Hudson developed the following equation: k1+k2 == log 1 To-r (135) in which t equals the time after dissolution, r, the optical rotation at zero time, r the rotation at the time t, and r the final or equilibrium rotation. The mutarotation coefficient, k1+k2, is usually expressed in common logarithms, but if the values of k, 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+k. 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 0.4 +0.2 0.0 -1.8 -1.6 -1.4 4 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 TIME-MINUTES FIGURE 107.-"Simple" and "complex" mutarotation curves. on a linear curve (see the curve for a-d-talosc 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-mit+Be-m+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 m, 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 m1 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. 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 In (140) 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: 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] °S 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 m1 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 m1, (8.03X10-3), in eq 140, and solving for r, at each of the times, t1, earlier than 29.7 minutes (including zero time), using T2+68.00, T. +46.34, and t2=29.7. (The equilibrium rotation, 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, t=1.9, and for d2 the values recorded at the later times, t2. By placing the average value of m2 (79.0X10-3) 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 = 903232 O- 50-30 the equilibrium rotation (46.34) and the calculated value of r1 at zero time (83.85), which has already been obtained by application of eq 140. The substitution of these values in eq 139 gives the equation." °S=37.51X10-008031 +3.25X10-0798+46.34. (142) If it is desired to use the natural logarithmic base, eq 142 is changed only by replacement of the base 10 by the base e, and by multiplying each of the exponents by 2.3026. The equation, which expresses the optical rotations as observed, is converted to a specific rotation by multiplying by the ratio of the equilibrium specific rotation to the observed equilibrium rotation. For example, in the mutarotation represented by eq 142, the equilibrium specific rotation of galactose was found from a separate experiment to be 80.2. The ratio of the equilibrium specific rotation to the observed equilibrium rotation is 80.2/46.34, or 1.7307. Multiplying eq 142 by this factor gives the mutarotation in terms of specific rotation. A summary of some mutarotation measurements, which have been conducted at the National Bureau of Standards during recent years, is given in table 149, p. 762 of this publication. The measurements reported therein were conducted as described in the following paragraph. (b) MEASUREMENT OF MUTAROTATION Mutarotation measurements are conveniently made in the following manner: The carefully purified sugar is powdered in an agate mortar and passed through a fine sieve. The weighed sample (about 2 g) is placed in a dry 100-ml glass-stoppered flask and about 50 ml of distilled water (preferably of known pH, or buffered with 0.001 N potassium acid phthalate,)2 at the correct temperature is added with agitation. The water can be added conveniently from a fast-draining pipette. Time, beginning with the addition of the water, is measured with a stop watch. After the sugar is dissolved, the solution is transferred to a water-jacketed polariscope tube and maintained at the desired temperature while optical rotation measurements are made. The work is preferably conducted in a room held at the temperature selected for the measurement; in any case, the water which circulates in the jacket of the polariscope tube should be held at constant temperature by a suitable thermostat. The optical rotations are measured directly after the solution of the sugar, and at such times thereafter as required to disclose the changes that occur. It is convenient for one person to make polariscope readings while another notes the times and records the results. It is usually advisable to make the observations in groups of 5 or 10 readings which (unless mutarotation is taking place rapidly) can be averaged for use in calculating the velocity constants. The method used for calculating the equations to represent the mutarotations and the mutarotation coefficients is given on page 444. The equilibrium specific rotation of the sugar is determined with a separate sample of the sugar. It is necessary to use the same concentration and temperature as those employed in the mutarotation measurements. The optical rotation was read in sugar degrees. 42 0.2041 g of potassium acid phthalate (NBS Standard Sample 84) dissolved in 1 liter of water. (c) VELOCITY AND EQUILIBRIUM CONSTANTS FOR THE MUTAROTATION REACTIONS As already mentioned, the mutarotations of certain sugars consist of two or more reactions which involve three or more substances in dynamic equilibrium. The calculation of the separate velocity and equilibrium constants for all the mutarotation reactions is scarcely feasible at present because the number and character of the reactions are not known. By postulating that the mutarotation involves only two isomers, the separate velocity and equilibrium constants may be calculated from ki+k2 (eq 135, p. 443), and the equilibrium constant kilka, which is obtained from the optical rotations by the equation Some values of k, and k2 thus calculated are given in table 60. (143) TABLE 60.-Equilibrium constants calculated from optical-rotation measurements, assuming that only two isomers are present in dynamic equilibrium (d) EFFECT OF TEMPERATURE ON THE MUTAROTATION RATE In accordance with the general behavior of chemical reactions, the velocity for the mutarotation of a sugar is accelerated by a rise in temperature. Between 25° and 35° C the rates increase from one and one-half to three times, depending upon the sugar and upon the character of the mutarotation reaction. The normal alpha-beta pyranose interconversions have higher temperature coefficients than the rapid mutarotation reactions (pyranose-furanose interconversions). The effect of temperature on the rate of mutarotation is represented most satisfactorily by means of the integrated Arrhenius equation, in which k' and k" are velocity constants at the absolute temperatures, T, and T2; R is the gas content; and Q is the heat of activation. In 1904 Hudson determined the effect of temperature on the velocity |