Wilhelmy published a paper, Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet (On the law, which takes place after the action of acids on cane sugar) in Annalen der Physik und Chemie in 1850 in which he gives the amount of dissociation of cane sugar (sucrose) into its simpler components (glucose and fructose) as a function of time. One can observe the change in the optical rotation of the sugar solution and determine the rate at which the amount of sugar decreases by means of an inversion coefficient, μ. Wilhelmy uses the optical rotation angle corresponding to the amount of sugar present to also specify its quantity, Z, present in the solution. When a portion, X, of the amount of sugar in solution is inverted it no longer rotates the light passing through the polarimeter X degrees to the right but instead rotates it μX degrees to the left. Wilhelmy explains as follows,

"The herein stated task will now be to determine, depending on whether and in what way M depends on the different physical conditions of the process, i.e., whether and in what manner M is a function of time, the amount of sugar, the amount of acid, the amount of Resolution agent, the quality of the acid, the temperature and air pressure. Answering these questions should be attempted in the following order.

The above however is yet an important point to discuss for the calculation of the experiments. The action of the acid on the sugar not only removes a portion of the dextrorotatory sugar but transforms it into opposite rotation. What therefore was originally the rotation =Z°, is, after a rotating X° amount of sugar, converted to =(Z - X - μX)°. To find out that given by observation quantity X itself, one needs to know μ. Hr. Biot has introduced for μ, i.e., for the amount of reverse rotation to the left, produced by a quantity of sugar which had a clockwise rotation of 1°, the name of the inversion coefficient. X then also gives Z = Z

_{0}- X. The determination of the inversion coefficients has caused some initial difficulties for me. In addition Hr. Biot has been wavering in his statements about the same. In his work on the sugar content of corn ¹) he gives for μ

for hydrochloric acid = 0.38

for sulfuric acid = 0.3867,

It can be said however that he himself noticed that he received very different results, which he thinks he can ascribe to different purities of the sugar. In a later essay ²) he then gives:

for sulfuric acid μ = 0.417

for nitric acid μ = 0.394

for hydrochloric acid μ = 0.38.

Thus it is initially to be noted that Hr. Biot, so far as I know, did not state for which temperature these coefficients apply; but since the rotatory power of galactose is dependent on the temperature, as well as the value of μ must be also differ according to the temperature of the reading. If one therefore wishes to know the value of μ for each temperature, one must first determine the law according to which the rotatory power of the galactose is dependent on the temperature."

¹) Comptes rend. 15, 529.

²) Comptes rend. 17, 757.

The results of Wilhelmy's observations are presented in Table III of his paper. The column headings are time, T, optical rotation, Drehung, the log of the relative change in the amount of sugar, Z, and the temperature. There is a footnote at the bottom of the page which gives a formula for computing the amount of sucrose, Z, present given the rotation angle, D. One can derive the formula as follows,

If we plot Wilhelmy's data we get a reasonably good fit to a decaying exponential function.

Supplemental (Nov 1): The value of the inversion coefficient used for the experimental data is μ = 0.3966. Wilhelmy refers to Biot's corn sugar in German as Schleimzucker which is literally translated as "slime sugar" but in the German Wikipedia it is another name given for galactose. Wilhelmy's definition of the inversion coefficient appears to be based on the relation Z

_{0}- D = (1 + μ)(Z

_{0}- Z) where the quantity or concentration of Z is given in units of 1° rotation. For problems in measuring the optical rotation associated with dissociation of complex molecules see the Wikipedia article on mutarotation.

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