Osmotic Pressure and Diffusion (the solute dissociates into two monovalent ions)

 

Let the velocities of the cation (+) and anion (-) in unit electric field (i.e. the mobilities of the ions) be (U) and (V) cm per second respectively. Now, each ion will be driven by the osmotic pressure, thus for the cation (F₁),

 

 

where (n) is the concentration per cc. of each kind of ion and (P) is the osmotic pressure of the solution.

The velocity of a cation due to the osmotic pressure is , that is,

 

 and for the anion

 

In most cases  and as a consequence if the osmotic pressure alone were the driving force the two ions would become separated and as this is not possible another force must be active. Consider some hydrochloric acid placed at the bottom of a tube of water.

 

The hydrogen ions would have a mobility (U), which would exceed the mobility of the chlorine ions (V). Under such circumstances the liquid at the top would be positively charged and that at the bottom negatively charged due to the more rapids movement of the hydrogen ions. An electric field would be set up which would reduce the velocity of the faster ions and increase the velocity of the slower ions. The final state would be reached where both ions traveled at the same rate. Assuming this state to occur the coefficient of diffusion can be calculated.

 

In the direction Ox, at any point let the let the electric field be (X) and the derivative  where E is the potential at that point. Now, a monovalent ion situated at that point would experience a force dynes according to the sign of its charge. Now, as the cation velocity under unit force is (U) cm per second, a force  will result in a terminal velocity of  cm per second. In the same way the terminal velocity

 

of an anion will be  cm per second.

 

The total velocity of a cation due to both osmotic pressure and the electric field will be,

 

                         and for the anion    

 

Now, it would be fair to assume that the number of ions of each kind crossing unit area per unit time would be the same  which will be equal to the product of their velocity and their concentration,

 

Thus,              

 

             Eliminating ,                    

 

Now although the osmotic pressure (P_a) according to Van’t Hoff is given by , in practice the actual osmotic pressure is less i.e.,  .

 

                 Differentiating,                    

 

     substituting for                

 

Now if the mass of each ion is (m₁) and (m₂), then the total mass crossing unit area per unit time will be,

                                                

 

and as (m) the mass of a molecule will be  then the total mass crossing unit area per unit time will be,

                                                     

 

Now according to Fick’s law the total mass crossing unit area per unit time is,

Consequently, the Diffusivity (D) is given by,