Transport numbers The fraction of total current carried by the ions of a specified type. The limiting transport number, t0±, is defined for the limit of zero concentration of the electrolyte solution. The relationship between transportation number and the mobility of an ion is: The relationship between transportation number and the conductivity is:

The measurement of transport numbers Moving boundary method: the motion of a boundary between two ionic solutions with a common ion is observed as a current flows. Indicator solution: Leading solution: The mobility of the M ions must be greater than that of N ions.

Conductivities and ion-ion interactions To explain the c1/2 dependence in the Kohlrausch law.

Hückel-Onsager Theory

24.8 The thermodynamic view of diffusion The maximum amount of work can be done when moving a substance from local x to x+dx is: When expressed with an opposite force: dw = - F dx Then one gets: Therefore: The slope of the chemical potential can be interpreted as an effect force, thermodynamic force. This force represents the spontaneous tendency of the molecules to disperse.

Connections between the thermodynamic force and the concentration gradient Since μ = μө + RTlnα One get Using concentrations to replace the activity:

Fick’s first law of diffusion revisit Fick’s law of diffusion discussed earlier was developed from the kinetic theory of gases. The flux of diffusing particles is due to a thermodynamic force arising from concentration gradient (i.e. the thermodynamic force is proportional to the concentration gradient). The drift speed is proportional to the thermodynamic force. The particle flux, J, is proportional to the drift speed. The chain of proportionalities (J ~ s, s ~ F, F ~ dc/dx) implies that J is proportional to concentration gradient.

The Einstein relation The flux is related to the drift speed by J = sc Comparing the above equation with the Fick’s law, one gets sc = -D (dc/dx) Express dc/dx in terms of F, one gets s = (DF)/(RT) The drift speed of an ions equals s = u E Therefore, u E = (DF)/(RT) = (zFED)/(RT) Reorganizing the above equation to D = (uRT)/(zF) (Einstein relation between the diffusion coefficient and the inonic mobility)

The Nernst – Einstein Equation Provides a link between the molar conductivity of an electrolyte and the diffusion coefficients. Can be applied to determine the ionic diffusion coefficients from conductivity measurement. For each type of ion λ = zuF = (z2DF2)/(RT) For electrolyte Λm = (v+Z+2D+ + v-Z-2D-)F2/(RT)

24.9 The diffusion equation

Derivation of the diffusion equation The amount of particles enter the slab in the time interval dt equals: JAdt, where J is the matter flux The increase in molar concentration inside the slab is: JAdt / (Al t) = J/l Consider the outflow through the right-hand side: -JAdt / (Al t) = J/l The net change is: Then

Solutions of the diffusion equation