Electric potential difference → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.06 × C.01 · Electric potential difference × Thermoelectricfamily: Thermoelectric

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c
• Thermoelectric figure-of-merit bound · formula-bound · hard, applies to this route
  η = η_Carnot · (√(1+ZT) − 1)/(√(1+ZT) + T_c/T_h)
• Onsager–Casimir reciprocity · constitutive-relation · listed, never decisive
  L_ij(B) = ε_i ε_j L_ji(−B): the cross-coefficients of coupled linear transport are equal up to the time-reversal parities ε of the variables and reversal of the magnetic field; L_ij = L_ji is the zero-field, even-parity case

Steps

1. 01
   Electric potential difference —drives→ Peltier effectestablishedclaim record

   electrical → thermal · loss to thermal

   • a current through a junction of dissimilar conductors

   solid · conductor required

   refs [1] [2]
2. 02
   Peltier effect —produces→ Temperature gradientestablishedclaim record

   thermal → thermal

   refs [2]
3. 03
   Temperature gradient —drives→ Seebeck effectestablishedclaim record

   thermal → electrical · loss to thermal

   ΔV = S · ΔT · Seebeck coefficient S [V/K]

   Open-circuit voltage between the ends of one conductor (absolute S) or across a couple of two conductors (S_A − S_B); ΔV and ΔT measured from the same reference end; S < 0 for electron-dominated transport, S > 0 for hole-dominated, so the polarity is material- and convention-dependent.

   • a spatial temperature difference across a conductor or semiconductor
   • a closed circuit of dissimilar materials, or a single material with ends at different temperatures

   solid · conductor required · temperature gradient required

   refs [3] [2] [4]
4. 04
   Seebeck effect —produces→ Charge carriersestablishedclaim record

   electrical → electrical · loss to thermal

   • carriers diffuse from the hot end and accumulate at the cold end

   refs [2] [5]
5. 05
   Charge carriers —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • carriers driven through an external load at the potential difference the source maintains

   refs [6]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 6

1. [1]
   J. C. A. Peltier. Nouvelles expériences sur la caloricité des courants électriques (1834)

   Annales de Chimie et de Physique 56, 371–386 · paper, no DOI · record
2. [2]
   H. J. Goldsmid. Introduction to Thermoelectricity (2nd ed.) (2016)

   Springer Series in Materials Science 121 · doi:10.1007/978-3-662-49256-7 · crossref ✓ 2026-09-19 · record
3. [3]
   T. J. Seebeck. Ueber die magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz (1826)

   Annalen der Physik 82(2), 133–160 · doi:10.1002/andp.18260820202 · crossref ✓ 2026-09-19 · record
4. [4]
   H. B. Callen. The application of Onsager's reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects (1948)

   Physical Review 73, 1349 · doi:10.1103/PhysRev.73.1349 · crossref ✓ 2026-09-19 · record
5. [5]
   C. Herring. Theory of the thermoelectric power of semiconductors (1954)

   Physical Review 96, 1163 · doi:10.1103/PhysRev.96.1163 · crossref ✓ 2026-09-19 · record
6. [6]
   N. W. Ashcroft, N. D. Mermin. Solid State Physics (1976)

   Holt, Rinehart and Winston · book, no DOI · record