shows that the groups attached to the carbonyl carbon greatly influence the equilibrium proportions of the open-chain and ring modifications. If the configurationally related acids are compared, it may be observed that acids which have like configurations for the a, B, y, and 8 carbons give similar equilibrium states. For example, equilibrium solutions of acids having the mannose configuration contain large proportions of the gamma lactones, while equilibrium solutions of acids having the glucose configuration contain larger proportions of the delta lactones and the free acids.

Attempts to determine the composition of sugar solutions have resulted in a number of methods for estimating the proportions of the constituents and the rates of change from one form to another, but none of these methods is entirely satisfactory. The methods may be classified roughly as (a) those which attempt to remove and study the separate constituents of the solution, and (b) those which depend on physical measurements of the solution as a whole. The removal of a constituent from the system by crystallization or chemical reaction causes an equilibrium disturbance which constitutes a limitation for all methods involving the separation of one of the constituents of the system. Measurements of physical properties, such as optical rotation, do not disturb the equilibrium, but usually the interpretation of the results is complicated, because the physical properties depend on all of the components in the solution and not on a single substance. When the components are available in the crystalline state, the information derived from freshly prepared solutions may be used to complement the results obtained with the equilibrium solutions. Information thus obtained is satisfactory for the more stable modifications in simple mixtures, but there is no really satisfactory method for determining the proportions of the labile components in complex mixtures.

2. CRYSTALLIZATION AND SOLUTION

If an equilibrium solution of glucose or another reducing sugar is concentrated and seeded with the alpha pyranose modification of the sugar in the absence of other seed, the pure alpha modification crystallizes from solution. The equilibrium disturbance caused by the separation of alpha crystals from solution causes the formation of more of the alpha isomer until finally all the sugar is converted to the alpha pyranose modification. Likewise, if the solution is seeded with the beta pyranose modification in the absence of alpha seed, the beta sugar crystallizes from solution until finally all of the sugar is converted to the beta pyranose modification. The reverse of this process, that is, the process of dissolving the sugar, has proved very useful in obtaining information concerning the equilibrium state [20, 21, 22]. If a finely powdered modification of a sugar is shaken continuously at constant temperature with a solvent in which it is only slightly soluble (so that the laws of dilute solutions apply), the solvent becomes saturated very quickly with the crystalline modification, but the sugar continues to dissolve at a slow rate. This apparent increase in solubility is caused by the conversion of the sugar into other modifications. If it is assumed that the solubility of a given modification is the same throughout the solubility experiment, the initial solubility of the crystals discloses the quantity of that modification present in the solution at any time, and the "maximum rate of solution" gives the rate of

formation of other modifications. Measurements with lactose have shown that the total sugar which will dissolve is substantially equal to the sum of the initial solubilities of the alpha and of the beta crystals. This indicates that the equilibrium solution of this sugar consists almost entirely of the two modifications. If equilibrium is established between only two modifications, alpha and beta, then, as Hudson has shown [23], the initial solubility of the alpha form (So), the final solubility (S), and the solubility (S,) at any time (t) measured from the beginning of the experiment may be represented by the equation,

1/t ln (S-So)/(St-So)=k2,

where k2 is the velocity constant for the rate at which unit concentration of the beta form changes back to the alpha form. Likewise, the solubility of the beta form is given by an analogous equation,

1/t ln (S-S)/(S'' — S')=k1,

in which the solubilities represent those obtained by measurements with beta crystals, and the constant (k) is the velocity constant for the conversion of the alpha form to the beta form. The sum of k2 and k obtained from solubility measurements with a- and B-lactose is in accord with the sum (ki+k2) obtained from optical-rotation measurements. The agreement of the results from the solubility and the optical rotation measurements shows that for lactose the equilibrium is satisfactorily represented by the two-component system, aß. The equilibrium constant, ki/k2, can be calculated (1) from k2 k1 and k2, as determined by the "maximum rate of solution", (2) from the initial and final solubilities by means of the relation, k1/k2= (S-So)/Se, and (3) from the optical rotations by means of the relation, k/k2= (ra—r∞)/(√∞—r3). With lactose, glucose, and many other sugars the three methods give like values. This is evidence that for these sugars the equilibrium state is represented within experimental error by a system containing two isomers in dynamic equilibrium. Certain sugars, however, give experimental results which are not in accord with a reversible system involving only two components.

3. MUTAROTATION OF FRESHLY DISSOLVED SUGARS

(a) MUTAROTATION EQUATIONS

In 1899 Lowry suggested that the mutarotation of nitrocamphor is caused by a reversible reaction [22]. Hudson showed that the two forms of lactose give equal velocity constants for their mutarotations and, in addition, that the maximum rate of solution of the crystals is in accord with the hypothesis that the change in rotation is not caused by different reactions but by opposite parts of one balanced reaction. By applying the mass action law to the reversible reaction k1 represented as a ß.

Hudson developed the following equation:

k1+k2 = = log

To-ro
r-r∞

(135)

1

in which t equals the time of dissolution, ro the optical rotation at zero time, r the rotation at the time t, and r the final or equilibrium rotation. The mutarotation coefficient, ki+k2, is usually expressed in common logarithms, but if the values of k1 and k2 are to be applied in kinetic problems, they must be converted to a natural logarithmic base by multiplying by 2.3026. The mutarotations of glucose, lactose, mannose, and similar sugars follow the course of a first-order reaction and give satisfactory values for the mutarotation coefficients. Mutarotations which give uniform values for k1+k2 may be represented by an equation of the type

in which [a] equals the specific rotation at the time t, A equals the difference between the initial and final rotation, C equals the final or equilibrium rotation, and m1 equals the mutarotation coefficient, ki+k2. If the mutarotation coefficient is expressed in natural logarithms, the equation is written as an exponential function of e, as follows:

in which k=(k1+k2) × 2.3026. If the logarithms of (r—r.) at various times are plotted against time, a linear curve is obtained provided eq 135 is applicable.

If eq 135 is not applicable, the mutarotation coefficient changes as the reaction proceeds, and the logarithms of (r-r) do not fall

LOG (ROTATION AT TIME t"- ROTATION AT EQUILIBRIUM)

<-d GLUCOSE

x-d- TALOSE

8 12 16 20 24 28 32 36 40 44 48 52 56 60

FIGURE 107. "Simple" and "complex" mutarotation curves.

on a linear curve (see the curve for a-d-talose in fig. 107). The sugars which exhibit complex mutarotations undoubtedly establish equilibrium states which contain substantial quantities of more than two modifications of the sugar. The mutarotation reactions fall in two classes: (1) Those which are relatively slow and appear to consist in the interconversion of the alpha and beta pyranose modifications, and (2) those which are relatively rapid and appear to consist in the interconversion of the pyranose and furanose modifications.

The prevalence of systems containing more than two sugar modifications in equilibrium is shown by the work of Riiber and Minsaas [12] Riiber, [24], Sørensen [25], Smith and Lowry [13], Worley and Andrews [26], Dale [8], and Isbell [10, 11, 15, 18, 27, 28]. Riiber and Minsaas showed that the changes in "solution volume" and refractivity, which occur during the mutarotation of galactose, can be explained by assuming that equilibrium is established among three modifications. From a study of the optical rotations, Smith and Lowry also came to the conclusion that the equilibrium involves three modifications, and they developed the following equation to express the optical rotation, a, at any time, t:

a=Ae-mt+Be-mat+C.

(138)

This equation, which is essentially the same as the one developed by Riiber and Minsaas, represents two consecutive reactions, as xyz. As applied to the sugar series, the equation is more or less. empirical, but it appears to fit the data for the complex mutarotations as completely as eq 136 fits the simple alpha-beta interconversion. Equation 138 can be expressed to the base 10, rather than to the base e, in which case the exponents m1 and m2 are in common logarithms rather than in natural logarithms. In this form the equation reads

a=A10-mit+B10-mat+C,

(139)

in which A is the change in optical rotation due to the slow or alphabeta pyranose interconversion, B is the difference between the initial rotation and that obtained by extrapolation of the slow mutarotation to zero time, and C is the equilibrium rotation. The exponents my and m2 are functions of the velocity constants for the separate reactions which take place during the mutarotation and represent the rates at which the slow and fast changes in optical rotation occur. In order to develop equations of this type from the experimental data (see p. 156 of reference 11), the mutarotation is divided into two periods, a short period beginning at zero time, during which a rapid change occurs, and a long period beginning at the practical completion of the rapid change. By applying the formula

to the data representing the long period (that is, the last part of the mutarotation), values of m, are obtained. It will be observed that m is the ordinary mutarotation coefficient measured for the latter part of the mutarotation and that a mutarotation which follows the simple unimolecular course gives rise to only one exponential term. The constant, for the initial rapid change m2 is calculated from the following equation:

log di t'1

(141)

in which d1 and d2 represent the differences between the observed rotations and those obtained by extrapolation of the long period back to the corresponding times. The extrapolation is accomplished mathematically by substitution in eq. 140

TABLE 59.-Mutarotation of a-d-galactose [11] 5.0 g per 100 ml at 20.0° C read in a 4-dm tube. °S=37.51X10.00803 +3.25 X 10.079 +46.34.

The following example is based on the data for the mutarotation of a-d-galactose, table 59, and is given to clarify this description. Column 2 gives the observed rotations at the indicated times. The calculation of my is begun at 29.7 minutes, as calculations started at earlier times give a drift in the constant. The values of m, given in column 4 are obtained by application of eq 140, using r1 = +68.00, t1 =29.7, and for r2, readings taken at later times. The slow reaction is carried back to times earlier than 29.7 minutes by substituting the average value of m,, (8.03 X 10-3), in eq 140, and solving for r, at each of the times, t,, earlier than 29.7 minutes (including zero time), using r2=68.00, +46.34, and t2=29.7. (The equilibrium rotation, r, subtracted from the calculated value at zero time, gives A in eq 139.) These values, subtracted from the observed rotations at the corresponding time, give the differences shown in column 5 of the table. The constant, m2, for the rapid change is then obtained by substituting the differences in eq 141, using d1=2.30, t1=1.9, and for d2 the values recorded a the later times, t2. By placing the average value of m (79.0×10-3) in eq 141, and using d2-2.30 and t2=1.9, the value of d at zero time, t1, may be obtained by solving the equation. The value so obtained (3.25) is that to be used for B in eq 139. The equilibrium rotation (46.34) is C'; A (37.51) is the difference between

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