# Thermodynamic Derivation of the Reciprocal Relation of Thermoelectricity

## TL;DR

This paper derives the Kelvin relation in thermoelectricity using a fully reversible thermodynamic approach, resolving past criticisms of earlier methods.

## Contribution

A new derivation of the Kelvin relation using equilibrium thermodynamics, avoiding assumptions of irreversibility.

## Key findings

- The Kelvin relation is derived using a fully reversible thermodynamic framework.
- The Onsager reciprocal relations are directly derived from equilibrium thermodynamics.
- The approach can be extended to other coupled physical phenomena.

## Abstract

The Kelvin relation, relating the Seebeck coefficient and the Peltier coefficient, is a theoretical basis of thermoelectricity. It was first derived by Kelvin using a quasi-thermodynamic approach. However, Kelvin’s approach was subjected to much criticism due to the rude neglect of irreversible factors. It was only later that a seemingly plausible proof of the Kelvin relation was given using the Onsager reciprocal relation with full consideration of irreversibility. Despite this, a critical issue remains. It is believed that the Seebeck and Peltier effects are thermodynamically reversible, and therefore, the Kelvin relation should also be independent of irreversibility. Kelvin’s quasi-thermodynamic approach, although seemingly irrational, may well have touched on the essence of thermoelectricity. To avoid Kelvin’s dilemma, this study conceives the physical scenarios of equilibrium thermodynamics to explore thermoelectricity. Unlike Kelvin’s quasi-thermodynamic approach, here, a completely reversible thermodynamic approach is used to establish the reciprocal relations of thermoelectricity, on the basis of which the Kelvin relation is once again derived. Moreover, a direct thermodynamic derivation of the Onsager reciprocal relations for fluxes defined as the time derivative of an extensive state variable is given using the method of equilibrium thermodynamics. The present theory can be extended to other coupled phenomena.