Osmotic Pressure and Diffusion (an undissociated solute)

 

Van’t Hoff proposed the Law that states that if the concentration of a solute at a point (A) is greater than that at a point (B) the osmotic pressure at point (A) is also greater than that at point (B). Thus, the osmotic pressure can be considered as a force giving the dissolved molecules acceleration from a point at high concentration to one at a lower concentration. Consider the forces due to osmotic pressure, in the direction of increasing (x) on an elementary cylinder of unit cross-section area as depicted in figure 12.

 

 

Figure 12. The Development of Osmotic Pressure

 

At (A) the force is (P) dynes where (P) is the osmotic pressure. At (B) the osmotic pressure will be,  dynes the difference being dynes.

 

Assuming there are (n) molecules per unit volume the number of molecules in the cylinder is (ndx) (the volume is (dx) cc as the cross-sectional area is unity). It follows that if (ndx) molecules are accelerated by a diffusing force of  dynes then the force per molecule will be,

 

  dynes (this is the force per molecule)

 

The viscosity of the liquid medium will impede the movement of the molecules so that they will reach a constant terminal velocity where the diffusing force will equal the retarding force (F₁).

 

Thus, the force  will produce a terminal velocity of cm/sec

 

It follows that the number of molecules crossing unit area near (A) per second is the number enclosed by a cylinder of length equal to the velocity and 1 sq. cm cross section,

 

                           i.e.,                      molecules per second

 

Now from Van’t Hoff’s law,               P = nk    or  

     Where (k) is the Boltzmann’s constant, which is the gas constant per molecule or

                                         () is the absolute temperature

                                  and (N) is Avogadro’s number.

 

Consequently, the number of molecules transported across unit area per unit time will be,

 

 

and their mass will be                    

 

                         where (m) is the mass of one molecule

 

Now Fick’s Law states,                 

 

    Where the symbols have the meaning previously ascribed to them.

 

                      Now,                             c = nm

 

                      Thus,                        

 

Thus if the mass transferred per unit area is considered then  (A = 1)

 

        And equating,                  

 

                                                                     

 

Thus, the relationship between the Diffusivity of a solute (D) and the retarding force per molecule (F₁) is established in terms of the temperature and the Boltzmann’s constant.