34 MINERAL SALTS ABSORPTION IN PLANTS 



but they are only able to do so if accompanied by anions which 

 must diffuse against the concentration gradient. At equilibrium: 



[M^]i-[A-]i = [M+]v[A-], 



where [A~]i and [A~]2 are the concentrations of anions on the left- 

 hand and right-hand sides respectively. At this point the electro- 

 chemical potentials on the two sides of the membrane are equal, and 

 the system has reached a condition of minimum free energy and 

 maximum entropy. The total number of positive electrical charges 

 on each side of the membrane is always balanced by an equal 

 number of negative charges, and therefore, at equilibrium: since 



[A-], = [M+], and [A-], = [M-]2 + [X+]2 



it follows that: 



[A-],<[A-]„and[M+]i>[M+L 



Similarly, in the situation represented in Fig. 7f, where immobile 

 anions (Y") are present; when the system reaches equilibrium: 



[A-],>[A-]2,and[M^]i<[M+L 



Since the concentrations of mobile ions (chemical potentials) are 

 unequal on the two sides of the membrane in a Donnan system at 

 equilibrium while the electrochemical potentials are equal, it follows 

 that there is an electrical potential difference between the two sides.* 

 This is sometimes called the Donnan membrane potential (Hober, 

 1947). 



In the examples discussed so far, only a single pair of mobile 

 ions was considered. If there are present in the system two or more 

 univalent cation or anion species wliich can traverse the membrane, 

 the position is a little more complicated, inasmuch as the ions 

 distribute themselves also in accordance with the principles of ionic 

 exchange. Thus, at equilibrium, the ratios of the concentrations of 

 various mobile cations to one another are the same on one side as 

 the other, and the same is true also of anions. 



Bi-, tri- and multivalent ions are distributed in Donnan systems 



*Ji—fi + ri)ZF where Ji is the electrochemical potential; //, the chemical 

 potential and Z, the valency, of the ion; F, is Faraday's number, and ip the 

 electrical potential. For further explanation of the relationships between fi, 

 fi and y, see Br5nsted (1937). 



