Use of Physical Properties with Kinetic Data

Instrumental methods have largely supplanted direct analysis. Any property directly proportional to concentration can be used, such as the angle of optical rotation, the absorbance at a fixed wavelength, or the area of an NMR peak. 

In the simplest cases, when reactant A contributes to the instrument reading Y, one can easily sense that the fractional change in Y is proportional to that in [A]. That is, provided the reaction proceeds to completion,  

The same is true, as will now be derived, for reactions that are more complex. Although the fact is not immediately apparent, this equation holds when other reactants and the products contribute to the value of Y. Consider any general reaction, 

Symbolize as y, the proportionality constant between species i and its contribution to the property (i.e., the partial molar volumes in dilatometry, molar absorptivities in spectrophotometry, etc.). Then at any time the instrument reading is 

The initial and final readings, when as usual A is limiting, are 

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