Debye-Hückel theory

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The Debye-Hückel theory (after Peter Debye and Erich Hückel ) describes the electrostatic interactions of ions in electrolyte solutions in electrochemistry .

This Coulomb attraction and repulsion forces lead to a deviation of the activity (effective concentration, formerly " active material ") of the ion species of their molar concentration in accordance with . The Debye-Hückel theory provides equations with which the individual dimensionless activity coefficient (sometimes also written as) can be calculated as a function of concentration, temperature and dielectric constant of the solvent.

Model presentation

Ion distribution in a solution

Oppositely charged ions attract, ions of the same name repel each other. For these reasons, ions are not randomly distributed in a solution, but have a certain short-range order in which anions are more likely to be found in the vicinity of cations and vice versa (Fig.). The solution's electrical neutrality is preserved.

In contrast to the ion lattice , ions cannot arrange themselves completely regularly in solution, because solvent molecules act as dielectric and weaken the Coulomb interactions, whereupon the thermal movement leads to a stronger distribution of the ions. On average over time, however, each ion is located in the center of a cloud of oppositely charged ions (indicated by circles in the figure). These ion clouds shield the charge of the central ion, which is the reason for introducing the activity as an "effective concentration" for ions.

Theory

Based on the above P. Debye and E. Hückel derived some frequently used equations by combining the Poisson equation with the Boltzmann statistics to describe the ion distribution.

Activity  coefficient f of the ion species  i

With

  • Charge number
  • Elemental charge
  • Solvent permittivity ( )
  • Boltzmann constant
  • absolute temperature
  • Radius (of the central ion, not the surrounding ion cloud)
  • Abbreviation in
    • Avogadro's constant
    • Faraday constant
    • Universal gas constant
    • Ionic strength in

The above equation is often referred to as the extended Debye-Hückel limit law, which can be summarized in its notation as shown below:

Constants of the Debye-Hückel equation for medium concentrations
Temperature
in ° C
A in B in
0 0.4883 0.3241 · 10 10
15th 0.5002 0.3267 · 10 10
20th 0.5046 0.3276 · 10 10
25th 0.5092 0.3286 · 10 10
30th 0.5141 0.3297 · 10 10
40 0.5241 0.3318 · 10 10
50 0.5351 0.3341 · 10 10
60 0.5471 0.3366 · 10 10
80 0.5739 0.3420 · 10 10

With

  • in
  • in

The range of validity is approx.

Radius of the ion cloud

The reciprocal of can be interpreted as the radius of the ion cloud:

This radius is also called the shielding or Debye length .

Debye-Hückel limit law

For ion clouds that are much larger than the enclosed ion (usually these are very dilute solutions with ):

this results in the equation most frequently cited for practical purposes:

This must be set if water at 25 ° C is used as a solvent. For other temperatures and / or solvents, it must be calculated using the equation given above (see also table).

Mean activity coefficient

Individual activity coefficients (or activities) can be calculated, but not measured due to the electrical neutrality condition. The following applies to the measurable mean activity coefficient of an electrolyte

For details see activity .

The activity coefficient is used, for example, in the law of mass action , to determine the solubility product and the increase in boiling point .

Debye-Hückel-Onsager law for the conductivity of ions

The Debye-Hückel law was used by Onsager in 1927 to determine the molar conductivity : according to Debye-Hückel, the oppositely charged ion cloud slows the migration speed of its central ion , which was previously calculated using Ostwald's law of dilution and Kohlrausch's law of square roots . The viscosity of the solvent has a major influence on the strength of the deceleration . The Debye-Hückel-Onsager theory improved that the ion mobility and the molar conductivity now depend on the concentration .

The following relationships can be established for dilute solutions (≤ 0.01 mol / liter) in water at 25 ° C:

  • for (strong) 1,1-electrolytes:
  • for 2,1 electrolytes (e.g. Na 2 SO 4 ):
  • for 1,2 electrolytes (e.g. MgCl 2 ):
  • for 2,2 electrolytes:
  • for 3.1 electrolytes:

Further improvements to the theory came through the mathematical descriptions of double and triple ions under E. Wickie and Manfred Eigen . With these models, the Debye-Hückel-Onsager law was extended to more concentrated solutions (≤ 1 mol / liter).

Directed movement in an electric field also disturbs the symmetry of the ion cloud. The asymmetry with braking effect resulting from the movement of the central ion is called the relaxation or asymmetry or Wien effect and the time period until the ions rearrange themselves is called the relaxation time . At high frequencies (above 1 MHz), which correspond to the relaxation time, the electrostatic braking effect of the ion movement does not apply . The molar conductivity or equivalent conductivity of the ions then reaches its maximum (at least the limit conductivity ) - even at higher concentrations.

According to the book chimica Volume II p. 148 (Eq. 8.56), according to Debye-Hückel-Onsager the following generally applies to all equivalent conductivities:

It contains and constants that only depend on temperature, the dielectric constant of the solvent and the valencies of the ions. is the ionic strength (mean concentration weighted quadratically according to the valences).

Since the molar conductivities are the product of the equivalent conductivities and the valence / charge exchange number of the ion (s), the equations for molar conductivities can be converted into equations for equivalent conductivities (there are then other constants in the model).

Empirical extensions for more concentrated solutions

Based on the extended Debye-Hückel limit law, relationships were obtained by adding further terms that can also be used for more concentrated solutions, in particular the equations of Guggenheim and Davies .

literature

  • Jerome A. Berson: Chemical Creativity. Ideas from the Work of Woodward, Hückel, Meerwein, and Others . Wiley-VCH, Weinheim et al. 1999, ISBN 3-527-29754-5 .
  • Calculation of individual activity coefficients of the ion types according to Kielland and Debye-Hückel, with a table of calculated values ​​for many ions and different ionic strengths; In: Udo Kunze / Georg Schwedt: Fundamentals of qualitative and quantitative analysis, Thieme Verlag Stuttgart 1996, p. 47 u. Pp. 320-322, ISBN 3-13-585804-9 .

Literature sources

  1. Peter Debye, Erich Hückel: On the theory of electrolytes . In: Peter Debye, Max Born (Ed.): Physikalische Zeitschrift . tape 24 , no. 9 . S. Hirzel Verlag, Leipzig May 1, 1923, p. 185-206 ( online at the Electrochemical Science and Technology Information Resource (ESTIR), The Electrochemical Society, Inc. (ECS) [PDF; 12.5 MB ]. Annotated English translation The theory of electrolytes by Michael J. Braus online at the University of Wisconsin. According to Braus, the use of the Greek letter κ (kappa, also written ϰ) came about through a typographical error, since an x ​​was used in the original.).
  2. ^ P. Debye and E. Hückel: Physik Z., 24 , 185 (1923).
  3. Hans Keune: "chimica, Ein Wissensspeicher", Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, Asymmetrieeffekt p. 148.