Reciprocal Relations
The relations and reciprocity to be discussed here have—at first glance—nothing in common with “mutual relationships” and “mutual obligations.”
These reciprocal relations are of a special kind and belong to physics and chemistry, or more precisely to thermodynamics. They were proposed and formulated in 1931 by the twenty-eight-year-old Norwegian physicist and physical chemist Lars Onsager. He was an outstanding scientist, and in 1968 he received the Nobel Prize in Chemistry.
The Nobel Committee’s citation reads: “For the discovery of the reciprocal relations bearing his name, which are fundamental for the thermodynamics of irreversible processes.”
Onsager’s discovery is regarded as one of the most important advances in twentieth-century science.
The reciprocal relations establish a linear dependence between thermodynamic fluxes and forces.
In simple terms, the Onsager relations assert that the matrix of phenomenological coefficients Lᵢⱼ, which connects the thermodynamic fluxes Jᵢ and forces Xⱼ (Jᵢ = Σ Lᵢⱼ Xⱼ), is symmetric:
This means that the influence of force j on flux i equals the influence of force i on flux j.
Such is this simple reciprocity.
At the Department of Physical Chemistry of Chernivtsi University, the potential and prospects of work in the new Onsagerian thermodynamics were appreciated among the first in the former Soviet Union. Many years have now passed, and one can confidently say that the celestial bodies were favorably aligned back then, and the right people found themselves in the right place at the right time.
In those distant 1960s, the department was headed by Professor Arkady Vladimirovich Pamfilov, and later by his students, who became renowned scientists: Professors Alexandra Ivanovna Lopushanskaya, Yarema Yuryevich Tevtul, and Vasily Vasilyevich Nechiporuk. A. V. Pamfilov and A. I. Lopushanskaya chose the thermodynamics of irreversible processes as their principal line of research, and over the course of thirty years the department—and the specialized laboratory organized later—made a fundamental contribution to science.
Specifically, the abstract principles of the linear and nonlinear thermodynamics of nonequilibrium systems were successfully applied to real electrochemical and kinetic processes. A world-renowned scientific school of the thermodynamics of irreversible processes was created, bringing together chemists, theoretical physicists, and mathematicians. Alexandra Ivanovna Lopushanskaya (AI, as her colleagues affectionately called her for short) was able to shape and “crystallize” (her favorite word) a unique scientific and creative microclimate within it. She was also able to foresee the future of the thermodynamics of irreversible processes—its development and its connection with other sciences: chemistry, biology, economics, and sociology.
In one of the first works, “On the Onsager Relations in Electrochemistry” (1962), Prof. A. Pamfilov and Prof. A. Lopushanskaya applied Onsager’s linear phenomenological theory to analyze the interrelation of fluxes of matter, heat, and electric charge in electrochemical systems. Their investigations provided a theoretical basis for cross effects such as the Seebeck and Peltier effects and confirmed the symmetry of the reciprocity coefficients.
The analysis showed that cross coefficients—such as those relating electric current to mass transfer, L₁₂ = L₂₁, and to thermoelectric effects, L₁₃ = L₃₁—are equal in pairs, which made it possible to calculate kinetic characteristics using measurable transport quantities.
In studying nonisothermal electrochemical systems, not only the reversible electrode processes but also the irreversible transport processes in solutions were taken into account.
Thanks to the precise physicochemical experiments of Ya. Yu. Tevtul (1960–1970), it became possible to quantitatively evaluate cross phenomena in the liquid phases of electrochemical systems and to calculate the entropies of transport. Such nonisothermal systems consisted of two “electrode/solution” subsystems—one maintained at temperature T₁ and the other at a higher temperature T₂, for example T₂ = T₁ + 10C. The liquid phases contained aqueous solutions of a number of inorganic and organic acids, as well as solutions of nickel, copper, and cadmium salts.
I carried out my first scientific work in the department together with my coursemates Ada Rader and Larisa Ivanova back in my student years. The paper was published in the journal Elektrokhimiya (Soviet Electrochemistry) in 1977. In laboratory number 43, on the second floor of the chemistry faculty, we experimentally determined the initial thermo-emf E (the Seebeck effect) as the ratio of the fluxes of electricity J₁ and heat J₂, to which the phenomenological coefficients L₂₁ and L₁₁ correspond:
The Seebeck effect shows that if a temperature difference ΔT is created in the system, a difference of electric potentials Δφ arises. This phenomenon corresponds to the symmetric (cross) Peltier effect—creating a difference of electric potentials in the system leads to the appearance of a temperature difference between the “electrode/solution” subsystems. There are two further symmetric effects—thermodiffusion, or the Soret effect, that is, a flux of matter due to a temperature difference, and a flux of heat due to a concentration difference (the Dufour effect).
The effects are interconnected, and the Onsager reciprocal relation reflects the symmetry of the phenomenological coefficients of two cross processes.
Such behavior is characteristic of all processes in nature. They “echo” one another through the cross phenomenological coefficients.
This is precisely the principal physical and philosophical conclusion of Onsager’s theory.
Reciprocity in Relationships
With the “reciprocal relations” everything is clear, but how is reciprocity established in relationships? Here, after all, there can be no chemistry, no physics with its symmetry—only lyricism remains.
It turns out this is not so. For the great philosopher Immanuel Kant, reciprocity in relationships meant a symmetry of respect and responsibility. And once again—symmetry, but on a different, ethical level. Both physically and morally, this points to recognizing in one’s partner exactly the same rights and the same freedom of influence as in oneself.
Did Onsager really come to an agreement with Kant? After all, Kant’s “symmetry of respect and responsibility” strongly resembles the reciprocal relations and the conclusion drawn from Onsager’s theory about the symmetry of the matrix of phenomenological coefficients—proving that a certain force j has exactly the same influence on flux i as force i has on flux j.
No one came to terms with anyone. Our world is so complex and yet so simple.
Boris Markovsky