Constant Concentration Gradient Across the Wall Separating Both Compartments

3 Constant Concentration Gradient Across the Wall Separating Both Compartments 

We have seen that when there is a constant number of molecules randomly flipping back and forth between two compartments, the stationary state is reached when the molecule concentrations in both compartments are leveled. Furthermore, this situation corresponds to thermodynamic equilibrium because the net average molecule flux between compartments vanishes, and because the system free energy reaches a constant minimum value. 

In this section we analyze a different situation that in which the molecule concentrations in both compartments are kept constant along time. Let c/ and ce respectively denote the molecule concentrations in compartments I and E, and assume without loss of generality that ce c/. Hence, according to Eq. (9.12), there exists a constant net average molecule current from compartment E into compartment I  

From a thermodynamic perspective, each time a molecule goes from the high to the low concentration compartments, its free energy decreases (recall that chemical potential can be understood as free energy per molecule in a given compartment). The free energy reduction per transported molecule is 

This energy reduction is ultimately dissipated as heat. If we consider the molecule current from to / by Eq. (9.21), we can deduce the following expression for the heat dissipation rate  

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