Arrhenius [13] proposed the hypothesis that some molecules in a reacting system contained sufficient energy to react, while some were inactive, and, if the system contained a constant amount of energy there would be an equilibrium between active and inactive molecules.

Thus

The displacement of the constant with temperature follows the van't Hoff equation

d log k Q
dt RT

in which Q is the energy of activation. If Q is constant over a wide range of temperatures, this equation can be integrated to the form

in which T represents absolute temperature, and R is the gas constant. Jackson and Gillis applied this formula to their velocity-constant measurements with satisfactory agreement.

The data are computed to a usable form in table 12. These data are reproduced to serve as a guide for general use. They are applicable to a concentration of 13 g of sucrose in 80 ml and will deviate slightly for different concentrations of sugar.

TABLE 12.-Time required at various temperatures for 99.99-percent inversion in the presence of 0.01, 0.1, and 0.7925 N hydrochloric acid as catalyzer

(c) INFLUENCE OF CONCENTRATION OF SUGAR ON THE CLERGET DIVISOR

The specific rotations of both dextrose and levulose vary with the concentration of sugar, and that of invert sugar likewise varies with concentration, as is shown by the Gubbe [14] equation

[a]20=-19.447-0.06068p+0.000221p2,

in which p is the percentage of invert sugar. Thus the basic values of the Clerget divisor discussed above are valid only for a concentration of 13 g of inverted sucrose.

Herzfeld applied to the basic value of the divisor the correction 0.0676 (m-13), in which m is the weight of inverted sucrose in 100 ml

of the solution taken for the invert polarization. This value of the coefficient has remained in general use to the present day. Steuerwald found a slightly higher value, 0.0717. Herles found 0.067 and Sazavsky

0.0677.

Jackson and McDonald [15] have recently measured this coefficient by observing the polarization of a series of solutions prepared by dilution of an invert-sugar solution over a wide range of concentrations. By this procedure assurance was had that all variables such as those arising from the inversion reaction itself were eliminated, the only variable being that caused by dilution. Two series of measurements were made. In one series each solution contained 10 ml of 6.34 N hydrochloric acid in 100 ml, the condition which prevails in the acid Clerget method; in the other series no substance other than dextrose and levulose was present, the condition of the enzymotic method of analysis. The results are given in table 13. The respective coefficients are shown in the following formulas:

(0.634 N HCl) P'=—(32.265+0.07935S)

(Pure water solution) P'-(30.994+0.08241S)

in which P' is the rotation calculated to 26 g of sucrose, and S is the weight of sucrose in 100 ml of solution. The relation proved to be linear between 2 and 26 g of sucrose.

TABLE 13.-Measurement of the concentration coefficient of invert sugar

(d) EFFECT OF VARYING TEMPERATURE ON THE CLERGET DIVISOR

The specific rotation of levulose varies considerably with the temperature of observation, while that of dextrose is very nearly independent of temperature. The specific rotation of invert sugar, and consequently the negative constituent of the Clerget divisor, are therefore functions of temperature. Clerget found that the divisor diminished 0.5° S for each degree increase of temperature above 20° C, and applied the correction-0.5t to his value 144.0, in which t is the centigrade temperature and 144.0 is the divisor extrapolated to 0° C. This does not imply that the value 144.0 is actually valid at 0°; it rather means that for relatively small deviations from 20° C, the correction is valid. If, as Zerban suggests, the basic value is defined as the reading at 20° C, the temperature correction becomes -0.5 (t-20). This value of the temperature correction has remained in general use to the present day. Tuchschmidt [16] in 1870 found

the value to be -0.50578t, but it is questionable whether the instruments available at that early date were capable of the precision required for so accurate a measure of the coefficient.

Zerban calculated from Vosburgh's observations that the coefficient for the half-normal (German) weight of sucrose would be -0.478 and for the quarter-normal weight, -0.466. Gillet [17] reported a value of 0.49 for the half-normal solution. Zerban states that the value -0.50 for final cane molasses at quarter-normal concentrations is considerably too high.

It is evident that considerable uncertainty attaches to the value of the temperature coefficient and that new careful measurements are urgently required.

The foregoing coefficients apply solely to the polarization of the inverted solution. Sucrose also has a definite, although small, temperature coefficient. The normal solution diminishes 0.03° S per degree increase of temperature, so that the negative temperature coefficients given above are to be increased to a higher negative value by 0.03° S when applied to the whole Clerget divisor. Pending further accurate measurements and general agreement, it appears necessary to use 0.53 for the temperature coefficient except in special instances where a different value is known to apply accurately.

In applying the Clerget divisor and its temperature coefficient to actual analyses, it is assumed that the solutions are made to volume and polarized at the same temperature, the saccharimeter wedges likewise being at this temperature. Zerban recommends that these readings be made at exactly 20° C in view of the uncertainty of the temperature coefficients. Evidently this difficult requirement can be met only by laboratories that have complete temperature control. It is urgent therefore that the temperature coefficients not only of the pure sugars, but also of the commonly occurring crude mixtures, be determined.

It frequently occurs that the two polarizations_differ slightly from each other in the temperature of observation. In such a case it is preferable to calculate the results from the temperature of the invert polarization alone and, whenever possible, to correct the direct polarization and the quartz wedges to this temperature. If the temperature of the wedges differs from that of the solution under observation, the reading can be corrected to the temperature of either solution by applying the temperature coefficient of quartz, namely 0.000148 per degree temperature per degree sugar. Since the effect of the coefficient is to lower the reading of the scale with increase of temperature, the apparent polarization is lower than it should be. Thus if a solution polarizes 100 S and the wedges are 1 degree centigrade higher than the solution, the reading must be increased by 0.015. Obviously these corrections need be made only for high polarizations and considerable differences in temperature.

If the solution for direct polarization is free from invert sugar, as is the case with beet products, and if made to volume and polarized at a temperature different from that of the invert polarization, it can be corrected to the temperature of the latter by

P=P+0.0003 P(t'-t),

in which t and t' are the temperatures of the invert and direct polarizations, respectively.

If, finally, the solutions for direct and invert polarizations were made to volume at the same temperature but polarized at different temperatures, the direct polarization (if free from invert sugar) can be corrected to the temperature of the invert polarization by

P=P+0.00061 P(t'—t),

in which the coefficient includes the changes arising from the change of rotary power of sucrose and the expansion of the solution.

(e) EFFECT OF HYDROCHLORIC ACID

Many dissolved substances affect the rotation of invert sugar, the greater number elevating it to a higher negative rotation but some altering it in a positive direction. Hydrochloric acid is most commonly used as the inverting agent, and its effect has been shown to increase the negative rotation to a higher negative value. Jackson and Gillis studied this effect quantitatively and found that the negative rotation was enhanced as the concentration of acid was increased, the relation being precisely linear up to 1.3 N hydrochloric acid and approximately so up to 2.5 N. They confined their measurements to a single concentration of invert sugar, namely that formed by the inversion of 13 g of sucrose in 100 ml of solution. In their formula, R and Ro represent the rotation at 20°C of 13 g of inverted sucrose multiplied by 2, m the grams of hydrochloric acid, and N the normality of the acidified solution.

R=Ro-0.5407m-Ro-1.972N.

(38) If we select for a stock hydrochloric acid one having a normality of 6.34 (d20 1.1029),

R=R-0.125v,

(39) in which v is the number of milliliters of 6.34 N acid in 100 ml of the solution polarized.

It is evident from these equations that the concentration of acid should be carefully regulated. Instead of the 5 ml of concentrated acid previously used, Jackson and Gillis recommended dilution of strong acid to 6.34 N or d20 1.1029. This constitutes a 1:1 dilution if the original concentrated acid contained exactly 38.8 percent of hydrochloric acid. As this is seldom the case, it is preferable to adjust the diluted acid to the concentration specified. This specification has been adopted by the Association of Official Agricultural Chemists [18]. Ten milliliters are used for inversion.

If in eq 39 the correction term which, evaluated for 10 ml of 6.34 N hydrochloric acid, becomes -1.25°, is applied, and if the experimentally determined values of R given in table 14 are then substituted, the equations can be solved for Ro, the rotation of invert sugar in the absence of acid. If no decomposition of invert sugar during the inversion reaction occurred, Ro would equal the rotation of pure invert sugar, or in other words, the Clerget divisor by the invertase method. For the acid inversion at 70° C, R, becomes -31.75; for 60° C, -31.93; for room temperature, -32.03; and for 4°C, -32.08. The accepted value for invert sugar by invertase inversion is -32.10. Evidently in all methods of acid inversion, decomposition of invert sugar occurs, but to a diminishing extent as the temperature of inversion is decreased.

(f) EFFECT OF VARIOUS REAGENTS ON THE ROTATION OF INVERT SUGAR

An effect on the rotation of invert sugar similar to that of hydrochloric acid is produced by neutral salts. It is thus not because of its acidity that hydrochloric acid enhances the negative rotation of invert sugar, but rather because it, like many salts listed below, is a dissolved substance which, conceivably on account of its high degree of solvation, produces an effect similar to an increase in concentration.

Jackson and Gillis [3] studied systematically the effect of various reagents on the rotation of invert sugar. They derived the formulas given in table 14, in which R is twice the rotation in saccharimeter degrees of the half-normal solution, and m the weight in grams per 100 ml of the substance (anhydrous) whose effect is measured. Further but less detailed measurements are given in table 15.

TABLE 14.-Effect of salts on the rotation of invert sugar

→ In their original article, Jackson and Gillis used the value -32.00 for invert sugar in the absence of reagents. However, they determined merely the slope of the curves upon addition of reagents. The substitution of the correct value, -32.10, does not affect their measurement of the slope.

It will be recognized that the list of salts is far from comprehensive, but the data show clearly that relatively large variations in the rotatory power of invert sugar can arise as a result of the admixture of salts and furthermore that the anion produces unpredictable effects. Under the column headed "molecular depression" is given the depression caused by 1 mole of dissolved substance, and under "equivalent depression" that caused by the salt calculated to valence 1. A very rough similarity in the equivalent depression caused by the sodium, potassium, and ammonium salts seems to occur, 1 molecular equivalent depressing the rotation by 25° to 40° S. Further reference will be made to this relation under a discussion of Saillard's modification of the Clerget method.

Attention should be directed to the well-known effect of basic lead acetate on the rotation of invert sugar. This is probably due to a chemical combination of the basic lead constituent with levulose and illustrates the necessity of acidifying the direct polarization of a crude substance which contains invert sugar and which has been defecated TABLE 15.-Influence of various reagents on the rotation of invert sugar