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In reading about the inversion of sucrose I came across the equation for the reaction rate constant below:

$$k = \left(\frac{2.303}{t}\right) \log\left[\frac{\alpha(0) - \alpha(\infty)}{\alpha(t) - \alpha(0)}\right] \tag{5}$$

What is the logic for difference of the rotations' sign for the logarithmic term? i.e conceptuatlly why is $\alpha(0)$ the minuend in the numerator and the subtrahend in the denominator?

I have seen other quantities like volume or pressure represented in this form but I don't understand the reason why reacting sugar would be presented this way.

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  • $\begingroup$ In a first order reaction can we use anything directly proportional to concentration rather than concentration itself because these come as a ratio (and hence units cancel) as in the expression as above. Your particular case is described in the answer by MaxW below. $\endgroup$ – porphyrin Aug 5 '18 at 10:10
  • $\begingroup$ The above formula is wrong. The denominator should be $\alpha(t) - \alpha(\infty)$ instead of $\alpha(t) - \alpha(0)$. $\endgroup$ – aventurin Sep 15 '18 at 22:36
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Not quite sure what your asking, but I think I understand. If I misinterpreted the question please let me know.

The hydrolysis of sucrose to a mixture of the component simple sugars, fructose, and glucose, causes the direction of rotation to "invert" from right to left. This is because sucrose rotates to the right, but the combination of the two simpler sugars rotate to the left. So the rotation at time zero is for the concentration of sucrose, and the rotation expected when the reaction is complete at time infinity can be calculated from the concentration of the sucrose.

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  • $\begingroup$ I still can't get the reasoning behind why the difference is directly proportional to the initial concentration and similarly the difference of rotation at infinite time and any time t is directly proportional to concentration at any other time $\endgroup$ – Scáthach Aug 4 '18 at 15:55
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    $\begingroup$ @harambe - Let's say that sucrose rotates $50^\circ$ to the right and the completely reacted product of fructose and glucose rotates $50^\circ$ to the left. There is thus a $100^\circ$ difference, so each 1% of the reaction will cause a $1^\circ$ change in rotation. $\endgroup$ – MaxW Aug 4 '18 at 16:00

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