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inversion heat at first nullifies and then overcomes the effect of solution; in Period III, solution is complete, distribution is uniform, and the heat effect is that of inversion alone. In order to determine the heat of inversion itself from a curve of this character, it is obviously necessary to extrapolate the curve of actual inversion across the entire interval, I + II. This is not altogether a simple matter. Generally speaking such a result may be accomplished by determining from the reaction curve in Period III an empirical formulation, extrapolating by this means, and modifying subsequently the extrapolated segment in accordance with the fact that during the period of concurrent effect spanned by this segment, one reacting substance is present in changing concentration. Disregarding for the present this final complication, it is clear that the first simple extrapolation itself is likely to be somewhat inexact, since it reproduces a segment of greater slope than that from which its formula was derived. The first essential precaution to be taken with respect to this procedure, therefore, would be to make the interval of concurrent effect as short as possible. In practice this would mean to effect solution as quickly as possible. One experimental necessity is thus at the outset clearly indicated. Assuming the task to have been adequately performed, there still remains the desirability of establishing a criterion. of normal behavior by means of which the complete theory of extrapolation may be simply developed, and the empirical curve, insofar as it departs from the behavior thus indicated, interpreted and, if necessary, corrected. Such a criterion, in this as in other cases, will be provided by the isothermal equation of reaction speed. In the case of the inversion of sucrose this equation is well established; and for any given conditions of temperature and concentration, its constants, which are not quite invariable, may be determined with high precision by polarimetric measurement, If now, the assumptions be made: first, that in this reaction the energy release is proportional to quantity of sucrose transformed, second, that negligible external work is performed; and third, that during the change the total heat capacity remains constant within measurement error, the temperature-time curve for the isothermal condition will be sensibly of the same form as that of speed. The first of these assumptions is justifiable only if the reaction heat is that of inversion alone; since the measurements which determine the speed equation are made with reference solely to the passage of the reacting system from sucrose and water to the equilibrium mixture of hexoses. It must, however, be tentatively made; and may be so made without much danger of subsequent embarrassment, inasmuch as any superimposed effects to be anticipated are likely to be too small to affect the theory of extrapolation beyond the limit of effective correction. The assumption itself, moreover, offers a possible opportunity for the detection of superimposed heats of consecu
tive or side reactions if these be measurable. As to the other assumptions made, the second may be granted at once; and the third also, when the heat capacities of factors and products, small in comparison with that of the whole system, are considered.1
On the basis of these assumptions, then, the simple theory of extrapolation may be safely developed.
The inversion of sucrose proceeds with negligible variation' after the manner of a monomolecular reaction. The familiar speed equation for this type of change,
(in which for a given weight of solvent, a is the initial weight in gram moles; x, x1, x2 the amounts transformed in gram moles, at the times t, t1, t2; and k the initial rate of transformation for unit mass) yields, on the basis of the assumption first made: namely, that temperature change under the conditions of experimentation will be proportional to quantity transformed,
and its integrals,
1 In actual determination, during the period of inversion (to which alone this analysis applies), the reacting system passed from sucrose in solution and water, to invert sugar in solution. The molecular heat of dissolved sucrose (14° to 26°) is 152.8 (Magie, Phys. Rev., 16, 381; and 17, 105 (1903)); that of water is 18.0, and that of dissolved invert sugar is (89.6 + 78.8) ÷ 2 84.2 (Magie, loc. cit.), in gram calorie units. The change in heat capacity per mole in this reaction, is therefore, (152.8 + 18) — 2 (84.2) 2.4, and for 50 g. of sucrose, a maximal quantity in measurement, 0.35 gram caloric units. Since the heat capacity of the whole system in actual determination was closely that of 1000 g. of water, the change in that capacity due to reaction thus amounted to 0.035 per cent. of the whole, and when not corrected for, caused the same percentage error in the measurement of the reaction heat. This error is beyond the limit of possible experimental accuracy, and is quite negligible. With other reactions, of course, the corresponding values might be large enough to affect the results of measurement. If this were ever the case, the effect of change in heat capacity on the form of the reaction curve could be corrected for with sufficient exactness by modifying the curve of observed temperature change in such a way that the difference between it and the derived curve varied in an approximately exponential manner from zero at the time when the temperature was minimal to the difference defined by the total correction at the time when the reaction came to an end.
2 That this was the case under the conditions of experimentation was shown by the results of accessory measurements of isothermal reaction speed under closely similar conditions of temperature and concentration.
where T, and T, are initial and final temperatures, and T, T1, T2, temperatures at the times t, t1, t2.
Now, if in actual experience solution were instantaneous, the lowest temperature reached by heat of solution alone-identical with that at which inversion began-would be determined by the point at which the inversion curve, extrapolated from the segment in Period III by Equation 2 cut the ordinate t = o. It remains to consider the effect of the changing sucrose concentration during the time interval covered by the extrapolation. In this period the inversion proceeds, not in accordance with the simple formula of extrapolation discussed above, in which the term (TT), corresponding to a in the speed equation, implies maximal initial concentration and, therefore, instantaneous solution, but at all times more slowly; its speed, initially zero, gradually increasing at first with positive and toward the end with negative acceleration, until solution is complete.
The relationships involved in this behavior are graphically represented in Fig. 2, to which subsequent discussion may be conveniently referred. In this figure, the complete curve of reaction obtained by correcting the curve of observed temperatures for temperature changes due to stirring and concomitant superimposed effects, and, if necessary or desirable, for change in heat capacity and other causes of independent variation, is represented by the heavily drawn line, abcdef. Of this, the branch ab represents the temperature change due to initial stirring and concomitant effects before the reaction is initiated; the very short interval be marks the initial lag, neglected in the present discussion, the ordinate to being drawn through c, the point from which the observed temperature fall becomes measurably significant; the initial temperature, the abscissa of c, is marked T1; the final temperature reached, T2; the minimal temperature, Tm. The inversion curve, extrapolated from Period III by Equation 2 (to be referred to subsequently as the simple curve of inversion), is shown by the line of dashes er; the point r marking its intersection with the ordinate to at the temperature T3. The curve of solution, which falls sharply from the initial point c with greater slope than that of the reaction curve, and which, after sudden flexure slightly above To rapidly approaches and soon coincides with the abscissa of this temperature, is represented by the lightly drawn whole line cgh. This curve, the form of which in measurement was not affected by the mechanism of solution, is here plotted from the results of accessory determinations of solution heats, collated in a manner yet to be described, and is very closely exponential. The curve of actual inversion, which lies at all times above
that extrapolated from Equation 2, is shown by the dotted line es; the initial point of which, s, lies on the abscissa T.. Finally, points on the reaction curve are designated TR; and those on the solution curve, the simple curve of inversion, and the curve of actual inversion, T', T" and
T", respectively. The time at which reaction begins (to) is marked to. These several designations will be used in the following discussion without further remark.
The problem of extrapolation is, obviously, to determine the temperature To. With reference to Fig. 2, we may write the following equations. For the curve of solution,
and for the simple curve of inversion,
T" = T2+
T2 + (T3 — T2)e−kt.
The rate of actual inversion is at all times proportional to the amount of sucrose in solution.' This amount will be, at any instant, the amount dissolved, less the amount inverted up to that instant. That is, the slope of the curve of actual inversion may be formulated:
where x and y are the amounts of sucrose dissolved and inverted at the time t; and where A is a constant proportional to the constant of inversion, k.
In this equation, since temperature change is assumed to be proportional to quantity transformed, the following relations obtain:
From this equation, by suitable transformation and integration, we obtain:
T"' = T2 + (T2 — To) 1/k -p (pe-kt — -ke-pt)
and since, if TR represent the observed temperature at the time t,
In this equation, all values save To are known; the temperatures, from observations made in each measurement, the constant k from available
It is for this reason that in actual experiment, mixture was accomplished by dissolving dry sucrose in acid. The relation between speed of inversion and hydrogen ion concentration is too complicated to make possible a simple mathematical treatment of the alternative procedure.