's values for the specific
heats of CO and CO2 and the value of Kunz for the specific heat of
charcoal were employed.
The value of R (in gramcalories per deg.) is 1.985. Hence
taking the above constants and this value of R, (19) reduces to
20235
In K  + 1.035 In T  0.001564 T + 8.604. (20)
Table 9 gives values of K calculated from equation (20) along
with values (marked Kob8 ) obtained from observed values of x
and T in tables 16. In the fourth column are the observed values
of x, the amount of CO in equilibrium with CO2 and charcoal at
temperatures from 8000 to 11000; and in the fifth columns the
values of x corresponding to the values of K in the third column.
ILLINOIS ENGINEERING EXPERIMENT STATION
The constants of equation (19) were calculated also by
the method of least squares, from simultaneous values of Kobs
and T, but the agreement was less satisfactory than by the first
method.
TABLE 9
VALUES OF K AND OF XZ
Temp. C.
500
600
650
700
800
850
900
925
1000
1100
1200
1300
1400
1500
1600
K (obs)
0.0065
0.022
0.059
0.094
0.136
0.283
K (cal)
0.000007
0.00013
0.00046
0.00137
0.0090
0.020
0.042
0.060
0.151
0.448
1.120
2.455
4.826
8.671
14.44
x . (obs)
0.526
0.738
0.871
0.912
0.939
0.971
x , (cal)
0.021
0.093
0.185
0.283
0.582
0.722
0.832
0.873
0.945
0.981
0.994
0.997
0.9985
0.9992
0.9996
x C (obs)
Boudouard
0.39
0.93
0.96
The agreement between "observed" and "calculated" values of
K and k in tables 8 and 9 shows that the changes of K and kc, with
temperature follow van't Hoff's laws. It is possible, therefore,
by means of equations (17) and (19) and the values of the con
stants of these equations given in tables 8 and 9, to compute K
and ki for any desired temperatures. Having the values of K
and k, and consequently of k2 (k2= 1 ) the per cent x of CO
corresponding to any time of contact t can then be calculated by
means of equation (13).
VII EXPERIMENTS AT 700°.
In Fig. 4 the curve for 8000 falls off very rapidly with increas
ing rate of flow of gas. At this temperature the gas velocity must
be exceedingly low to obtain the equilibrium percentage of CO.
THE FORMATION OF CO IN GAS PRODUCERS
At temperatures below 8000 it was practically impossible to reach
equilibrium with a finite gas velocity. A great number of experi
ments were made at 7000 but the results were too inconsistent to
admit of mathematical treatment. Some of the observations are
given in the following table:
TABLE 10
OBSERVATIONS AT 700 C. (CHARCOAL).
t = Time of 1 % CO/100
contact in seconds 
86.9 0.0115 0.012
86.2 0.0116 0.155
99.9 0.0100 0.077
23.4 0.0426 0.004
15.0 0.0668 0.014
7.11 0.1405 0.009
9.71 0.103 0.022
5.60 0.178 0.006
5.02 0.199 0.008
4.18 0.239 0.012
These results show that, except at exceedingly low velocities,
the amount of CO formed was never greater than one or two per
cent.
VIII. EXPERIMENTS WITH COKE AND COAL.
The experiments with coke and coal were conducted in the
same manner as with charcoal. The material was crushed to
pieces about 5 mm on a side. The constants a and y of equation
(13) were obtained for each temperature by the method given in
the appendix. Tables 1115 contain the results of the observa
tions with coke. In the last column of each table are given the
values of x, the percentage of CO formed, calculated from equa
tion (13).
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 11
RATE OF FORMATION OF CO FROM CO2 AND COKE
AT A TEMPERATURE OF 900° C.
ki = 0.00231
k2 = 0.03686
TABLE 12
RATE OF FORMATION OF CO FROM C02 AND COKE
AT A TEMPERATURE OF 10000 C.
ki = 0.02323
k2 = 0.3591
Time of contact 1 % CO/100 % CO/100
in seconds  Observed Calculated
t t
123.2 0.0081 0.784 0.866
80.25 0.0125 0.644 0.795
33.25 0.0301 0.529 0.527
18.72 0.0535 0.320 0.350
6.37 0.1571 0.139 0.138
4.101 0.2439 0.115 0.091
3.072 0.3258 0.092 0.069
1.983 0.5045 0.063 0.045
Time of contact 1 % CO/100 % CO/100
in seconds. Observed Calculated
142.0 0.0070 0.276 0.278
80.20 0.0124 0.131 0.169
43.91 0.0228 0.094 0.096
24.82 0.0403 0.057 0.056
16.11 0.0620 0.049 0.037
9.575 0.1045 0.026 0.023
3.741 0.2671 0.008 0.009
THE FORMATION OF CO IN GAS PRODUCERS
TABLE 13
RATE OF FORMATION OF CO FROM CO2 AND COKE
AT A TEMPERATURE OF 1100° C.
ki = 0.1335
k2 = 0.5296
Time of contact 1 %0/100 % CO/100
in seconds 1 0/100 CO/100
t seconds Observed Calculated
90.00 0.0111 0.971 0.971
29.92 0.0334 0.854 0.955
13.20 0.0758 0.661 0.817
6.765 0.1476 0.556 0.592
3.198 0.3135 0.317 0.346
1.784 0.5606 0.304 0.211
1.660 0.6030 0.240 0.1942
1l590 0.6299 0.221 0.190
1.462 0.6840 0.214 0.177
0.962 1.0399 0.133 0.121
TABLE 14
RATE OF FORMATION OF CO FROM CO2 AND COKE
AT A TEMPERATURE OF 1200° C.
ki = 0.4095
k2 = 0.6718
Time of contact
in seconds
t
18.92
12.70
8.250
2.402
1.582
1.080
1
0.0528
0.0788
0.1213
0.4160
0.6320
0.9260
% CO/100
Observed
0.989
0.978
0.953
0.685
0.439
0.335
% CO/100
Calculated
0.987
0.983
0.956
0.624
0.460
0.357
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 15
RATE OF FORMATION OF CO FROM C02 AND COKE
AT A TEMPERATURE OF 13000 C.
ki = 1.483
k2 = 0.7313
Time of contact 1 % CO/100 % CO/100
in seconds  Observed Calculated
8.860 0.1129 0.999 0.997
4.149 0.2415 0.979 0.997
2.100 0.4760 0.932 0.955
1.130 0.8850 0.834 0.816
The results with coke are shown graphically in Fig. 6. The
curves for 900', 10000 and 11000 are considerably lower than
the curves with charcoal for the same temperatures, except for
very low velocities.
.10 .20 .30 .40 .50 .60 .70 .80 .90
RECIPROCAL OF
TIME
Fia. 6.
1
OF CONTACT, 
1
     
THE FORMATION OF CO IN GAS PRODUCERS
TABLE 16
RATE OF FORMATION OF CO FROM C02 AND ANTHRACITE COAL
AT A TEMPERATURE OF 11000 C.
ki = 0.119
k2 = 1.410
TABLE 17
RATE OF FORMATION OF CO FROM CO2 AND ANTHRACITE COAL
AT A TEMPERATURE OF 12000 C.
ki = 0.2374
k2 = 0.1767
Time of contact 1 CO/100 % CO/100
in seconds  Observed Calculated
47.05 0.0212 0.997 0.993
10.39 0.0964 0.856 0.901
5.070 0.1971 0.715 0.688
2.845 0.3516 0.423 0.472
1.592 0.6270 0.310 0.309
Time of contact 1 % CO/100 % CO/100
in seconds Observed Calculated
34.20 0.0293 0.8780 0.912
9.370 0.1069 0.6>10 0.657
5.415 0.1848 0.4770 0.472
3.301 0.3026 0.3020 0.322
2.439 0.4101 0.2650 0.251
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 18
RATE OF FORMATION OF CO FROM C02 AND ANTHRACITE COAL
AT A TEMPERATURE OF 13000 C.
k = 0.5791
k2 = 0.2016
Time of contact 1 % CO/100 CO/100
in seonds T Observed Calculated
12.40 0.0806 0.999 0.997
6.030 0.1659 0.965 0.968
3.600 0.2779 0.824 0.876
2.980 0.3358 0.809 0.822
1.908 0.5249 0.663 0.668
1.070 0.9350 0.503 0.462
The observations with anthracite coal are given in tables 16,
17, and 18, and are illustrated graphically in Fig. 7.
.10 .20 .30 .40 .50 .60 .70 .80 .90
1
RECIPROCAL OF TIME OF CONTACT, 
FIG. 7.
Here the curves fall off even more rapidly than the curves for
coke in Fig. 6. With very low velocities, that is, when the time
THE FORMATION OF CO IN GAS PRODUCERS
of contact is sufficient for the reaction to reach equilibrium, the
percentage of CO formed is practically the same with each of the
three forms of carbon. As the rate of flow of the gas increases,
the effect of the difference in the reaction velocities becomes more
appreciable.
TABLE 19
VALUES OF X, FOR CHARCOAL, COKE, AND COAL.
Temp. C. x. (cal) xx (obs) x. (obs) x. (obs)
Charcoal Coke Coal
900 0.832 0.871 0.875 ......
1000 0.945 0.939 0.886
1100 0.981 0.971 0.968 0.914
1200 0.994 ...... 0.987 0.994
1300 0.997 ...... 0.996 0.997
Values of x co the percentages of CO in equilibrium with CO2,
and charcoal, coke, and coal respectively, are given in table 19.
The values in the second column of this table were calculated
from the values of K in table 9, by means of the equation
x2 = RT
K
1  x p'
A comparison of Fig. 5, 6, and 7 shows that the reaction
velocity is greatest with charcoal and lowest with anthracite coal.
The temperature coefficient of k,, the coefficient of reaction ve
locity, was determined for coal and coke in the same manner as
for charcoal. The "observed" and "calculated" values of k/ are
shown in tables 20 and 21.
The constant k2 in the equation
d[CO]
dt [ k [C02]  k2 [CO]2
is the coefficient of reaction velocity of the reaction
CO,2 + C = 2 CO
taken from right to left. At the temperatures of these experi
ments, 800°13000, the carbon produced by the decomposition of
ILLINOIS ENGINEERING EXPERIMENT STATION
CO is in the form of lamp black, regardless of the form of carbon
present in the reaction tube, viz: charcoal, coke, or coal. At any
one temperature, therefore, k2 should be the same in all three
cases. From a comparison of tables 16, 1115 and 1618 it will
be seen that there is considerable deviation in the values of k2
for the three forms of carbon used. This is doubtless due in part
to experimental errors. There is a further consideration, how
ever, to which attention should be called, viz: that the reaction
in question is not reversible. The lamp black produced by the re
verse reaction
2 CO = CO2 + C,
is not identical physically with the form of carbon, charcoal, or
coke that is consumed in the formation of CO. Consequently the
law of chemical mass action is not strictly applicable. In the
systems under consideration, equilibrium would not be reached
until all the carbon has been transformed to lamp black.
TABLE 20
VARIATION OF k1 WITH TEMPERATURE (COKE)
47220
In ki =  0.009699 T + 45.597
T
TABLE 21
VARIATION OF ki WITH TEMPERATURE (ANTHRACITE COAL)
31972
In ki =  + 0.02272 T  56.607
T
THE FORMATION OF CO IN GAS PRODUCERS
IX. APPLICATION OF EXPERIMENTAL RESULTS TO THE PROCESSES OF
THE GAS PRODUCER AND BOILER FURNACE.
As stated in the introduction, the experiments here described
were undertaken primarily to determine the temperature neces
sary for the formation of high percentage CO gas in the fuel bed
of the gas producer, and to ascertain the conditions that govern
the formation of CO in boiler furnaces. The results here pre
sented indicate that the amount of CO formed in the gas pro
ducer depends on three factors: (1) the temperature; (2) the
depth of the hot portion of the bed; and (3) the rate of flow of
gas through the bed. Stated in a more concise form, the per
centage of CO formed depends on the temperature and the time
of contact of gas and carbon, i. e., the average time required for
a molecule of gas to pass through the fuel bed. The variation of
the percentage of CO with the rate of flow of gas is illustrated
in Fig. 4, 6 and 7. The curves for coke, Fig. 6, may be taken as
representing the conditions in the fuel bed of the producer. At
13000 C., for example, with zero velocity (time of contact = o)
1
practically all the CO2 will be converted to CO; when t= 0.5,
t
(time of contact t = 2 sec.), 90 per cent CO is obtained; and
when t = 1 only 80 per cent CO is formed. In a fuel bed one
foot in depth, since
1  velocity of gas _ v
t depth of fuel bed I
a time of contact of t = 2 sec. corresponds to a velocity 0.5 ft.
per sec. and t = 1 to a velocity of 1 ft. per sec. At 1300° C.,
then, in a fuel bed one foot in depth, with a velocity of 0.5 t. per
sec. 90 per cent of CO would be formed and with a velocity of
1 ft. per sec. 80 per cent. In a fuel bed two feet in depth, the
gas velocities corresponding to the same percentage of CO would
be twice as great. In other words, for given conditions of tem
perature and quality of gas, the depth of bed and velocity of gas
ILLINOIS ENGINEERING EXPERIMENT STATION
must vary proportionally and their ratio v must remain con
stant. A fuel bed one foot in depth and a gas velocity of one foot
per second should yield the same percentage of CO as a bed two
feet in depth with a gas velocity of 2 ft. per sec.
It is impossible to determine accurately the velocity of the
gas through the producer fuel bed, on account of the difficulty of
estimating the magnitude of the passages through the bed.1
The velocity lies probably between 0.5 and 5.0 ft. per sec.
The right half of the curves in Fig. 6 lies within these limits and
therefore corresponds approximately to the conditions of pro
ducer operation.
0
A
600 700 800 900 1000 1100 1200 1300
TEMPERATURE, DEGREES C.
FIG. 8.
1400 1500 1600
In Fig. 8 is shown graphically the variation with tempera
1
ture of the amount of CO formed with different values of 1. The
t
'When any given number of pounds of air is passed per second through a fuel
bed of given dimensions, the velocity through the bed will increase as the per
centage of voids is decreased. Thus the velocity will be much higher with slack
coal than with uniformly sized nut coal. Further, the per cent of voids will
be influenced by the amount of coking and clinkering.
THE FORMATION OF CO IN GAS PRODUCERS
ordinate is the per cent of CO in gas containing initially 21 per
cent CO2 (air in which the oxygen has been converted quantita
tively to CO2). The abscissa is temperature in degrees Centigrade.
1
The upper curve, t = 0, represents the maximum amount of
CO which could be produced from air. The intersection of the
curve for any velocity with a given horizontal line, for example,
the line for CO = 30 per cent, gives the temperature required
to form that amount of CO with the particular velocity. Thus
to obtain 30 per cent CO with a velocity of one foot per sec.
(length of bed = 1 ft.) will require a temperature of 13600 C.,
and with a velocity of 2 ft. per sec., 1435'. The curves of Fig. 6
and 8 indicate that the temperature of the producer bed should
not be less than 13000 C.
These investigations demonstrate that a very high tempera
ture is necessary for the production of CO from CO2 and carbon.
There are other considerations, however, which are opposed to
the operation of the fuel bed of the gas producer at extremely
high temperatureabove 13000 C.: A high temperature of fuel
bed means that the gases will leave the producer at a high tem
perature and thus lower the efficiency of the producer. The gain
in capacity will therefore be accompanied by a loss in efficiency,
unless the heat of the gases can be used efficiently for generating
steam and preheating the air blast. Also a high temperature fav
ors clinkering. In the application of the results of these experi
ments to commercial producers and furnaces it will be necessary
of course to consider the various questions that are involved.
Various explanations have been suggested to account for the
presence of small amounts of carbon monoxide in the flue gases
of boiler furnaces. Perhaps the one most generally accepted by
engineers is that the oxygen of the air first unites with carbon
to form CO2 and that as this gas passes up through the hot fuel
bed it combines with carbon in accordance with the equation
CO2 + C = 2 CO.
ILLINOIS ENGINEERING EXPERIMENT STATION
Assuming this to be the correct explanation, then the ques
tion to be solved is what conditions are favorable to this reaction
and what conditions will tend to retard it. In the preceding para
graphs it has been shown that the higher the velocity of the gas
and thinner the fuel bed, the less will be the percentage of CO
formed. A heavy fuel bed in the boiler furnace would therefore
favor the formation of CO. Also, the greater the supply of air to
a given depth of bed, the less should be the tendency to form CO.
X SUMMARY AND CONCLUSIONS.
1. The rate of formation of CO in the reaction,
CO2 + C = 2 CO
has been determined with charcoal from 8000 to 1100' C., with
coke from 9000 to 13000 C., and with anthracite coal from 1100° to
13000 C.
2. The differential equation for the velocity of incomplete
reactions
dx a + 1
dt = k a  x )  kx2
has been solved for given values of Ak' and k,, and it has been
shown (in the appendix) that the method is applicable to other
cases.
3. Van't Hoff's laws for the variation of equilibrium con
stants and coefficients of reaction velocity with temperature have
been applied to the values of k, and K obtained in these experi
ments, and a close agreement between observed and calculated
values has been found.
4. By means of the equations expressing the laws referred
to in paragraphs (2) and (3) it is possible to compute the per
centage of CO formed at any temperature and with any time of
contact.
5. It has been shown that for the production of a high per
centage of CO gas, the producer fuel bed should have a tempera
ture of 1300' C. or over, and that increasing the depth of the hot
THE FORMATION OF CO IN GAS PRODUCERS 35
portion of the bed will increase the percentage of CO, and conse
quently the capacity of producer at first rapidly and then more
and more slowly.
6. To minimize the production of CO in the boiler furnace the
fuel bed should be thin. Increasing the velocity of the gas will
tend to decrease rather than increase the percentage of CO formed.
THE FORMATION OF CO IN GAS PRODUCERS
APPENDIX
ON THE COMPUTATION OF THE CONSTANTS OF THE REACTION
EQUATION
BY CHARLES N. HASKINS.
1. REDUCTION AND INTEGRATION OF THE DIFFERENTIAL EQUATION.
The differential equation is
dx a+ 1
d = k, ( a  2 x )  k2',
where 100 a = % ( CO + CO2 ) at time t = 0, (1)
100 x = % CO at time t,
t = time in seconds,
and k, and k2 are the two constants of the reaction the values of
which are sought. The initial condition is that
a = 0, when t = 0. (2)
To integrate, we introduce a new variable z and new con
stants a, y, defined by the relations
a +1
2 i) x b 2 az
( 2 (2 (3)
a _ ( 1 + (, 7 1 (a +1 (1)
S= , 1 a +1 a ( 1  2 )
The differential equation becomes, under these substitutions,
38 ILLINOIS ENGINEERING EXPERIMENT STATION
dz = 2) (4)
with the initial condition z = 0, when t = 0. (5)
Integrating, we have
In + _ = 2 at ; (6)
and solving for z,
e at  e at
z = reat + e at = r tanl at; (7)
whence, substituting in (3),
2 ar tank at
a + 1 1 (8)
2 )( ± Y
2. THE EQUATION FOR y AND THE CRITERION FOR THE EXISTENCE
OF ONE AND ONLY ONE ROOT.
We have (equation 8) an expression by means of which the
per cent of CO at any time t may be computed if the constants
y and a are known. We now wish to determine y and a from two
pairs of observed corresponding values of t and x. Let these two
pairs be (t,, x), and ( t2 x2), and let t2 > t1.
Then since a is known we may compute
(a + 1)\ a (+ 1)
1 () a' 2  (a + 1)
2 1 22
and have
In r + z= 2 4at , In +2 2 ,42, (9)
Y  Zl r .
from which y and a are to be determined. Eliminating a, we
readily obtain
In 2 = t2 In (10)
T  22 t1 Z
'The symbols In x, log x will De used to denote the natural and the common
logarithm of x, respectively.
THE FORMATION OF CO IN GAS PRODUCERS
The determination of y and a depends therefore on the solution of
this (transcendental) equation in 7.
Consideration of the function
U (r) = tl In r +Z t In + (11)
r  z2  1
and of its derivative
U',()  2 ( 2 tl) (12)
shows that the equation
r +22 2 + Z1 (10)
7  z2 t1 7  Z
has a root r > Z2 when and only when
t2zi  t122 > 0, that is, z2 t2 (13}
21 t1
If a root exists there is but one, and it satisfies the in
equalities
Z2 < r < z z t2z t2 2 t11 (14>
t2 21  t1 Z2
The inequality (13) furnishes a negative criterion for the
applicability of the differential equation (1) to a reaction under
investigation. For if the reaction is governed by equation (1)
and if the observations are made with sufficient accuracy there
must exist a 7 satisfying equation (10) and hence the inequality
(13) must be satisfied. If, then, this inequality is not satisfied,
and hence no such y can be found; then either the assumptions
involved in (1) must be invalid, or there must be errors in the
observations. On the other hand, if (13) is satisfied we can only
conclude that (1) may be applicable, and we proceed to deter
mine whether it is so by computing y and a and comparing the
values of x computed by means of (8) with the observed values
of X.
40 ILLINOIS ENGINEERING EXPERIMENT STATION
r+ Z2 t2 Y + zi
3. SOLUTION OF THE EQUATION In  = In
'z2 1] r 3z
If the selected pairs (t,, xi), (t2, x2) of observed values satisfy
the criterion
22 t2
 < t (13)
z1 t\
we compute y as follows. Passing, for convenience of computa
tion, from natural to common logarithms, we have
r + z2 t,2 + z2
log  log (lOa)
r  Z2 1 r Z1
Assume now a value for y, say y = 71,
where z2 < r1 < Z Z2 (t2 2 tl Z)
t2 e3  t Z2
Then we may compute a quantity log N1 by the equation
log Nj = t log" + , (15)
Determine now a new value of 7, y = 72, by the relation
log r2+ 2= log N (16)
that is,
r2 + Z2 _1 I (16a)
orI
, + 1
r, = z N I_ 1 (16b)
Proceed now to determine a new approximation y3 from 72 in the
same way that 72 was determined from 71, and continue the pro
cess until its repetition produces either no change, or a change
which is negligible compared with the experimental errors. It
will be found that in general the process converges fairly rapidly
and only a few repetitions are necessary.
'This computation may be abridged by the use of Gaussian logarithms.
THE FORMATION OF CO IN GAS PRODUCERS
Suppose, then, that y has been found by this process. Then
a is computed by either of the relations
1 r +z 1 r+z2
a In , a  In (9a)
t, 2  Ze t r Z ,
or, what amounts to the same thing, if N is the last of the numbers
N1 N2,.... used in computing y,
1 T + ze 1 1
a r n Z=2 InN  log In 10,
t2  z2 t2 t
(17)
2.3026 log (17)
a = log N.
t2
4. COMPUTATION OF THE REACTIONCONSTANTS kl, k2, AND VERI
FICATION OF THE REACTIONEQUATION,
When the constants a, y have been computed we can find the
original constants ki, k2 by the relations
2 a
k = , k = (a +)a (1 r) (3)
2/ a 2 r
To determine the applicability of the reactionequation (1) to the
case in hand, we have now only to introduce the values a, y, just
found into the equation
z = 7 tank at, (7)
2 a r tank at
x = (a+1 + r tank at (8)
and compare the values of z or x obtained with those found by
observation. For this purpose equation (7) is the simpler, espe
cially when the z's corresponding to observed values of x have
already been computed.
ILLINOIS ENGINEERING EXPERIMENT STATION
5. CORRECTION OF THE CONSTANTS a, y BY THE METHOD OF LEAST
SQUARES.
The constants a, y obtained above are determined by two pairs
of observations only. It is of course desirable that all the obser
vations be used in fixing their values, as on account of experi
mental errors the values obtained from different pairs of observa
tions will in general not be identical. We proceed, therefore, to
correct the constants by the Method of Least Squares.
Let ( t,, x, );( 12, x2 ); * * .(, xn,) ben observed pairs of values of
t and x. The problem is then to determine a and y in such a way
that if
In (r + zi) 2 a t,  a, (61)
then
n2 = i 2 (18)
1
shall be a minimum.
In order that a and y shall make nv' a minimum they must
satisfy the socalled normal equations:
nd(62) n d8 n r i
2 da 21 \2 j  InY iz0/
n d(a) _. da
2 dr dr = (19)
1 dr
S i  zi 2z  ati 0.
These equations may be written
n n  \
2. a E .,+ _ = o . (20)
1 2 z2 1 rzi ( + zi
THE FORMATION OF CO IN GAS PRODUCERS
As their exact solution in their present form is impracticable on
account of their complexity we replace them in the usual way
by a system of approximately equivalent linear equations, by
making use of the fact that the quantities u, v by which a and y
differ from ao, yo respectively are small compared with
ao, 7o.
Substituting
a = ao + U
r = ro + V, (21)
we have
/ V
ro + zi (2
I + V
zi In ro  zi (22)
2 zi Yo + z
(1 2 Z2) ( +  ( + 2ro V V
0 i
S(o + zi 2To V
zE  In  \ 2 y o v + v/
1 (2  z ) 1 + /
Expanding the logarithms by Maclaurin's series and neglecting
terms of the order of uv and v2 in comparison with those of the
order of u and v we have
n V + V t =Z i In ro + z
n^ ^+ 2 i L In  2 a0 t,
1 1 0 i 1 To  zi
(23)
ti zi z o + z)
" i  +v = In 2 aoti1
0o 1 (r,z?) 1 (oz) roo ,
Expressing the natural logarithms in terms of common logar
ithms we have, putting
ILLINOIS ENGINEERING EXPERIMENT STATION
Ai  tj , Bi ? Z Ci =log ro + zi  Mt,
TO  i 7o  Zi
K= n210 = 1.15129 , M   1 0.86859 ao ;
n n n
, Fi A? + v J Ai Bi K> Ai C , (24)
1 1 1
n n n
u , Bi A, + v Ei B = B K BK a , (25)
1 1 1
or in the usual notation of the method of least squares  .
u[AA] + v[AB] = K[AC],
(20)
u [B A] + v[B B] = K [B C].
From these equations u and v are readily computed and hence
the corrected values
Sao+ U(21)
7 = ao + V
are found.
6. APPLICATION OF THE METHOD OF §§ 3, 4 TO OTHER EQUATIONS
OF REACTION VELOCITY.
The well known equation'
dx
S= k (1  )  (27)
with the initial condition x = 0, when t = 0, is reducible
by the substitutions
xz
z= rr^ x +  ,
a = k k2, k r , k (28)
r = k2
to the form we have considered, viz.:
'Cf Nernst, Theoretische Chemie, 5 Aufl, p. 564; 2d English edition, p. 568.
THE FORMATION OF CO IN GAS PRODUCERS. 45
dz a (4)
S (r ) (4)
with the initial condition z = 0 when t = 0. Its .integral is,
therefore,
In + 2 at, (6)
or z = r tank at, (7)
r tank at
or 1 + r tank at ' (8)
and the constants are determined by the equation
r + z t2 + z1
In =  Inr ' (10)
r  Z tl r  z
if the criterion
Z2 t2
zl tl
(13)
is satisfied.
The more general equation'
dx
 ki (a,  x) (b,  x)  k2 (a2 + x) (b2 + x), (29)
with initial conditions x = 0, when t = 0 is reducible by a
substitution of the form
S (30)
1+z
where p' and a are constants depending on a,, b,, a2, b2 but not on
1c, k2, to the equation
dz (  z'). (4)
The initial condirions, however, are now not
t = 0, z = 0, (5)
but t = 0, z =  zo ; (5')
and hence the integral and the equation determining the con
stants are more complicated.
'Cf. Nernst, Theoretische Chemie, 5 Aufl. p. 543; 2d English edition, p. 542.
ILLINOIS ENGINEERING EXPERIMENT STATION
The equation determining y is
r + Z2 2 1 + z 2 + zo
Irn 2  In  ( 1 In (31)
r r Iz \ 1 0
and the criterion for the existence of a solution is
t,2 z  tl z2  (t2  t0) z0 > 0. (32)
The solution, if existent, is unique, and is determined by
process similar to that of § 3.
7. CONCLUSION.
The analysis in the preceding sections furnishes a simple
negative criterion (13) or (32) for the applicability of the differ
ential equation (I) or (29) to a given reaction and, in case this
criterion is satisfied, provides a straightforward method of corn
puting the numerical values of the reaction constants k, and k2
from any two pairs of observed corresponding values of x and t.
It renders unnecessary, moreover, except as a matter of control
any observations of equilibrium conditions.
The detailed discussion of the more general equation (29) is
reserved for subsequent publication.