HANS H. USSING 



In order to make the point clear, let us consider a model system which, in an exag- 

 gerated form, illustrates the problem. The system consists of two compartments, / 

 and 0, which are separated by a 'membrane'. The 'membrane' is largely imper- 

 meable, communication of solvent between / and being possible only through a 

 number of pores which have the shape of small osmometers with the semi-permeable 

 membrane facing towards / and the long narrow stem opening into 0. Compartment 

 / contains sucrose dissolved in heavy water, whereas the outside medium is pure 

 ordinary water. Owing to the osmotic effect of the sucrose, water will be sucked 

 through the semi-permeable membrane and water will be replenished via the stems 

 of the osmometers. If the area of the semi-permeable membrane is large and the 

 diameter of the stem is small, the linear rate of water flow in the stems may easily 

 exceed the diffusion rate of water. Consequently, although water can easily pass 

 from to /, the heavy water of the inside solution can never reach although it 

 passes easily enough through the semi-permeable membrane. This model system, as 

 already mentioned, represents an exaggerated picture of something that will always 

 occur in pore membranes. 



Let us now consider a simple pore membrane which is impermeable, for instance, 

 to sucrose. At the boundary of the membrane adjoining the sugar solution events 

 are governed by the ideal law. The net flux arises as the difference between the water 

 diffusing out of the sugar solution and that diffusing into it, and we can write 



M J M out = fl w(o)/ fl w(i); 



but, since the water phase filling the pores is pure water, it will only flow to replenish 

 that lost by osmotic suction in so far as a difference of hydrostatic pressure is built 

 up between the ends of the pore. In other words, that part of the water transfer 

 process concerned in overcoming the internal friction in the membrane phase is 

 governed by the laws of laminar flow and not by the laws of diffusion. Now these 

 laws are of a very different nature. For a pore of given length the amount of water 

 which can diffuse through in a given time under steady-state conditions depends on 

 the area, or in other words, on the radius to the second power. Laminar streaming 

 through a cylindrical pore, according to Poiseuille's law, is proportional to the 

 radius to the fourth power. We can put this a little differently and say that for a 

 given area diffusion is independent of the number of pores in which this area is 

 divided up, whereas the flow of water is proportional to the second power of the pore 

 radius. 



It is quite easy to express these considerations in mathematical terms. I shall not 

 take your time by developing the expressions, but shall confine myself to presenting 

 a few of the resulting expressions. It turns out that the following expression is gener- 

 ally valid for a semi-permeable membrane: 



M A [ Xo i 

 ln-^=-^ -Ax .... (i* 

 M out AJo a 



* Footnote : In , . '° , indicating the ' one-sidedness ' of the process, has the dimension of a potential. J« is a 



■M ut 



I rxa i m . 



' current strength ', whereas -=— / -.dx is the diffusion ' resistance '. Thus the whole expression is analogous 



Dvi J o A 



to Ohm's law. 



36 



