198 



SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS 



threshold energy E 



Energy E 



Figure 8-2. Maxwellian Distribution of Energies in Molecules. 



lision need be a fruitful one. At a higher temperature, T 2 , more molecules 

 have the necessary threshold energy to react, and therefore the rate is faster. 

 Experimentally, Svante Arrhenius, about 1889, observed that the rate in- 

 creased exponentially with the temperature. Since in solutions, the con- 

 centrations do not vary appreciably with the temperature, the temperature- 

 dependence is practically all in the rate constant, k. Thus 



k = Ae- E *' RT 



where A is a constant in moles per liter per second, E* is the threshold 

 energy in calories per mole, R the gas constant (1.987 cal per degree per 

 mole), T the temperature in degrees K, and "e" is 2.71828, the base of 

 natural logarithms. Taking logarithms of both sides 



In A = In A - E*/RT 



or, changing to the base 10, the more familiar system: 



log k = log A - E*/2.303RT 



Hence a graphical plot of experimental results of rate measurements at dif- 

 ferent temperatures plotted as log v vs \/T has a slope of -E*/2303 R; and, 

 since R is known, the value of the threshold energy, E*, can be determined 

 (see Figure 8-3). 



Table 8-3 gives values of E* for different kinds of processes. E* is often 

 called energy of activation as well as threshold energy, and the measured value 

 can often aid in the characterization of the rate-determining step of a 

 process. 



