154 GROWTH PRINCIPLES AND THEORY 2 



where the v's are the reaction rates, i.e. the fluxes, the A's the chemical affinities 

 (De Bonder), i.e. the thermodynamic forces. 



4. The reciprocal relations are based vipon the principle of microscopic reversibility 

 stating that in a system in equilibrium the ffuctuations of particles away from 

 equilibrium have the same frequency as those toward equilibrium. This principle 

 imposes restrictions vipon the possible ways in which a system remains in equilib- 

 rium, i.e. macroscopically unchanged, from which the Onsager relations can be 

 derived. 



From these principles a thermodynamics of irreversible processes and (with certain 

 restrictions) of open systems and steady states can be derived as a consistent theory. 

 As may easily be seen, this theory is particvdarly useful in phenomena where there 

 is an interaction of two or several processes: therinodiffusion, thermoelectricity, 

 thermoosmosis, electrochemistry, the fountain effect of Helium II, etc. The theory 

 leads to a solution of many problems where classical theory was insufficient. 



Here only some consequences of a general nature will be indicated which 

 elucidate the new aspects opened up by irreversible thermodynamics. From the 

 standpoint of the theory discussed above, Prigogine (1947; Prigogine and Wiame, 

 1946) has summarized these general consequences as follows: 



a. Open systems tend toward a steady state which is defined by minimum 

 entropy production. 



b. During the development of an open system toward the steady state, the 

 entropy of the system may decrease. 



c. Steady states are, in general, stable. Hence if one of the system variables is 

 altered, the system manifests changes in the opposite direction. The principle of 

 Le Chatelier, therefore, is valid not only for closed systems but also for open 

 systems which are the site of irreversible processes, provided that these systems 

 are in a steady state. 



While statements b and c appear to be generally valid for open systems, 

 statement a, known as Prigogine's theorem, was subject to considerable discussion 

 {e.g. Haase, 1951; Denbigh, 1952; Bertalanffy, 1953; Jung, 1956; Haase, 1957; 

 Foster, Rapoport and Trucco, 1957, with survey of literature) . It can be shown — 

 and this was in no way unknown to Prigogine — that steady states correspond to 

 states with minimum entropy production if and only if several rather severe 

 restrictions apply, namely i. Linearity of the phenomenological laws; 2. Validity 

 of the Onsager reciprocal relations for all fluxes and forces; 3. Constancy of the 

 Onsager coefficients. 



For this reason, Prigogine's theorem applies to transport processes such as, 

 e.g. the Knudsen effect : An ideal gas is enclosed in two compartments which are 

 kept at different temperatures and connected by a small opening. Then eventually 

 a steady state is reached where macroscopic mass transport disappears and the 

 pressures are different in the two compartments. Thus the final steady state 

 (unequal pressures and inhomogeneous distribution of the molecules) shows a 

 higher order than the initial state (equal pressures and homogeneous distribution). 

 Calculation shows that entropy of the gas decreases in the process, and that 

 entropy production is a minimum in the steady state. 



