2634 LOWRY AND JOHN: STUDIES OFCCLXIX-Studies of Dynwnic Isomerism. Part X I I .The Equations f o r Two Consecutive Unimolecular I)Changes.By THOMAS MARTIN LOWRY, D.Ss., and WILLIAM THOMAS JOHN,B.A., B.Sc.THE problem considered in the following pages is to determine thecourse of a chemical change which proceeds in two stages, eachreversible and each obeying the unimolecular law. Two cases areconsidered :(1) That in which the total quantity of material is constant, asis usually the case when isomeric changes are studied in solution.(2) That in which the concentration of one of the constituents iskept constant, as, for instance, when isomeric change takes place in asolution saturated with one of the isomerides.Work i n this direction has already been done by Harcourt andEsson (PI& Tvccns., 1866, 156, 193) and by Mellor (Chemical Xtaticsand Dynamics, Chapter V), who has given the equations for two con-secutive non-reversible actions : X - Y - 2 (Zoc.cit., pp. 98, 114),and has calculated the course of the action in one particular case.P. V. Bevan (Phil. Trtzrts., 1904, A. 202, 71) has given the equations,and has calculated one series of values f o r the case X T2 Y 3 2DYNAMIC ISOMERISM. PART XU. 2835Finally, Rakowski (Zeitsch. physikal. Chem., 1906, 57, 321) has investi-gated the general case of n consecutive unimolecular actions, and hasplotted series of curves for the special cases X - Y - Z++ andX - Y Z 2. General equations for the case X L Z Y Z 2 weregiven by Rakowski, but no further investigation was made.The inquiry, of which the results are now described, was begun in1903, at which date the majority of the solutions referred to abovewere not available.It was hoped that the study of the equations forthe action X Y z 2 might throw light on the question of theexistence in aqueous solutions of dextrose of a substance intermediatebetween a- and /3-glucose (compare Trans., 1903, 83, 1314). At therequest of one of us, the equations shown on p. 2642 were then workedout by Mr. H. Klugh, of the Central Technical College, for the casein which the concentration of one of the constituents is kept constantby contact with the solid, This case, so far as we are aware, has notbeen considered by any other investigator. The comparison of theoryand experiment presented, in the case of the sugars, difficulties whichwere sufficiently great to prevent the utilisation at the time of theinformation which had been obtained; but the recent discovery of aseries of inflected mutarotation curves rendered urgent the study of thecommoner case in which the sum of the concentrations is constant, andled to the detailed inquiry recorded below.Previous investigators have shown that under certain conditionsthe intermediate substance Y increases to a maximum concentrationand then decreases again, whilst the concentration of the final product2 gives rise to an inflected curve when plotted against, t.Our owninquiry has included the study of the intermediate substance, butspecial attention has been paid to the inflected curves for the finalproduct, and a method has been devised whereby these curves may becharacterised readily by drawing or calculating the intercepts of thestationary tangent on the lines which indicate the initial and finalconcentrations of the product.General Solution.Case I.If the concentrations of the three substances X , Y, and 2 be repre-sented by x, y, and x, and the four velocity constants by k,, k,, k3, k,,as shown in the scheme :k, k,it Compare Walker, Zcitsch. yhysikul, Chevz., 1899, 28, 1772636 LOWRY AND JOHN: STUDlES OFthe f undamcntal equations are :dx - _ - - k l x + k2y dt9 = + kla: - (k, + k4)y + k3zdtI n the case of unsaturated solutions in which the total concentration isconstant : x + y + x = const. = 1, the constant being taken as unity inorder to simplify the algebraical working.The assumption has alsobeen made that the experiments are carried out with materialsinitially homogeneous, so that when t = 0 , z=1, y=O, z=O, and inaddition dz/dt = 0.The solution of the differential equations is as follows :x = xx) { - m2 e - ? I L ] t + ml e -7)2?tm2 - ml "2 - ml 9n2 - m1 m2 - "1z = z m { - 2 2 . e - m i t + - J - s - m 2 t + l mn ~ , - ml m2 - mlwhere :mlm2 = k2k3 + klk3 + k,k4.These functions undergo a remarkable simpli6cation when one ormore of the velocity constants is reduced to zero. Thus, if the firststage of the action is izon-reversible, k,=O, and the m functions arereduced to the simple form :if the second stage of the action is non-reversible, k , = 0 , andif both stages are non-reversible, k, = k, = 0, andor vice versa.m l = k , ; ~122=ik3+k4;m 1 = k 1 + k 2 ; m 2 = k 3 ;ml = ZC, ; m2 = k3DYNAMIC ISOMERISM.PART XII. 2637Under these con<itions, the original equations can be given in termsof the velocity constants without making use of m, and rnB. But thesimplified equations, as investigated by Mellor, by Eevan, and byRakowski, are of very little value in the study of dynamic isomerism,since they can only be applied to non-reversible isomeric changes, andare inapplicable in all those cases in which the original substance canbe recovered from its solutions by recrystallisation. We have thereforebeen obliged to devote our attention to discovering exactly how muchinformation can be obtained from the study of t.ha experimental curvesin the general case in which both actions are reversible and all thefour constants are finite and unknown.Porm of the y t Curves.Case I.The chief feature of the yt curves is the occurrence under sqme con-ditions of a maximum concentration, followed by a decrease to thelimiting value yor! . The condition dy/dt = 0 gives for the co-ordinatesof the maximum the values :m - k'ml - k3log eL?, 1 tm = -rn2 - nilIf k, lies between m1 and rn2, the ratio is negative and the"1 - k3logarithm imaginary; the yt curve then runs up steadily from theorigin to the limiting value without passing through an intermediatemaximum.were greater than m2 or less than ml.The former condition we have proved to be impossible.As regardsthe relative magnitudes of k, and m,, we have found that when k, isequal to m,, k3 = k, = ml, whilst k,$rn, gives k3zk1. The occurrence ofa maximum in the yt curve thus depends only on the relative m a p i -tude of k, and k, ; if k,>k,, no maximum can be developed whatevervalues ase assigned t o k, and k4, whilst if k,<kl, a maximum alwaysappears.It is noteworthy that when k3 = k,, the e-mlt factor of the yt equationvanishes and the curve assumes the simple logarithmic form character-istic of a single unimolecular action.The dependence of the form of the yt curve on the constants k, andk3, and the small influence of the constants k, and k4, naturally extendto the simpler cases i n which k2 or k 4 = O ; they were pointed out byRakowski as applying under these conditions, but the general casehad not been investigated previously.A maximum would occur i2638 LOWRY AND JOHN: STUDIES OFPorm of the z t Curves.Case I.The curves connecting x and t are characterised by an initial“period of induction,” when = 0, and by a, point -of inflexionwhere dtz = 0. I n dealing with an experimental curve, the existenceof a true period of induction may be demonstrated by the constancydxd 2 Xt =1 ‘21.00.80.60-40-21’0Iof the initial values of x, but its duration is merely a question of theperiod which elapses before the sensitiveness of the methods ofmeasurements allows of the detection of the gradual change in thesevalues and has no quantitative value whatever.The inflexion has,however, definite quantitative features which may be recognised readilyin the experimental curves. Of these, we attach special importance totwo features which are independent of the actual velocity of change,a point of some importance in dealing with changes which depend oDYNAMIC ISOMERISM. PARr XII. 2639the presence of a catalyst or impurity, and proceed with differentvelocities in the case of samples of different degrees of purity. Thesefeatures are the co-ordinate x i of the point of inflexion, especiallywhen expressed by the ratio zi/aoo and the ratio t a / t b of theintercepts ta and t b of the stationary tangentand x = x c i o .d2z The condition x2=0 gives for the pointupon the lines z=Oof inflexion the co-ordinates :Unlike the co-ordinates for the maximum in the y t curve, theseexpressions are always real, and the point of inflexion is a regularfeature of all the zt curves.The tangent a t the point of inflexion is given by the equation :2 - zi = m(t - ti),whereThe intercepts, obtained by substituting x = 0above equation, are given byta = ti - ~ l / mt b = ti + (zoo - xi)/m.The intercept-ratio is then found by substitutionIt will be noticed t h a t the ratios 2i/zm and taltband x=zoo in theto bedepend directly onthe ratio m2/ml, but that none of the individual velocity constants arepresent in the f o r m u l ~ by which the values of these ratios are deter-mined.The experimental study of the xt curves can therefore beused to determine the ratio m2/mn1, but is only indirectly of value indetermining the magnitudes of the individual velocity constants. Theexpressions which give the ratios zi/xoo and ta/tb in terms of m, andm2 are too complex to be solved easily, even when the numerical valuesof these ratios are known ; we have therefore calculated the followingtable, from which the values of inz/ml corresponding with any givenvalues of Z i / x q and ta/tb may be determined by interpolation 2640 LOWRY AND JOHN: STUDIES OFTABLE I.m$m, or m4/m36'05'04'03.02.52-01 '41'21 -11.0zilza,0-1850,1980'2130.2300'2400'2500-2610'2630.2640,264t a / l b0-06070.06670'07200.08000.08410.08810'09240-09340-09380'0939The limiting values, when mz/ml = 1, are2 - = 1 - - = 0.264,za e!!.= 0.0939;3it follows, therefore, that if an experimental curve gives values inexcess of these figures it cannot be due to two successive unimolecularchanges, but probably depends on some more complex sequence.Having determined the ratio mz/rn: from the ratios z~/xoo and t & , ,it would be easy to deduce from ti = -----loge-2 the individual values of m2 and m,, but these would probably be found tovary widely according to the amount of catalyst or impurity presentin the experimental material; it is for this reason that we have laidspecial emphasis on determining the ratio m2/ml and the ratiosk, : k, : k, : k,, rather than the absolute values of these quantities.A sa, trace of catalyst cannot alter the character of the final equilibrium,the ratios kJk, and k,/k, must be independent of the speed of the action ;a similar statement would probably be true of the ratios k,/k, : k3/k4and m2/ml if the quantity, and not the nature, of the catalyst werechanged, but an alteration in the relative speeds of the two stages ofthe action might be produced if a different catalyst were introducedinto the system.1 mm2-7n1 mlTyansposition of Constants. Case I.On examining the equations given above, it will be seen that theequations connecting x andt, as well as the equations forthe co-ordinatesof the point of iuflexion and for the stationary tangent, do not containany of the individual velocity constants, except in so far as these serveto determine the values of ml, m2, and x , .It therefore follows thatwhen x/xa is plotted against t, the course of the curve is determinedentirely and exclusively by the values of ml and m2. From this factsome important conclusions may be drawnbYNA MIC ISOMERISM. PART XII. 26411. The m functions are syml~zctrical it& refvence to k,k2 und k3kl. It istherefore possible to interchange k, aud k2 with k, and k4 withoutaltering in the slightest the courae of the st curve. Thus :(a) On comparing the three equilibria2 1 1 1 1 32 1 1 1 1 21 z z l z Z 1 1 z z l Z z 1 1 zz 1 zz 1x-,in which the final proportions are the same throughout, but the relativevelocity of the two changes is altered, it is noteworthy that the firstand third give identical curves for z and t, although these differwidely from the curve for the second equilibrium; the three yt curves(which involve k,) are also entirely different from one another,(6) In the case of the equilibria:1 12 12 z 1 - 12 11 11 .- 2 = 22 11 21 ~ ~ 2 t - 41 11 2l - 1 Z z - a1 21 12 - 2 - 11 22 14 ~ 2 ~ ~ 1the form of the curve obtained by plotting z/zm against t is in everyrespect precisely the same for the two members of each pair, in spiteof the alterations which are produced in the values of zoo by thetransposition of the constants. It is a noteworthy contrast that thealteration of relative velocities on passing from0.5 1 1 11 1 2 12 ~ 1 ~ 1 t o 2 z z - 1alters completely the form of the xt curve, in spite of the fact that theultimate proportions of the three isomerides remain unchanged.2.If k1 = k, the m functiom contain the other two velocity constantsonly in the fopm k,+k,. It is therefore possible to increase k2 a t theexpense of k4, or vice versa, without affecting in the slightest the formof the curve for x/xa against t. Identical curves are therefore givenby the equilibria :2 2 2 2 2 2 2 23 1 2 2 1 3 0 43 ~ 2 ~ 1 2 = 2 = 2 1 - 2 ~ 3 0.-2-.4,in which the equilibrium is gradually dispIaced in favour of 2 at theexpense of X , until X disappears altogether, owing to the non-reversibility of the change X -+ Y.Even more remarkable is the fact that when k,=k,, the abovetransformation can be made without affecting the form of yt curve,which remains unaltered (for instance) over the whole range from2 2 2 24 0 0 24 ~ 2 ~ 0 to 0 .2 2 4 .* The final proportions of :F, 9, and c arc licre shown by simple integers, forexample, 1 ; 1 : 1, instead of the actual values, 0-33 : 0.33 : 0'33 of xw : ya, : x m 2642 LOWRY AND JOHN: STUDIES OFI n the former limiting case the third isomeride 2 is not formed at allsince k, = o ; the y t curve has therefore the simple unimolecular form,and this form is retained throughout the whole series of equilibria.It is noteworthy that the above transformation leaves the yt curvesunaltered, and not merely the curves for y/ym against t, yoo (unlikezoo ) remaining constant in value throughout.General Xolutiolz.Case 11.The fundamental equations for a saturated solution in which theconcentration of the original substance is kept conetant by contactwith the solid are :5 = I 9 dt = k, - (k, + k4h)y+k3zthe initial concentration being taken again as unity in order tosimplify the algebraical working.The general solution is :mrn4 - m3k k2 = xgg { L e - n t 3 t + - 3 - e - m 4 t + 1k, --L 4 YCQ =& ; "00 -q3'm4 - m3wherem3 = ${ (k2 + k, + k4) - J ( k , + k3 + k,)2 - 4k&3),m4 = S((k.2 + k3 + k4) + J(k2 + k, -k k*)2 -- 4k&,),m3m, = k2k3'Porm of the Curves. Case 11.The condition, dy/dt=O, for a maximum in the gt curve leads inthis case to the equation :tmax.= ____ 1 1og:2?4,m2-m, k2-m3The logarithm is imaginary if k2 lies between m4 and rn3. This, wefind, must always be the case : the development of a maximum in theyt curve is therefore impossible, whatever values may be assigned tothe velocity constants.The equation for the st curve is remarkable in that it has preciselythe same form as in Case I, the only difference being that the limitingvalue, x S , and the m functions, m3 and m4, are derived in a differentway from the velocity constants k,, k2, k,, and k4. The difference canbe expressed very simply by taking the expressions for the unsaturatedsolution of Case I and making k, = 0 in order to convert m1 and m2 intom3 and m4, but this transformation has no experimental significance, aDYNAMIC ISOMERISM.PART XII. 2643k, appears in the expression for xcL, . It is, however, important to noticethat the ratio m4/rn3 may be determined from the experimental valuesof zi/zm and tali* for a saturated solution by the same equations andformulse that were used to deduce 9nz/ml in the case of an unsaturatedsolution, the table on p. 2640 being equally applicable in either case. Thealteration in the form of the m functions has the effect of renderingthem unsymmetrical in reference to k, and k2 and k, and k,; as aconsequence of thi8 alteration an interchange in the relative velocitiesof the two stages of the action cannot be made without altering theform of the x t curve.Transposition of Constants. Case I T .The most important transformations in Case I1 are as follows :1.The m functions do not contain k, ; moreover, this constant entersinto the equations for yt and zt only as determining the values of yx,and xa. The curves for y/ym against t, and for z/xcL, against t,are therefore entirely independent of $. Identical curves are thusgiven by the equilibria :- 1 11 11 - 1 - 1and so on, up to the limiting case when k1 has a very high value andthe action becomes non-reversible.Theseconstants can therefore be interchanged without affecting the form ofthe x t curves, the value of zY) being unaffected by the transposition.Identical zt curves are therefore given by the four equilibria :2. The m functions are symmetrical in reference to k, and k,.1 1 1 2 2 1 2 22 1 1 1 2 1 1 12.-1--.1 2 z 2 Z z l l z z l - 1 l z 2 Z z 1 ,the identity of the first pair with the second pair being established bymeans of the first transformation.This second transformation cannotbe effected in the case of the yt curves, which depend in a special wayon the value of k,.Numerical Values.I n order to illustrate the form of the curves, values have beenworked out for y and x against t in the case of the three equilibria :2 12 11-.-1_.1 1 11 11 . - 1 = 1 1 21 21 - 1 - - ; - 1both for unsaturated solutions (x: + y + x = 1) and for saturatedsolutions (x = 1). It will be noticed that the final equilibrium is thesame throughout, but that there is an increase in the velocity of thefirst or of the second stage of the action in the first and lastcases.Case I.-In considering the unsaturated solutions one of the mostnot,able features is the maximum in the yt curve when the velocit2644 STUDIES OF DYNAMlC ISOMERISM.PART XII.constants are 2211, the first stage proceeding twice as rapidly as thesecond. When the two stages are equally rapid, the curve, for con-stants 11 11, is of the simple unimolecular type. When the constantsare 1122, the growth of y is checked by the increased velocity withwhich it passes into x, and the curve falls below the preceding oneof unirnolecular type. The x t curves show the usual " period of induc-tion " or horizontal tangent at the origin, and also exhibit points ofinflexion; it is noteworthy that the curve becomes more inflected ifthe two stages are made to proceed with unequal velocities, but, thatidentical effects are produced by accelerating either the first or thesecond stage of the action ; doubling the velocity of either stage raisesthe point of inflexion from 21.7 to 23.0 per cent.OF the final value, andimreases the ratio of the intercepts of the stationary tangent from0.075 to 0.080.Case 11.-The six curves that are plotted for saturated solutionscall for but little comment. Two of the yt curves, those for constants1111 and 1122, intersect a t t = 3 (approx.), in addition to beingcoincident at the origin, t =0, and a t t = The x t curves are alldistinct, the inflected character of the curve being increased byaccelerating the first, and decreased by accelerating the second, stageof t h e action, the ordinate of the point of inflexion being changedfrom 0.175 to 0.187 and 0.146 respectively, whilst the intercept ratio,0,058, is increased t o 0.062 in the former, and decreased t o 0.045 inthe latter, case.TABLE XI.Numevical Etlues.Case I. Case 11.Y/Yoo ' z/za. Y / Y a ' Z/Z, * ---- t. 2211. 1111. 1122. 2211. 1111. 1122. 2211. 1111. 1122. 2211. 1111.1122.0.1 0.471 0259 0'248 0.025 0.013 0'173 0.091 0.087 0.009 0.005 0.0090.2 0.753 0'451 0.418 0'082 0.046 0-303 0.166 0-154 0.055 0'016 0'0290-4 1.014 0.699 0.624 0.232 0,155 0.478 0.282 0.254 0.098 0.055 0.0890.6 1.091 0-835 0.737 0.383 0.260 0.584 0.367 0.328 0.177 0'104 0.1560.8 1.102 0.909 0.807 0'512 0.356 0'654 0'433 0.388 0.258 0-158 0.2241.0 1.091 0.950 0.855 0.619 0-473 4 0.705 0'486 0.441 0.335 0.213 0.2871.2 1.075 0-973 0.889 0,703 0.562 0.744 0-531 0.489 0.406 0.267 0.3471'4 1'060 0.985 0.915 0'769 0'638 0,776 0.569 0-532 0.470 0'319 0.4011 - 6 1'047 0.992 0 934 0.821 0.701 0.802 0.603 0.572 0'528 0.367 0.4511.8 1.03i 0 995 0 949 0.861 0,754 2 0.825 0.634 0'608 0.580 0'413 0.4972.0 0'845 0'662 0.641 0.626 0.455 0.5402.5 1.015 0-999 0 979 0.943 0.877 0.884 0.736 0.711 0.721 0.550 0.6303-0 1.008 1'000 0'989 0.970 0.925 0.914 0 770 0.768 0-792 0.636 0.7034.0 1'002 - 0.997 0.991 0'972 0.952 0.843 0'850 0.884 0.746 0~8085.0 1.001 - 0.999 0'998 0.990 0.973 0.893 0'904 0.935 0'827 0.87610.0 1-000 - 1*000 1.000 1~000 0.999 0'984 0'990 0.997 0'974 0'986t , o r t i 0.7i8 No No 0549 0'380 No No No 0-623 0'861 0'569'I"o1-5- 1'102 max.max. 0'230 0'217 niax. max. max. 0'187 0'175 0'146Ym G-3t, - - - 0.151 0.103 - - - 0-164 0.225 0.140ib - - - 1.858 1 380 - - - 3'623 3.861 3,0691.029 0.998 0'960 0.892 0.798- - - 0.080 0.015 - - - 0.062 0'053 0'04THE DJNITRO-DERIVATIVES OF DIMETHY L-P-TOLUIDINE. 2645Summary ccnd Conclusions.1. Equations are given for the changes of concentration which takeplace in a reversible chemical action which proceeds in two stages,each obeying the unimoleculsr law. When the total concentration isconstant the intermediate form may pass through a maximum concen-tration, but this is] not possible when the concentration of theinitial form is kept constant, for instance, by saturation with the solid.I n each case, however, the growth of the third form is represented bycurves which exhibit a period of induction and a point of inflexion.2. The occurrence of a maximum concentration of the intermediateform depends exclusively on the velocities with which it is producedfrom the other two forms, and is independent of the velocities withwhich i t passes into these forms.3. The inflected curves showing the growth of the concentration ofthe final product are independent of the individual velocity constants,except in so far as these determine the value of certa in ‘ ‘ m ” functions,involving in the case OF unsaturated solutions all the four velocityconstants, but in the case of a saturated solution only three of them.The ratio of the two ‘‘ m’’ functions can be deduced from the concen-tration at which the point of inflexion occurs, or by drawing thestationary tangent and measuring its intercepts on the lines show-ing the initial and final concentrations of the product. If theconcentration at the point of inflexion is greater than 26.4 per cent,.of the final concentration, or if the ratio of the intercepts is greaterthan 0.0939, the curve cannot be due to two consecutive unimoleculttractions and must depend on some inwe complex sequence.130, HORSEFERRY ROAD,WESTMINSTER, S.W