ON ISOTHERMAL REACTION VELOCITY I N HOMO-HETEROGENEOUS SYSTEMS I N T H E ABSENCE OF SOLVENT; WITH SPECIAL REFERENCE TO T H E CONVERSION O F FUSED MALEIC ACID INTO FUMARIC AND MALIC ACIDS BY KRISTIAN HOJENDAHL*
General Theory The reaction considered under the above heading may be represented by Liquid ----+ Solid or Gas. Gas --k Solid or Liquid. the latter at constant pressure. The actual reaction may be uni-or multi-molecular according as one or more molecules of the reactant are involved in it. If the reaction takes place between two (or more) mixed reactants the treatment, which follows, wilI apply only to those oases in which stoichiometrically equivalent quantities of reactants initially compose the mixture. For the sake of clearness we shall consider the case of a liquid (fused) reactant, which is giving rise to one or more slightly soluble resultants. Under these conditions we can regard the liquid phase as saturated with the products formed. Consequently the concentration of these products is constant and equal to the solubilities of the products. During the reaction the concentration remains constant, but the volume of the liquid phase decreases. As the concentration of the reactant is constant its activity must also be constant. Therefore the probability k, that a molecule can react in unit of time is CORstant. From this it follows, that the total number of molecules which react during time dt is -dN=Ntk dt where Nt is the number of reactant molecules present a t time t. The fraction of the total number of molecules in the system or, what amounts to the same thing, the fraction of originally reactant molecules No, which react during unit time is
If the molar fraction
(2)
is considered as representing the concentration
(volume-concentration has no meaning when applied to the composition of a *Danish Government Ramsay Memorial Fellow.
FEAC'IICN V I L O C I ' I Y W17H FUSED MALFIC ACID
759
heterogeneous system) the above expression is identical with the usual expression for reaction-velocity for a reaction of the first order. As IC is a constant even if the actual reaction is multimolecular it will be seen, that the reaction even then will appear to be of the first order. If more than one compound is simultaneously formed from the same reactant both the probability ICI that a molecule of the reactant can form a compound I and the probability kII; that it can form another compound namely 11, must be constant. The fraction of the original reactant molecules, which in time t has formed compound I is
-J'"(z)
=
I(
g ) k I dt = kI/(?)
t
t
dt
t
Whilst the fraction which during time t has formed compound I1 is
Hence the two compounds are formed in the constant ratio
(3
T h i s shows that the Wegscheider test is bound to hold fo; two simultaneous homo-heterogeneous reactions, whatsoeoer the actual order of the reactions m a y be. The kind of reaction to which the foregoing consideration may be expected to apply may be illustrated by two cases:I . The reaction between ethyl-bromide and dimethylaniline (work upon this system is in progress). 2. The decomposition of molten maleic acid. This is considered in the following section.
Conversion of Molten Maleic Acid Into Malic and Fumaric Acids Maleic acid melts a t 130-13'I and decomposes above that temperature with measurable velocity to form several products among which fumaric acid is the predominant one. Fumaric acid is precipitated during the reaction. According to Skraup' W c acid and maleic anhydride are formed as well. Tanatar2 confirms the main results of Skraup. Skraup finds, that maleic anhydride is formed in greater amount than would be expected from the equation: z maleic acid, + maleic anhydride+malic acid Skraup determines maleic anhydride as the part dissolved in some benzene present in the heating tube. Probably the presence of benzene has affected the process. In the light of subsequent results obtained by the writer, it would appear that maleic anhydride is formed to a minute extent by the following reaction. maleic acid maleic anhydride+water
a
Monatsheft, 12, 107 (1891);14, 501 (1893). Ber. 27, 1365 (1894).
760
KRISTIAN HOJENDAHL
This equilibrium is disturbed if maleic anhydride is removed by benzene. The water formed simultaneously can furthermore cause the following reaction : maleic acid+water 4 malic acid wherefore also malic acid is formed in greater amount if benzene is present. This probably accounts for the curious results of Skraup. By heating with benzene for 2 hours a t 120" he obtains 3 2 % anhydride and IO% malic acid, while by heating without benzene for g hours at 130' he only obtains 12% malic acid. Weiss and Downs' have investigated mainly the transformations of fumaric acid and maleic acids in aqueous solutions. Warburg2 has investigated the photochemical change: fumaric a c i d a m a l e i c acid. He finds that fumaric acid attacks glass, a phenomenon the writer has also observed .
Application of the Theory of Homo-Heterogeneous Reaction Velocity The following four reactions probably take place in molten maleic acid: (I). maleic acid + fumaric acid (11). z molecules maleic acid +malic acid+maleic anhydride. (111). maleic acid maleic anhydride +water (IV). maleic acidfwater +malic acid The equilibrium for (111) is so much in favour of maleic acid, that this reaction and (IV), which depends upon it may be neglected at present. If k in equation (I) is not taken to be a constant then ( I ) is a general equation for any reaction. In this case we therefore can write the velocity of reaction (11)above thus
e
Let Nft, Nat and Nht denote the number of molecules of fumaric acid, malic acid and maleic anhydride respectively, which during time t are formed from No molecules of original maleic acid. Then: N, = No- Nft- Nat- Nht Nt varies rapidly, due to the formation of fumaric acid by reaction (I), which is a swift reaction. kz depends on the concentration of maleic acid in the liquid phase, This does not contain fumaric acid but all the other components. The molar fraction of maleic acid a t time t with respect to the liquid phase only therefore is: Nt Nt fNat+Nht Nat and Nht are formed by the slow reaction (11) therefore Nat and Nht are small compared with Nt and also with the decrease in Nt. The molar 1
J. Am. Chem. SOC.44, 1118(1922).
* Sitzungsber. preuss. Akad. Wiss. 50, 964 (1919).
REACTION VELOCITY WITH FUSED MALEIC ACID
761
fraction therefore will remain nearly unity and kz will with approximation be a constant. This means that reaction (11) as well as reaction (I) approximately will be of the first order. For the same reason the Wegscheider test will hold for the same two reactions. Analytical Methods As to the analytical methods all previous investigations have determined fumaric acid as that part, which does not dissolve in water, allowance being made in some cases for the solubility of the acid in water. For the determination of malic and maleic acids several methods have been developed, but the authors themselves state that these methods zre not very satisfactory. Therefore a method has been developed depending mainly on electrical conductivity and titration of total acid. Briefly the method consists in taking the melt (0.05 gram molecule) dissolving it in so far as it will dissolve in 2 5 cc of water a t 25' . This takes several days. It is then filtered, the filtrate diluted with conductivity water to I or 1 / 2 litre and conductivity and titration-measurements carried out on this, the malic part, of the analysis. A certain amount, of fumaric acid is dissolved in this part which amount is obtained from solubility data. The undissolved residue consisting mainly of fumaric acid is then dissolved in hot water and diluted to I or I / Z litre. This is sufficient to keep the fumaric acid dissolved when cooled. On this part of the analytic mixture, which is termed the fumaric part the titer and the specific conductivity are determined in a manner quite similar to that employed in the malic part of the analytical mixture. The experimental values for specific conductivity and total acid are then employed in the formulae deduced in the following.
The Specific Conductivity of Mixed Aqueous Solutions of Inactive Malic, Maleic and Fumaric Acids Arrhenius' has considered the ionisation of mixed electrolytes and outlined the theory of isohydric solutions. F. Barmwater2 has calculated the conductivity of mixed solutions of two weak acids on the basis of the mass-action equation. He arrives a t a third order equation. In the treatment given below equations of the third order are to a large extent avoided. The terms occurring in the various equations employed are as follows:c = the specific conductivity of the solution. 1000
K = -= a constant experimentally determined to be
2.5 75
PW c,,c, and cf denote the concentrations (moles per litre) of malic, maleic and fumaric acid respectively. Z. physik. Chem. 2, 284 (1888). Ibid. 45, 557 (1903).
762
KRISTIAN HOJENDAHL
k,=3.99X10-~; k,=1.34X1o-~ and k f = g X 1 0 - ~denote the first stage dissociation-constant for the acids. ya, ym and yf denote the degrees of dissociation of the three acids respectively in the mixture. 7 =the total acid (moles per litre) determined by titration. To render the treatment feasible it is necessary to make certain reasonable assumptions. In the first place the conductivity has to be determined a t concentrations, at whiqh the gas laws hold. In the second place the second stage of dissociation has been neglected as being very small compared with the first. In the third place the mobilities of the three acid anions have been assumed to be equal. This of course only holds because of the very similar size and constitution of the acid anions. In consequence of this the specific conductivity of the mixture may be put proportional to the concentration of hydrogen ions. The proportionality constant is K. Therefore UK= cay,+ cmYm+ CtYf. (1) Applying the law of mass-action to the dissociation we get:
and similarly for the other two acids. Substituting uK for the concentration of hydrogen ions: yaaK _ _--ka 1 -Ya
or ka and two similar equations. aK+k, Substituting these values of y,, y m and yf in equation Ya=-
g
K
c,ka = e aK+k,
cmkm aK+km
(I)
we obtain :
Cfkf +-aK+kf
(3)
One more equation is needed. This is obtained by titrating the solution: 7 = ca+ Cm+Ci Whence : c, = 7- ci- c, Inserting this in equation ( 3 ) we obtain:
uK=-
c:, .ka
aK+k,
km aK+k,
+ (7-ci)
(5.)
cakm I C f k f . aK+km aIi+kf
From this the following formula for the determination of the concentration of malic acid in the mixturc is obtained:
REACTION VELOCITY WITH FUSED MALEIC ACID
(7-Cf)krn+---cf kf uK+krn uK+kf km ka uK+km uK+ka
ca =
7 63
uK (6)
As mentioned earlier cf is known from solubility measurements. cm is then calculated from ( 5 ) . The expression to be used for the fumaric part of the analysis is obtained from the above by changing the suffixes and neglecting the very small magnitudes containing c, It is:
km
uK+krn
kf. uK+ki
To verify the formula deduced, conductivity measurements of mixed aqueous solutions of maleic, malic and fumaric acid were carried out. The ordinary Wheatstone-bridge method with telephone was employed. The results are given in the following two tables. The temperature is 25Ofo.1 Cm
0.0495 0.0445 0.0396 0,0347 0.0297 0.0248 0.0198 0.0148 0.oogg
0.0000
0.0049 0.0099 0.0148 0.0198 0.0247 0.0296 0.0345 0.0395 0.0444 0 0493
0.0050 0.0000
'
Cf
Cm
0.005
0.045 0.035
0.005
Experimental Calculated conductivity Conductivity
Ca
Ca
0.000
0.010
0.005
0.025
0.020
0.005
0.01g
0.030
0.005
0.010
0.035
0.005
0.005
0.005
0.000
0.040 0.045
0.007746 0.007293 0.006850 0.006310 0.005740
0.007750
0.007270
0.006785 o ,006272 0.005730
0.004550 0.003880
0.005140 0.004515 0.003840
0.003I 2 3
0.003IO5
0.002373
0.002377
o.001630
0.001645
0 .005150
Experimental conductivity
Calculated conductivity
0.007266 0.006247 0.005og4 0.003843 0.003I43 0.002439 0.001746
0.007265 0.006293 0.005178 o.ooggoo 0.003198 0.002459 0.001748
That the experimental conductivities in the latter table are rather low is probably due to the attack of fumaric acid on glass as observed by Warburg.
764
KRISTIAN HOJENDAHL
The Reaction-Velocity Measurements The materials employed were kindly presented by Messrs. Weiss and Downs. The maleic acid was purified by recrystallisation from acetone solution. The investigations were carried out in a gas heated and electrically regulated thermostat containing liquid paraffin. The bulb of the thermoregulator was filled with air and the temperature was kept to 0.5" The thermometer used was specially made and was calibrated by the National Physical Laboratory (England) to 0.1" Owing t o the effect of fumaric acid on glass as stated previously the vessels containing the reacting mixture were test tubes of fused quartz. The walls of these were very bad heat conductors. Hence the charge (5.8 grams maleic acid+ I gram fumaric acid) was fused over a bunsen-burner, whereupon the tube was immersed in the thermostat and left there for a given time, after which the tube was taken out and the contents cooled suddenly by adding 2 5 cc of water. The mode of analysis has already been described. The results of the measurements are collected in the following tables and diagrams. TABLE I Temperature I 39.6"C.
.
time sec
546 3270 6980 12420
fumaric acid 0.052
malic maleic maleic kuni kuni acid anhy- acid fumaric malic dride 0.008 0.008 0.93'2 ~ . ~ X I OI -. ~~ X
0.213 0 . 0 4 2 0.042 0,703 6.98 0.368 0.080 0.080 0.472 6.91
0.511,
0.105 0.105 9 . 2 7 ~7 . 2
average
7.2
2.2
.so
3.2
.40 I .S O X
5 .o
I
Wegscheider test
I O -1~. 6 X 1 0 - ~ 6 . 5
1.54 I
x
kbi
malic
5 so7 4.60 4.77 4.85
TABLE I1 Temperature 149.8"C. time sec
990 3480 7200
14190
fumaric malic maleic acid acid anhydride
maleic acid
kuni
fumaric
kuni
kbi
malic
malic
Wegscheider test
0.138 0.033 0.033 0.796 1 1 . 4 X r o - ~3 . 4 X 1 0 - ~ 4 . 8 X 1 0 - ~4 . 2 0.312 0 . 0 8 2 0 . 0 8 2 0.523 10.9 3.3 6.4 3.78 0.490 0.131 0.131 0.248 1 2 . 5 3.1 11.9 3.75 0.657 0.153 0,153 0.038'. 4.3
average I I . ~ X I O ~- ~. ~ X I O - ~ 4 .o The values for maleic anhydride are calculated on the assumption that malic acid and maleic anhydride are formed in equal amounts according to the equation 2 molecules maleic acid +malic acidfmaleic anhydride. Reaction velocity data were likewise obtained in glass vessels. They are not as exact as those quoted and are consequently omitted here.
'
REACTION
VELOCITY WITH FUSED MALEIC ACID
765
Discussion of Results It will be seen, that the measurements give results, which are in close agreement with the theory. Especially the curves for malic acid are unimolecular with an exactness equal t o the experimental error. It is surprising that the fumaric acid especial-
ly in the latter table shows a deviation from the unimolecular scheme, which is definitely outside the experimental error. This deviation may be due to the reaction : Maleic acid maleic anhydride+water the equilibrium for which at the lower temperature 140' is entirely in favour of maleic acid, as shown by the applicability of the preceding theory. At I '05
DIAGRAM 2.
it may be shifted sensibly from left to right in the earlier stages of the run. Later on this reaction will go back again partly because of the disappearing
766
KRISTIAN HOJENDAHL
maleic acid and partly because of the maleic anhydride formed by the reaction: z moles maleic acid --t malic acid+maleic anhydride At 140' the dissociation of maleic acid into anhydride and water is considerably smaller although appreciable. The velocity-constant for fumaric acid therefore is a bett,er constant. At 160Oon the other hand the dissociation is so prominent that,the vapour pressure of water +anhydride is equal to I atmosphere. At this temperature maleic acid boils with decomposition to water and maleic-anhydride vapour.
The Infra-Red Absorption Band The critical increment or energy of activation of the reaction: maleic acid +fumaric acid can be determined by help of the Mnrcelin-Rice equation:
d loge kuni -- E dT RT2 Inserting the average values of kunifor fumaric acid from the tables the following value is obtained: E = I 5800 cal From this the position of the infra-red absorption-bandl is calculated using the equation: E=hv A wave length of X = I . 80p is obtained. This has been verified experimentally using a Hilger infra-red spectrometer and a radiomicrometer2. As source of radiation a Nernst glower was used. . A cell containing fused maleic acid was obtained in the following manner. Fused maleic acid was placed between two heated sheets of quartz and allowed to cool. The sheets were then placed between the Nernst glower and the spectrometer. After some time the central part of the film of solid maleic acid melted. By regulation of the current through the Nernst glower it was arranged, that the central part was kept just molten. The edges of the film were kept solid by screening the radiation, the solid maleic acid thus formed a part of the cell. As solid fumaric acid was formed in the melt no attempt was made to obtain any extinction-coefficient. The result is shown in the appended diagram (3), which shows the deflections observed in the case of three samples of maleic acid. The fourth curve shows the deflection observed for the case of the Nernst glower itself. In this case the width of the slits has been reduced so that the observed deflections might be comparable with the other readings. Curve I V gives the shape which the deflection curves would possess, if no selective absorption were present. Such curves are represented by dotted Lewis: J. Chem. SOC.109, 796 ( 1 9 1 6 ) . See Taylor and Lewis: J . Chem. SOC. 121, 665 (1922).
,
REACTION VELOCITY WITH FUSED MALEIC ACID
767
lines. The divergences represent the selective absorption. The observed bandheads viz. 1.80,1.85 and 1.7jp are in good agreement with the calculated value 1 . 8 0 ~ . No other band was observed in the region 0.8 to 3 . 5 ~ Beycnd 3 . 9 there seem to be more bands, but the deflections for this region were too small to give any trustworthy values. I n connection with this work it was observed, that maleic acid is dimorphous. The crystalline film formed by the solidifying of the fused maleic acid changes after a short time into a white,aggregate of crystals. Dimorphous maleic acid has been observed crystallized from different solvents by Biilmann and Lund.
On the Absolute Value of the VelocityConstant of the Transformation of Maleic to Fumaric Acid It is of interest to compare the observed u n i m o l e c u l a r velocityconstant of the formation of fumaric acid from fused maleic acid with that calculated on the basis of the radiation hypothesis of thermal change and also with the empirical equation of Dushman, the latter being known to agree satisfactoriiy with the unimolecular (but catalysed) velocities for the decomposition of gaseous phosphine and gaseous nitrogen-pentoxide DIAORAV respectively . As pointed out by W. C. M. Lewis‘ on the assumption that the rate of absorption of energy by a Planck oscillator is continuous it follows that the unimolecular velocity-constan t should be given by: RT
kuni= 2 .46j X 1 0 - ~ ~ n ~ ~ ~ ~ Dushman’s equation is: . -E RT
kuni
=
In the case now under consideration the experimental values of E and v axe :
E = 15800 cal; v = I .667X10I4 per sec. Dushman’s equation gives: ku,i=7.8X106 Phil. Mag. 39, 26
(1920).
768
KRISTIAN HOJENDAHI,
Lewis' equation gives for n = I : ku,,i=3.2X10-*
whilst the experimental value is:
k",i
= 7 . 2 x 10-6
The observed velocity-constant lies nearer the equation of Lewis than that of Dushman, but there is no quantitative agreement. It is the first known case in which the observed velocity constant is less than that given with the radiation-hypothesis using Planck's oscillator. It is to be concluded apparently that the rate of decomposition is much less than the rate at which active molecules are formed, i.e. the majority of the active molecules become de-activated instead of decomposing. It is possible that in a homogeneous liquid state we are here dealing with a case in which collisions are significant for deactivation.
Summary (I). An expression for the reaction-velocity in homo-heterogeneous systems has been outlined. The expression arrived at shows that the process will apparently be of the first order even if the actual reaction is multimolecular. Hence the Wegscheider test necessarily holds for two simultaneous homoheterogeneous reactions. Catalysis generally will give a deviation from the first order. (2). An expression for the electrical conductivity of aqueous mixed solutions of maleic malic and fumaric acids has been deduced and tested experimentally. An analytical method based on electrical conductivitymeasurements and titration of acid has been developed for the analysis of a mixture of maleic, malic and fumaric acids. ( 3 ) . The rate of formation of fumaric and malic acid respectively formed from maleic acid in the molten state has been determined experimentally at the temperatures 140'C. and 15o'C. In accordance with the theory outlined above both of the reactions: maleic acid +fumaric acid and 2 molecules maleic acid -+ malic acid +maleic anhydride are found to be of the first order. (4). For the case of the reaction maleic acid +fumaric acid the critical increment and the position of the infra-red band have been calculated. The latter has been determined experimentally. A good agreement with the calculated value is obtained. ( 5 ) . The observed velocity constants have been compared with certain theoretical expressions. The author wishes to express his thanks to the Ramsay Memorial Fellowship Trust and the Danish Committee for the award of a grant. Muspratt Laborator!/ of Physical and Electrochemistry, lrniversify of Liverpool.
