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The Kelvin relations are a cornerstone of thermoelectric theory, demonstrating that the Seebeck, Peltier, and Thomson effects are not independent phenomena but are deeply interconnected aspects of the same underlying transport process. Lord Kelvin derived these relationships by applying the laws of thermodynamics to a thermoelectric circuit, treating it as a reversible heat engine. His derivation, while insightful, predated the more rigorous 框架 of irreversible thermodynamics.

Later, Lars Onsager’s work on reciprocal relations for irreversible processes provided a more general and solid foundation for the Kelvin relations. The Onsager reciprocal relations, based on the principle of microscopic reversibility, confirm Kelvin’s results. The relations are immensely practical. For instance, it is often easier to measure the Seebeck coefficient (S) and its temperature dependence than it is to directly measure the Peltier ([乳胶]Pi[/latex]) or Thomson ([latex]mathcal{K}[/latex]) coefficients. Using the Kelvin relations, one can calculate [latex]Pi[/latex] and [latex]mathcal{K}[/latex] from measurements of S, which is critical for characterizing new materials and designing efficient devices.