Viscous Force Acting on an Ion in Solution

Striking advances in science sometimes arise from seeing the common factors in two apparently dissimilar situations. One such advance was made by Einstein when he intuitively asserted the similarity between a macroscopic sphere moving in an incompressible fluid and a particle (e.g., an ion) moving in a solution (Fig. 4.63). 

The macroscopic sphere experiences a viscous, or drag, force that opposes its motion. The value of the drag force depends on several factors—the velocity v and diameter d of the sphere and the viscosity / anddensity/ of the medium. These factors can all be combined and used to define a dimensionless quantity known as the Reynolds number (Re) defined thus  

When the hydrodynamie eonditions are sueh that this Reynolds number is much smaller than unity, Stokes showed that the drag force F opposing the sphere is given by the following relation 

The real question, of course, centers around the applicability of Stokes law to microscopic ions moving in a structured medium in which the surrounding particles are roughly the same size as the ions. Initially, one can easily check on whether the Reynolds number is smaller than unity for ions drifting through an electrolyte. With the use of the values 10 cm, = 10 cm s , 10 poise, and / = 1, 

However, the hydrodynamic problem that Stokes solved to get F = 67cn/v pertains to a sphere moving in an incompressible continuum fluid. This is a far cry indeed from the actuality of an ion drifting inside a discontinuous electrolyte containing particles (solvent molecules, other ions, etc.) of about the same size as the ion. Furthermore, the ions considered may not be spherical. 

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