938 CONCENTRATION FACTORS CHAP. 27 



(27.79) (p„,,. - P)/P^,,, = e-const.[CO.] 



(analogous to Mitscherlich's and Baule's revised formulations of Liebig's "minimum law" 

 of fertilization). A similar formulation was suggested by Brackett (1935): 



(27.80) ^— ■ " ^ = const. ,-![C0.] -(P/6)+<^1 



max. 



The correction term P /b in the exponent was intended to account for the fact that the 

 carbon dioxide curves approach asymptotically, at low [CO2] values, a straight line with 

 a finite slope rather than the axis of ordinates. The correction term c was attributed to 

 respiration. The exponential as a whole was supposed to represent the "true amount 

 of CO2 available at the site of photosynthesis" — the first two terms describing the supply 

 by diffusion from outside, and the third that by respiration. The occurrence of a con- 

 centration factor in the exponent, unusual in reaction kinetics, was associated with 

 Beer's law of light absorption. However, since carbon dioxide is not the light-absorbing 

 species in photosynthesis, it is not clear why its concentration should appear in the ex- 

 ponent, and the equation of Brackett (as well as that of Harder) is not likely to represent 

 more than an empirical approximation formula fitting some of the experimental results. 



If it is true, as asserted by Smith (1937) that the carbon dioxide curves 

 definitely cannot be represented by hyperbolae, the question arises as to 

 what could cause their deviation from the hyperbolic form, and lead to a 

 more sudden saturation. It is easy to show that the relation between P 

 and [COo], which was expressed by quadratic equations in the several cases 

 when two reaction steps were assumed to lead from the external carbon di- 

 oxide to the complex ACO2, will be represented by a cubic equation if a 

 third intermediate process is added {e. g'., B + AC02-^ BCO2 + A; BCO2 + 

 light products -^ photosynthesis). Since the equation representing 

 [BCO2] as a function of [CO2] will contain higher powers of [BCO2], but 

 only the first power of [CO2], the carbon dioxide curves will deviate from 

 hyperbolae in the direction of a slower (rather than of a more rapid) ap- 

 proach to saturation. 



An obvious way to obtain carbon dioxide curves that approach satura- 

 tion more rapidly than hyperbolae is to assume carboxylation equilibria 

 involving two (or more) carbon dioxide molecules, e. g., 



k 



(27.81) 2 CO2 + A . ' ^ A(C02)2 



K 



or more generally : 



(27.82) nCOi + A , ' ^ A(CO.>)„ 



In this case, the thermodynamic carboxylation equilibrium condition (as- 

 suming that the several carbon dioxide molecules in the complex are inde- 

 pendent of each other) is: 



(27.83) [A(C02)„] = XaAo[C02]V(l + A^fCOa]") 



