Onsager equations

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

These equations are known as the Debye-Hiickel-Onsager equations. 

The variation of equivalent conductance with concentration occurs according to the modified form of the Onsager equation as given below ... 

MN = M+ + bland a fraction a of the molecules MN may be dissociated into ions M+ and M". With strong electrolytes such as, salts and strong acids in water, the substances are known to be fully ionized, so that a = 1. The Onsager equation in these cases takes the following (original) form ... 

With weak electrolytes, such as acetic acid and ammoniacal water, a is less than 1, and a modified form of the Onsager equation is complied. It must be emphasized that modified and original forms of the Onsager equation are only complied in fairly dilute solutions. Several interesting observations and deductions follow from the features corresponding to different electrolytes. 

As already mentioned, the criterion of complete ionization is the fulfilment of the Kohlrausch and Onsager equations (2.4.15) and (2.4.26) stating that the molar conductivity of the solution has to decrease linearly with the square root of its concentration. However, these relationships are valid at moderate concentrations only. At high concentrations, distinct deviations are observed which can partly be ascribed to non-bonding electrostatic and other interaction of more complicated nature (cf. p. 38) and partly to ionic bond formation between ions of opposite charge, i.e. to ion association (ion-pair formation). The separation of these two effects is indeed rather difficult. 

If an external field is present, the procedure would be the same except that now in place of 0(i) and (n) one would use the corresponding probabilities of electron escape as given by the Onsager equation in the presence of an external field (see Sect. 9.5). 

Various treatments of these effects have been developed over a period of years. The conductance equations of Fuoss and Onsager l, based on a model of a sphere moving through a continuum, are widely used to interpret conductance data. Similar treatments n 3, as well as more rigorous statistical mechanical approaches 38>, will not be discussed here. For a comparison of these treatments see Ref. 11-38) and 39>. The Fuoss-Onsager equations are derived in Ref.36), and subsequently modified slightly by Fuoss, Onsager and Skinner in Ref. °). The forms in which these equations are commonly expressed are... 

Onsager equation phys chem An equation which relates the measured equivalent conductance of a solution at a certain concentration to that of the pure solvent. 6n,sag-ar i,kwa zhan )... 

Figure 3 Escape probability as a function of the initial electron-cation distance. The lower broken curve is calculated from the Onsager equation [Eq. (15)]. The numerical results for different mean free times x were taken from Ref. [22]. The unit of x is rJ(ksTlmf, where m is the electron mass. The upper broken curve was calculated using the energy diffusion model. (From Ref. 23.)...

The Kirkwood-Onsager equations can be generalized to include multipole moments higher dian the dipole, leading to the expression... 

The Debye-Huckel-Onsager equation for the dependence of the ordinary diffusion conductance on concentration has been extended by Falkenhagen, Leist and Kelbg [3] to apply to more concentrated solutions and their equation for a 1 1-electrolyte may be written... 

Dielectric constants are determined for pure liquid dimethylsiloxane oligomers. Mean-square dipole moments, calculated from the Onsager equation, are in good agreement with predicted values based on the RIS model (S 117) with neighbor dependence and chain conformational energies obtained in an independent analysis of the random-coil dimensions of such chains. In addition, the observed temperature coefficients of are in qualitative agreement with calculated results. 

In dilute solution the rare earth salts behave as 1 3 electrolytes and obey the Onsager equation in a modified form up to a concentration of 0.01 N. The behaviour of various rare earth salts, such as chlorides, bromides, nitrates and perchlorates has been examined [209—212]. The equivalent conductivity data for the rare earths is compiled in Table 11. Extensive ion-pair formation has been observed for rare earth sulphate solutions. 

A much better representation of the dielectric behavior of polar liquids is given by the Onsager equation... 

When association occurs, the Onsager equation must be modified to include only the free ions. Assuming the ion pair makes no contribution to the conductance, equation (18.58) becomes... 

Sometimes, the conductivity of the solution may decrease due to the formation of electroneutral ion pairs. Under these conditions, the Fuoss-Onsager equation can be used to calculate the molar conductivity (A) of associated electrolytes [57] ... 

Up to concentration of 2 X 10-3 gram-equivalents per litre there is a satisfactory agreement between the results calculated from the Debye-Hiickel-Onsager equation and the actual values of conductance of uni-univalent electrolytes. The validity of this equation has been verified even for uni-bivalent electrolytes, while for bi-bivalent electrolytes there are greater deviations to be observed. 

Another equation which has received use is the Wishaw-Stokes equation, which is discussed in Robinson and Stokes [32], This equation is based on the extended Onsager equation and is corrected for viscosity in a Walden rule sort of term ... 

Both the slope and intercept point in Fig. 23 change with increasing extent of conversion, which may be due to chainging NJs and a changing interdipole correlation. To simplify the analysis, an empirical modification to the Onsager equation was used to analyze the data ... 

Vm is the molar volume of the compound). The Debye-Clausius-Mossotti equation is applicable only to nonpolar gases at moderate pressure, and to nonpolar solvents and solutes in nonpolar solvents. For polar gases and polar solvents the -> Onsager equation gives more precise data. 

Debye-Huckel-Onsager equation Onsager equation, - Debye-Huckel-Onsager theory... 

Debye-Huckel-Onsager theory — (- Onsager equation) Plotting the equivalent conductivity Aeq of solutions of strong electrolytes as a function of the square root of concentration (c1/2) gives straight lines according to the - Kohlrausch law... 

Onsager equation — (a) - Debye-Hiickel-Onsager equation, see also - Debye-Hiickel-Onsager theory. 

Onsager equation — (b) The Onsager equation is an extension of the Debye-Clausius-Mosotti equation, and it relates the - dielectric constant er with the -> dipole moment p ... 

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