1248 THE TEMPERATURE FACTOR CHAP. 31 



Here, n is the number of degrees of freedom participating in the activation ; 

 for n = 2, expression (31.6) is reduced to (31.5). (Formula 31.6 is useful 

 for even n values only; a generalization applicable to odd n values involves 

 gamma functions instead of factorials.) 



In the "activated complex" theory of absolute rates, the concentration 

 of the activated complex is assumed to be proportional to -yT e "' 

 (where ¥„. is the free energy of formation of the activated complex from the 

 reaction partners), and the rate of its transformation into the final reaction 

 product, proportional to V^, so that the rate equation has the form : 



(31.7) V = cTe-Pa/RT = ce^JR Te-EJRT 



where Sa and Ea are the entropy and the heat of formation of the activated 

 complex, respectively. The factor c in (31.7) contains — in addition to the 

 usual concentration factors — almost exclusively universal constants, and 

 therefore has approximately the same value for all reactants. Conse- 

 quently, wide variations in the velocity constants of reactions having 

 similar activation energies must be attributed, in this theory, to differ- 

 ences in the factor e^"^^ — i. e., in the entropy (probability) of the activated 

 complex. In the collision theory, on the other hand, variations of this 

 kind are attributed to so-called "steric factors" (the fraction of collisions 

 with the required energy that result in configurations making the reaction 

 possible) . 



Whichever theory is used, the theoretical rate constant is always, to a 

 first approximation, an exponential function of Ea/RT, since all other 

 temperature-dependent factors in the above equations — except for the 

 term in parentheses in equation (31.6) when n is large — change so slowly 

 with temperature that they can be considered constant in the narrow range 

 over which the velocity of a reaction is susceptible to experimental study. 

 This is particularly true of reactions in living systems, which are restricted 

 to the "biokinetic" range of approximately O^to 40° C. (Between 270° 

 and 300° K., T changes only by 10% and \/T only by 5%, whereas most 

 reaction rates increase in this range by as much as a factor of eight or ten.) 



According to all three formulae, the specific temperature dependence 

 of a reaction is determined solely by its activation energy, Ea', to a first 

 approximation, Ea can be calculated from the values of the rate constants 

 at two temperatures, by means of the equation : 



(31.8) Ea = -4.57 f(^) (cal/mole) 



In equation (31.8), log f is a linear function of l/T; in (31.3), a linear 

 function of T. However, in a narrow range of temperatures, the hyper- 



