Prediction of
REACTION RATES Farrington Daniels UNlVERSlTY OF VISUWNSm, MNIISON, VIS.
Ekamination of about thirty recently published, accurate unimolecular gas-phase reactions shows that the simple Arrhenius equation, k
-
Sixty per cent of these reactions ham frequency factors, a, of 10” to Id’ corresponding to RT/Nh. Larger frequency factors suggest chain reactions and smaller mlues often indicate that there may be an appreciable entropy of actimtwn. The kinetics of the rearrangement of certain allyl groups illustrates the f a c t that the requirement of a n improbable
shape in the formation of the actimted molecule decreases the entropy. Empirical and theoretical rules f o r predieting reaction mtes are examined and illustrated, including the Eyring equation: RT k-K-e
is applicable.
N ORDER to react, chemical
substgnoea must 6mt be SGtiVated by the introduction of energy. In ordbry
rapidly moving m o l e c b t h m whioh have energy in the range from a b u t l0,OOO to 100,OOO doriea per mole. The w d known distribution of energy in a group of molecules i indicated in Figure 1. The pmportion of these faat molecules is exceedhgly d, but their pre8ence accounts for the pmgres of chemical resotions. The numbs of rapidly moving molermles inoresea exponentially ae the tempsrsaua is ind , and in a rough way the quantitative application of tbie Atim the empiriosl appmximstim that the rates of many Wd reactions mt Mom hpmatum double or tsebfe for 6 100 c. rise in hperatlue.
AS
--w
Nh
I n checking these rules it is importani t o look f o r ‘‘hidden” constants and t o avoid disturbing factors such as trace catalysts, dissolved air, moisture, light, and wall effects. Rate measurements which depend on a nonspecific property, such as total gas pressure, may be unreliable. atamic distsnoea before new products can be formed. The mum of a bimolecular reaction is indicated in Figure 2. Such complaxea have only a brief exiatenm, surviving wmetimes for about 10” w n d , but their pro+ c8n be calcu-
C
4 . - /\
D
-
I _ -
A
A
B
B
A
B
Figure 2. BbnObeuhr Ractios, ShOaGinn Formatiool of.4ctivatur Cornpbi
lated indirectly Byen if they csnnot be isolated and mew. ured. There isan ~uilibriumc o n c e n b t m o f t b a a a t i vatadlllohuh which osabe calC~WbytbeOrdineryRllea
.
o
f
~
~
c
a
aFo
-
:
-RTLnK
(1)
Ma- A X - T U . .
.
,
.
.
m % i d v a i e d molecdes m y be regarddm de6nita . . c o O n p k mW&and am called by Eyring the “ d v W l compW’; .€tis~ ~ ~ E StoWapply Y energy to mwd b r e .ctents.tcptherinto an aativated c o m p k with suitableintar’
504
‘
K
g*n’ -8’
ERT
(2) ~
(3)
We cannot measure equilibrium constant K for activated mateculee any more than we can .meaaure K‘ for’.the equi-
’.
INDUSTRIAL AND ENGINEERING CHEMISTRY
VoL 35, No. 5
1
e*@,
liirium between h e , carbon, and at room temperah; but we can calculate it from heat content AH and entropy AS by formnlaa 1,2, or 3 just aa we do for benrspra
y
Pnducts
.
lime
F i y m S. A a i m t h n Enorff and Heat qf b t f o n
We cannot mess~reAH dorimatziaally because we are unable t~ imlata the tleeting activated complex, but we can
dculate it aa we d d t e AR thermodynamicallyby plotting the logarithm of eqnilibriumconstant K agsinst 1/T and t&ing the slope of the line. In kinetice we plot the logruitbm of the d o n rate, k,sgainst 1/T and ddtetEe heat of activation from the slope of the line. Thia heat of actition usually bears no relation to the best of d o n , which is concerned with the d&rerm in energy between reaotanta and pmduds without rederenee to interm* 8tagw (Figure 3). A mweeaful-formula for predicting reaction rate% is now available. Eyring's equation (6) follow6 directly from ststis t i d meohsnios: (4)
It is usually sssumed that the probab~tyconatant K knot far from unity, and the evaluation of the heat of activatiou A" and the entropy of activation A S becomes the important taek of theoretical kinetics. Before exploring this equation further, it will be profibble to coneider the empirical Arrheniun formula which p d e d it: -Q
(5)
k=aem
Here the activation energy,Q, ia usually obtained by determining the specificrata constant k at mer4 temperaturesand taking the dope of the line obtained by plotting In k againat 1/T. The fmqmcy fador a has the eame dimemions aa k, since B-Q/'~ is dimensionless. In nnimoledar reactions it oftenhas values of about lo1', which is of the order of tude of the vibration fmquenciea of stoma in a molecule. Thin formula has dwell in recording &ply and accurately the in9uenc-aof temperature on the specificrateof moetahemid reactions whether in the gaa p b or in solution. An empirical equation can always be i m p r o d by intrcduoing more constsnts, and so a t am p waa intrcdud into the Arrheniu equation wbich allowed for orientation at coEdon- lock-and-key &e&: k
-
INDUSTRIAL REACTION RATES
or about 101s and that representing the entropy of activation,isredly aprobabilityfaotorwhichcanbew~ected with upatid srrangempm. T6a fsctors K(RT/Nn)cas''R are grouped together and called the ''frequenoy fsctn". In bimohuhr d o n a the d u e of the frequmey fnetor is not far fmm the number of collisions. It variea with the molecular weight and with the temperature but is um$y in the neighhhwd of lox1, when the c o n m t r a h am torpresaed in molea par liter. In many unimolecular gakphasa d o n a there is little change in entropy on going into the activated state, BO that kY" is nearly unity. Then the frequenay fis about 101s. In most unimol& reactions one of the molecular bonds Simply breslrs without having to form a Snothermolede, 80 there is no change in need for a pmhability term. There are, however, a few exceptioIlk One lime to be fairly typical (4) is the marrmgement of l-cy~le hexenyl aUyl malonitrile wbioh can refractive index meamnwnenta. It reaction in which substituting into Epustion 4,
Solving this equation, the entropy of activation AS'is -11.7
entropy units. Tbia decrease in entropy on activation ia in agreement with the fact that the reaction seema to d for a special spatial arrangement. In other words, mme of the degrees of freedom are frozen in the formation of the activated complex. Apparently the mymgement is accomplished by the formation of an intarnal rmg aa shown in Figure 4. Several other malonitrilesand relstedcompounds with similar remmgem.enta show tbe m e derrease in entropy of activation. The dflemc-a in mamngement rata of the different compounds depends chi+ on the heat of activation;
H CtHi CN
C
I I H
I W N I E-CLH I C-H
I H-C-E \
-Q p a s im
Examining Equation 4 (fromEyring), it is seen that RT/Nh baa a value at mom temperatwe of 8.3 x 6.0 X
May, 1943
1w x a00
lo'* X
6.5 X
lo-"
INDUSTRIAL AND ENGINEERING CHEMlSTRY
H c . H s CN
I I
NC -
I I
H-C-H
H-C
H
H-C-H
4
1
1
even a smsll change in Q mskes an enormow ditierenae in k. The exphnential charmfar of th~equation greatly magdies any error in the activation energy. Theoretically, if we know the energies of the reacting molecules at all p d b l e interatomic distances before and after paeshg through the activated complex, we can 6nd the minimum d u e of energy required to bring abaut the cbange. This is the activation energy. The energy conbur lines for a procans similar to that shown in Figure 2 are indicated in Figure 5. Moleoules AB and CD applosch esoh other, form an activated complex, and separate into AC (not shown) and BD. The wtivation energy in the height of the lowest “energy pass” between the two valleys which correspond to the stable coniigurations of the reactants and the h a 1products. This procedure givw us the desired means for predicting reaction rates 4f only we can find a way to draw in correctly these eqrgy-contour lines, particularly in the region of the energy ‘‘pw”. The means for calculating these energies are quite unsatisfsct4ry. One approach is the 8emiernpirid calculation of Eyring. In thia calculation there are four ammptions: ‘1: It is aenumed that a ploleeule, rnore’partieularl the e v a t e d compkx, behaved m If I t were made up of eevedatom
Interatomic D/stoncc
psus.
2. Energg-distancs curves are calculated for each pair of atoms. These FranekCondon c m can he calculated empirically by the Morse uation They we of the general type ahown in crocn section at”B rFi& 51. Thev mcalculated fmm
F i y r a 5. Gntour Ma S M n g E - q y at T r e n t Interatomic & t a w as ~ o h l e A Appmacha Molecule CD and R-ta to Giw DB
G.r . o o L l o ~.hmn R.nck4kdon -ne.
but there in a p d e h between heat of activation and entropy activation,an increase in heat of activation being BC companied by an increase in entropy of activation. If tbe reaction involves chains, the nver-all &ion will go too faat for its normal activation energy, and the frequency fnctor will be too large. The existence of chains is fairly common at higher temperatures, and a chain reaction may still look like a 6rahrder reaction. For example, in the decompo&on of a hydrocarbon, the reaction may go through free radi d s in the following manner (8):
of
CHa + CH, + C&+CE, + GE,
Wa
(1) (2)
C&
+
When a large supply of resotants is prasent and a steady state in reached,
dt
&CE,
-
kiccim
- koarC C , ~+ k c m
-
-
~ C C U .CC~E,
=0
From these rate expremions it can be shown that, since the first reaction is slow compared to the other reactions which involve free radicals, dt
= k,caa.
dg -
kcaim
(7)
This is the equation for a firahrder reaction. ACTNA’L’lON BNBllClES
The difficulties in calculating frequency factors in the Arrhenius equation are bad enough, but the diiKiculties in calculating aotivation energies m much more serious hecaw 906
pair is obtained from calorimetric measuramenta or from apeetmEO ic measuremenis the diatance between atoms in the normal m o L is obtained h m electron di6ractlon or x-raya’ and the or band nbmtion freauencv in obtained from Raman. &. spectra Butthe- n & o n i s ~ r n @ i c a l a t W ’ a n d i s ~ t o b e in enur by hto-% of oslones m the nei&hrhood of dissociation energy. Errors m t h calcnlahom ~ togather with an im roved formula kva been (8). bll the attractions b e t w 2 E E i C naira. taken two at a time, are combined with the help of Emborder tion calculations. Even for foar atom with six p o s s i b e pairs, the computation in approdmste and tedious. Fortunately, it in not neaeaanr to cover the whole moutaioside with contaw lines: they areneeded only in the region of the lowest pess bet& th6 two valleys. With a little pmtiw the amount of calmuhtioncan
t.
ry,
be m t l y r?luced. d job of calculation with n o m It is 1 7) a x ahduuds o rrulers muiw with miections of the ener&ked.at d i 6 e r d distances dong ihehoale. A aetting is m e.and the energies are added up. ,Aftq perhaps thirty m t tlngs it 18 posslble to 6nd the one whch gwea the lowaat total enem-melv. the activation enem. 4.-AU pah8-d atoms are held t&-ther both bv homowlar or quantum mechanical athotion (thekind of at&tion @ociated with the eleomn pair) and also by ionic or ele&rwtaticattraction. In some moledes the homomlar form ia the crreater. in
attraition is due ta homopolar binding and which in due to dectroatatic binding? H in awh a h p l e moleoule that it is ruegible to make reiia&%%h tedioua. calculations on the different types of attractiod. In‘the hydro@ molecule it turns out that 88 r cent of the total athction 18 homo and 14 cent is eEtmstatic; Eyring thst the &2cent aL& to all atom paira. This d o n is not very e&, hut no tter If the true value IS 10 or 20 per approach has et bean o&ed. cent instead of14, aerious errors will he introducedinto the calculation of specific rate constant k. The uncertainties in the calculation of activation energies by Eyring’s semiempirical method are su6iciently great to justify simpler rules, even though they reet on I- secure foundations and can be regarded only as empirical. In the Morse curves the largestfactor is the heat of dissociationof the atom pair, the other two factors being relatively leas imprtmt. Let us neglect them entirely. However, we cannot take all the heat of dissociation of the broken bonds because, in the activated complex, mme new bonds are formed too, and tbe energy of their formation in the complex must be
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 36. No. 5
1N ID U S T R I A 1. HE.\ C T I O N R A T E S
+
considered. Turning to the reeation Ha L -+ ZHI, which seems to be our beat known bimqledar reaction,Hirschfelder
(7)finds that about 0.28 of the energy required to b d the bonds givp the known energy of activation, and he . w m w that this %per cent canbe oarriedover to fdd o n s : AB+CD+AC+BD
-
AEwwsu,
+ DCD)
= 0.28 (DAS
(8,
where DAB,DCD energies repuiredto diesmiate m h g mol* d e s AB and CD into atom These dissocistion heatd of atom pairs are subject to considerable uncertainty BS 'one tziw to assign a d a t e bond enerKv to them, ' of the molecule in whici~they are found. N e v e A E are useful for approximate wtimates; a few of these estimates are included in Table I.
ondly, the theories may be e b k e d through a &at&id study of the h e number of readion rate determ$ptiona recorded in the li&tur8. It may appear simple to check the calculations of activation by reterenoe to the literature, bit it ia not simple. Unfortunately, the reBction rate data are m inaoourste and the mech& 80 oompliwted tbat mnchreaaloulating and diRcarding of complicatd data and evaltlatiq of q p r b e n t d accuracy are y to determine whetherthetheoriea are in agreement mth expmkentd facta. The greateat need now in chemical kinetics is for accurate data on uncomplicated unimoleculsr and bimolewlar reactions over wide temperature intervals. The di5cultiea inherent in the experimental determbtion of activation energiw are due to &equate techhue and to incomplete methods of calculation. Theee points may now be elaborated.
e
EXPERIBISM*L tdSTHOD8
In molecules which can have wed di5mt BLectmnio configurations, such BS exist when conjugated double bo* are present, the phenomenon of mawmnce wurs and the heat of dissociation may deviate largely f q m that calculated by Table I. A c e to Hirschfelder (7)in the dill simpler c88e of atom reactions of the type, A + BC- AB C (exothermic) (9)
..
Activation energiw are evaluated by ddmmwng the rate conatant9 at mereut tempera(uree. U n f o r t u n s ~ in , gas resetions (which are sometimes thougbt to be the simplest) mod of the reaction rates have been followed by the total gaa pressures at hquent time interwla Tim LE a nonspeci6c method in which a oertein reaction in amumed to take plsee; on the basis of this assumption the amount of original mahiddemmpondis eslrmlatedfmmthetotdpmi+ m. Sometimes this metbod givea correct d t a , but if there are competing reactions or sdsorptions on the wall 01 if the d o n does not alwayn p d as amumed, them may be
+
the heat of activation is equal to 5 per cent of the energy of breaking the bond: AIfaetiv.~ 0.06 DBC The activation heat is quite small anyway, 80 that a close calculation is not signi6cant. All these methods seem to be fairly eatisfadory in the dosen or 80 c a w where it is now poesible to check them with experimentally determined activation energies such BS in the following resotions (7): HI +I*; HS Br,; L CL; Brz m,etc. According to another useful dation for making &tea of activation energiea the activation energy in endothermic reactions m& be at least as great as the endothermic heat of resetion. It may, of conrse, be much greater. According to a recent article (6)the activation energy may be calculated by the formula, Q 2.8681 7 (35.5 m.465 )
-
+
+
+
-
-
0.
where the spectmcqic heat of dissociation, D., of a bond in a molecule undergo+ unimolecular decomposition is dated to a freqnency of vibration, T, and the ~ r m w W c abond l energy, 00,by the relation: D. = 1.4295 +Do (11) A nomograph for solving Equations 10and 11is available (11). In general, there are two ways of t6sting the theories involved ih the prediction of reaction rates. First,they may be, checked for very simple &ems such BS
H
+ E ru.
+
Hr
-+
H
where all the computations can be made rigorously even though laboriously. This type of reaction is usually too simple to be of much practical inkreat in ohemistry. SeeMay, 1943
b e r m F.Cmr
F i p m 6. FreqFmtor a Obtained from Tmenty-3ht S Z k , Unimobcuhr Gas Reactioru wbU. in Chemical Abstracts, 193641
sarious errors, particularly when the meaeurements are carried down to low preealms. It is poEsible that the pmgreas of chemical kinetics bas been bandicappd beeause the experimental reeulta were more complicated than they needed to be-mnplicated by inadequate methods of meawtrement rather than by the nature of the chemical d o n s t h d v e a . One example may be cited. For 6 i b n yeam it hss been 80 capted tb& photosynthwis in the gmwing plant b aneffioienf prow8 rqniring ~ o m e t h i qles% than four qusntaper atom 01
NDU.STRIAL A N D E N G I N E E R I N G CHEMISTRY
carbon produced in the plant. It is now known that the d u e is about ten instead of four. Apparently the difficulty lay in the fact that in the earlier megsuraments the total pressnre wae measured,and it ww BBwuned that the ineraaas in pressure was due to the evolution of o w , but under the special conditions of the experimmt an unexpectd evolution of carbon dioxide took place also. If meaeuraments of total pressure are to be used in rate m-enta, they must be checked by chemical analyak. SpciIic methods for determining the concentration of just one of the readants or pmdncta are much needed. Phpical methods which do not dieturb the regding Bystem are best, such 88 the absmpton of light by one of the substances pregent and the application of Beer's law. Thin method can be extended to the i n f r a r e d and the ultraviolet. New analytical took which are speoiiic for a single compound are eagerly sought. Such an instrument is the dropping mercury elecW e or polarugraph which, by previous calibration, givea simple galvanometer madings proportional to the c o n c e n b tion of one of the rcactsnts or pmducta. The m o d of amp lea followed by quick chilling is not a bad method for following the course of the reaction, if the re. action is going so slowly and the chilling with solid carbon dioxide or other mean~ is so rapid that there is no appreciable change in concentration during sampling. In another eatis factory technique m a l db u l b of reacting material are started together, and each is chilled quickly and analyzd after a difTerent time inbrval. Again, flow methods can often be recommended. It is true that the calculation of the time a given molecule &ys in the heated chsmber may be complicated, but often this d i 5 d t y may be lesa than the difficulties involved in the other methods. Thin method has the a d m that it is possible to accumulate &cient quantitiea of materisl for satisfactory ehemidanalyais. In memuring reaction rates it is necesswy to be on constant guard against super6cialinfluenoes which may &ect eriously the course of the reaction. Among them may be counted walk, moisture,air, and light. The wall e5ecta in gas reactions may be m wrious 88 to dqualify the argument that gaspha€a reactions are the eimpleat to interpret and the beat for testing theorieu of chemical kinetics. There may be a considerable contribution by a hetemganeous reaction on the walk; or fresradicals from intermediate stsgee in the decomposition may be trapped by the walls. If activation energieu are to be dculated, it is particularly important to have the conditions of the walk always the name at all k p e r a t d t h e r scrupulously clesn or deeply coated. Traees of moisture may give profoundly altered effectsdue to the p m c e of polar moleoulea on the walk in gae reactions, or to the presence of an ionizing solvent in nonpolar o&c solvente.. Again the presence of hydrogen ions may cauea appreciable catalysis, and it is likely that some c.wa of catalytio behavior of d t s such 88 calcium chloride and dnc chloride am d y due to hydrogen ions produced by hydmlyak with traceaofmoisture. Oxygenoftheaircan caw many diffioulties. It resds readily with free rsdioals and can be responsible either for inhibiting a reaction or for derstingit- Forexa~nple,thephotabro& tionofoinnamicacidincarbntatrmhh 'de s8
is speeded up more than tenfold by simply drawing out the diesolved air with an aspirator. The e5ecta of the catalysts or inhibitors may be greatly magnified in the CBBB of c h i n reactions. Daylight in an ordinary laboratory is Bometiunes su5cient to caw appreciable reaction when chain reactions are involved. DIFFICULTIES OF CALCULATION
Most laboratory reactions are mixed reactions, complicated with competing reactions, coasecutive reactions, or reverse reactions; and it is not m y to determine the extent of these complications. Only a few of these complex reactions are simple enough to permit calculation of the over-all rate by intqtation of the differential equations involved. A mechanical integrator would often be helpful. Unfortunately one cannot be mre that a given solution is a unique solution. There may be emera1differentcombinations of re. actions that will Bive the m e o v e d rate. Many reaction rata data have been oversimplified by forcing the data to fit a simple first- or Sscondader equation or by taking the measurementa over too Bhort a time interval; It is true that, after a reaction is three fourths completed, the calculational emre involved in determining the rate constant become very large,and the situation may become Beriouely complicsted by the accumulation of decomposition products. However, if the rate measurementa are carried over only a short fraction of the total reaction time, the conclusions may be entirely inadequate and the calculation of the apparent activation energy may be seriously in error. The difficulty of calculating signiscant activation energiea for mixed reactions is evident. The graph obtained by plotting log k for the o v e r d mixed reaction against 1/T may he a straight line, and the apparent activation energy obtained fmm ita slope is useful in predicting the influence of tempem thus ture on the mixed reactions, but the activation ene obtained is not Bignificsnt in checking theoriea of xrnical kinetics, or in studying the influence of molecular structure or
INDUSTRIAL AND ENGINEERING CHEMISTRY
Val. 3s. No. 5
aubatituting groups or solvent8 on the rate of the reactions reactimare d l y d, and the gss reactionsare free fmm solvation decta which sometimescomplicateeither the activaIf the OM o d reaction involves the summation of tion energy or the frequency fsrtor or both. two separate reactions I and E,then the apparent activation C M A6sirads from 1936 to 1941 contain about 2.500 energy includes the activation energy of both I and II. itema on d o n velocity; of theee,400 or 500 will be intuest. Often I is much slower than II, and the activation energy of I1 then becomes negligible in cornpaxison with the activation ing fmm the stenapoint of cl~emicalkinetica, and h u t 200 will have utilisable data. There are perhaps 300 diRerent reenergy of I. Only in such caaee does the observed activation energy posseae theoretical signilicance. A check of the fm- actions &h d l y discernible Arrhenius equation constants. (In Borne caaee there will be constants for several dXemnt requency factor with theoretical values is a helpful criterion aa actions in a @inglearticle.) o f these m, less than 100 will to whether or not the reaction is Simple and the calculatd have Simple clean*ut data. Among thia group, the MI uniactivation energy applicable to an uncomplicated reaotion. moleoular gae d o n s &own in Figure 6 were selected. Most of them were determined by meamramen? of total pret~ 8u18.
In a aurvey of 6rst-order reaotiom, one is impreseed with the large number in which the solvent takes a h t part in the chemical reaction. As already explained, they are really
The apparent frequency factor aa well aa the apparent activstiOn energy may be subject to complicationswhich prevent r6gnificant compsriaon with theory. In unimohnlar reaotionS the unita of concentration cancel out, but in bimoleculsr reactions the nnmerical values of k and a depend oh the unita and the stsndard state chosen. Most of the data so far recorded in the literature have been Bxpreaeed in concentrations of molea per liter or moles per ec. The former seema preferable au being closer to normal experimental conditions. It must be r e m e m M that the frequency factor is a catch-all for "bidden constsnta" as well an for experimental errorein the dehmhation of the activation energy. One of the commoneat examplea of thea bidden constante is found in rasctions, Bornetimea called "pendo-nnimokdar", aUOh 88 hydrolynk r e a o t i q in which the solvent water rewts with a substance d,but becaw water is present in such lsrge ex(56 mol- per liter), ita concentration doas not appear to change. The d o n a are really bimolecular and should have a frequency fsctor of the order of 1011 instead of 10" 80 often found in unimolecnlar readions, but into the experimentally observed frequency factor has bean absoibed the term giving the concentration of water. Thus,
-
kCa.0 C A
h v d .
-
SURVEY
k (66)
-Q
6t.vd.e
rn
OF LITERATURE
With a broader whation of all the factore involved, it is worth while to examine the literature for checks on the dgni6canca of the activation energy and the accuracy of ita d e termination. Just the eimplest caaee can be reviewed berei. e., unimolecular reactions. The entropy in thm M.Y.
1913
PRACTICAL APPLICATIONS
The hest equation for predicting reaction rates is: k (55)C A mole, liter-', m.-l
- - koem
bimolecular laactim with frequency faotore appropriate for bimolecular reactiom, but they are & d i e d 80 6mt ordw. R e a d i o ~of thie type have been responsible in part for a failure to & how well the theory of unimolecular ~ O t i O n s sgeee with experimental faate. In Figure 6 most of t h e gsbphsse unimoledar reactions give a frequency factor fmm 101' to 101' comsponding to &T/Nh, and show that in all probab~ty: (a) The activation ene~& have been reliably megaued; (6) the reaction megsuIBd in not a mixed d o n with several di5emnt steps contributing to the obseoved overaU reaction rate; (c) the entropy change is small when the moleoule is converted into a m o h l e of the "activated complex". In the few reactions with a higher value of the frequency factor, there may be chain reactions which I d to an abnormally rapid reaction. The importance of the entropy term &e*'n waa slow to be realised bemuse it waa hidden in the o b 4 frequency factor. The experimentaUy determined value of the frequency factor w80 so inaccurate that any a b n o d t i e a went unnotid. Another c a w for abnormal frequency factore in bimolecular reaations isa failure to e x p r e ~ ~ t hreactantsin e terms of their stendard states, such aa moles per liter or moles per cc. I t may happen that the real reactants are intermediates which sre preeent only in very low concentrations. The spec& rate constants may be reproducible enough and the activation energies correct, but perhap the obetved frequency factor should include a large term to convert the reactsnta into standard statea. The entropy term calculated from the observed aativation energy and the observed frequency factor may then be entirely wrong.
AS* -AB* k = -RT e T e F
Nh
-0 (13)
For most nnimolecular gagphase reactions,a may be taken aa 10" unlem a chain regction is involved or unlw there is a considerable change in the shape of the molecule. The graph of Figure 7 is obtaind on thia assumption. The slanting limes represant6rat-orderapmi6c reactionrateconstanb,k,exprewd in seconds, and the activation energy Q is plotted against the absolute temperature. From Figure7 it is easy to estimatethat the decomposition of nitrogem pentoxide with an activation energy of 25,000 cdories per mole will have, within the range 0' to 100' C., specific reaction rates k betwean lo-' and 10-8. Values of 10-1 to 10-6 corraspond roughly to half-lives varyinn from seconds to months, which c o r n the practical range of most menm-
INDUSTRIAL AND ENGINEERING CHEMISTRY
. ,
..,
.
.
.
.
menta in chemical kinetics. Ethyl bromide, which is known to deoompose with endy megsurable rates (perhaps 10-9 at 400' C. into ethylene and hydrobromic acid, mnst have an
With the esmiempiricalcalculations it is passible to calculate activation energy of about 55,OOo dories. Tllw estimah the activation energy in each of the several reactions for both are in good agreement with exp€Timental facte. direet addition and for the atom chain (IO). When the actiT w o more illustrations of Fisure 7 may be dered. Revatiou energy is considerably leas for one reaction than for the ferring to Table I and rdizing that the activation energy in other @ble reactions,it may be concluded that thin reaction endothemic d i o n must be at I& 88 great aa the heat will predominate. For example, reaction a (Figure 8) with of reaction, it is expected that the deemupwition of ethyl an activation energy of 25,000 dories will take place in preferchloride will q u i r e a tempemhue between KOoand Booo C. ence to reaction E with an activation energy of about 60,000 Again, any prooess involving a direct rupture of the carbon calories per mole. Rythermore, them e m s to be little difhydrogen bond at a co%t of 9 2 , Wcalories per mole must have ference in t k two caam betwesn the activation energy for a slow, limitingstep which WIB quire 8 stiu Jligb6T temper.+ direet addition and for the atom chain. In reaction 6, howturn. Such &tiom can be completely neglected at ordinary ever, the calculated activation energy with chlorine is 80,Mx) temperatm. hen though these estimates of reaction rates calories by the bimolecular d o n and 67,000 calories by the are rough, theJT are particularly helpful in eliminating the coratom chain. Although the chain mechanism predominates reaponding reactions from d d e r a t i i o n when a numbr of in the decomposition of the chloride, both mechaniems Beem competing reactions are p m t . equally possible for the decomposition of the iodide. "he dadations of FEgure 7 are &illess l eatiefsctory for Many ionic reactions are so fast that their rates cannot be nnimolecular d o n s in solution. A distribution cnrve of a large number of ht-order reactions recorded in the literature m e a s i d . The ion itself may be regarded 88 being in an activated atate. Some ionic reactions are slow and here the h l a r to that of FFgura 6 still &owe a maximUminthe neightheories of Br@nsted(8and Bjermm are uaeful, not for d c u borhood of lot1 to 10'6 but it iS much 5tter and broader. law absolute rates, but in predicting the e5ect of electrolytes combinstion with the solvent may w i l y change the conetsnta and other f@ra on tbe reaction rate. of the equation by large amonnta. In photochemical reactions a howledge of the quantum For biimokdar reactions, is not negligible 88 yield (molecules acting per quantum absorbed) enables one to it often is in unimolecnlar d o n s , and a no longer haa calculate the d o n rata when the intensity of the photothe value of about 10" which is equivalent to RTfNh. The chemically active light is known. In some reactions, where retwo te3lns together come out to be of the order of the collision versal of the reaction occurs or competing reedions are preafrequency between molecules. When concemtrstione are exent, the quantum yield may be considerably lees than unity; p d in moles per liter, the term a in many bimolmlar in chain reactions it is greater than unity. Some of these resetions often hss a value in the neighborhood of 10" or 10'1. However, this short cut for estimating aonstsnt a ( ~ 8 quantum yields may be estimated by analogy when they haw not hean reoorded in the literature. Aa in the CBBB of thermal 10'' in a bimolecular d o n is not very eatisfactmy. The catalytic impurities may seriously d e c t the rates of activation energy for the bimoleoular reactions can be 4- &ne, the reaction. mated roughly from heata of dissociation as given in Eqm The fact that most I.eactionsare complex and contain intertions 8, 9, and 10 or from the Bemiempirical calcnlation of mediate reaction producte suggeste that rapid chilling of a reEyring. acting %ystean should provide a chemical meana for obtaining In general, it is eafer to predict +dative values thanabsolute certain desired producta. Applications of this technique have dues, and the estimation of difierenees in madon rates is not yet hean fdly explored. more reliable than the estimation of the actual rate. Some of Although most chemical reactions are complicated with the errore due to inadequate data or mathematid appro& competing,consacutive, and mveraa reactions, and thus lead matione are the m e in each of the reactions being compared, to fractiod ordera in the rate equation and to time lags and and they tend to cancel out. changes in the rate expression during the c o w of the reacThe Beparation of isotopes may be taken aa an example in tion, the &tiou tbat all chemical reactions take place in which nearly all the factors except thm depending on ms88 simple nnimolecnlar and bimolecular (and occneionally tercaneel out, at least to a certain eatent. using rite aemimolecular) step leads to simplification in predicting reaction empirical method of Eyring it can be calculated (3)that the rates. The formulas for estimating the rates of the separate activation energy for rupturing the carbon-carbon bond, unimoleeular and bimolecular reaction steps are not yet satis C'LC1', is 29 gram-dories lees than that required to rug factory and the mathemstical calculations are often extremely ture the C1LC1*bond. This 1 4 to a ratio of the two v e complicated, but the problem is clearly dehned. of 1.05. preliminaryexparilmity conetanta kd- d:/kolsmenta on the fractional fermentation of B U ~ e~ e mto be in accord with this calculation. If these measurements are later LITEUATUBB CITED confirmed, they will constitute a stmng support for the semi(1) Altar. W.B.. and H.E.,J . C h a . Phw., 4.1331 (1986). empirical calculation. (2) Brglnated. J. N.. Chm. Rm., 5. 231 (19%): LsMer, V. K., The addition of the d8erent halcgens to a given umtuIM..lo, 179 (18.32). rated compound provides another opportnnity for making (3) Daniels, F.. "Chemical Kinetioa". p. 249, Itbans, ComeU Un*. b I(l3R. . . . . .. . . predictions of reaction rates under conditione where some of (4) Fwtsr. E.G., Cope. F.J.. and Daniels, F.. forthcoming publiosthe uncertainties tend to caneel out. A halogen may add to tion. ethylene in a number of different ways, as shown in Figure 8. Fusseai,P., and Wurick, E.,J . Phva. Chem.. 46.630 (1843). The halmn mav add directlv or it mav dbv a free radical GIa&one. A. C.. Lkdler, F.J., and Eyring, H.E.,"Theory of Fate Rooefa?d". New York, M&sw-€iiU B w k Co.. 1941. chain of &e typki Hiraohfelder, J. 0.. J . am. Pb.. 9, 646 (1841). Hulburt, E.M..snd Hirsohf.ldar. J. 0.. Ibid.. 9,61(1941). Rice. F. 0.. and Henfdd, K. F.,J . Am. Chem. &., 56, 286 x, 2x (1834).
m,
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X 510
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CHAHa
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'(10) Sherman, A., Suthedsnd. R. O., and Quimby, 0. T..J . C h m . Phw., 4,732 (1838). (11) W.rrioL, E..J . C h m . Bdwxtim. 20. 134 (1948).
INDUSTRIAL A N D E N 0INEERING CHEMISTRY
Vol. 33, No. 8
