When two electrolyte solutions of different concentration meet, ions diffuse across the junction and, because the ion species migrate at different speeds, a diffusion (liquid junction) potential builds up. Max Planck's 1890 treatment of this process yielded a further, purely thermal corollary: the energy balance of isothermal electrodiffusion closes as if each migrating gram-ion (a mole of ions of one species) transports a fixed heat equal to RTRT (Planck's osmotic constant expressed in heat units, about 569 gram-calories per gram-ion at 18 degrees Celsius), so a diffusion front cannot remain isothermal; the dilute side warms while the concentrated side cools, by about 0.0077 degrees Celsius each, i.e. about 0.015 degrees Celsius peak-to-peak, for an ideal-dilute normal (one gram-equivalent per litre) hydrochloric acid solution diffusing into pure water. Planck judged a thermoelectric detection at least qualitatively not impossible and wrote that experiments on the question were unknown to him. In the later literature the interpretation of the carried heat was subsumed into the ion-specific heat-of-transport framework of irreversible thermodynamics (Eastman, Agar), and the concentrated-solution, multidimensional-geometry, and charging-transient extensions Planck left open have all been developed into mature theory. What no publication reports, from Planck's day to now, is the experiment his paper actually proposed: direct, spatially resolved observation of the transient temperature profile at an electrolyte diffusion front, and its quantitative account for a real, partially dissociated, non-ideal solution, in a regime where current literature values of the governing ionic heats of transport are mutually discrepant. The problem asks for exactly this measurement and quantitative description.

Background

An electrolyte is a substance that dissociates in solution into positive and negative ions. When two solutions of the same electrolyte but different concentrations are brought into contact, thermal agitation drives each ion species down its own concentration gradient (diffusion). Because in general the positive and negative species do not diffuse at the same speed, their fluxes would separate charge; the electrostatic field produced by the slightest charge separation pulls the lagging species and holds the leading one back, so the solution settles into a state of near-perfect charge neutrality in which both species migrate together at rates fixed by their electrochemical-potential gradients. The steady electrostatic potential difference across the transition zone is the diffusion potential or liquid junction potential (LJP). Max Planck, in two companion papers of 1890, gave this theory its quantitative foundation (what are now called the Nernst-Planck equations): the companion paper treats the potential itself, while the 1890 paper on the excitation of electricity and heat in electrolytes treats the energy and heat balance of the same process under the ideal-dilute assumptions of van 't Hoff's osmotic theory: each ion species behaves as an ideal gas exerting osmotic pressure (van 't Hoff's law), the solutions are fully dissociated, the medium is infinite and one-dimensional, and the mobilities are constant at uniform temperature.

Planck's central bookkeeping result (his Section 5) is that the energy balance of isothermal electrodiffusion closes exactly as if every gram-ion (the mass of Avogadro's number of ions of one species) carries the fixed heat R=RgasTR = R_{gas} T with it as it migrates, irrespective of sign or species, numerically about 569 gram-calories per gram-ion at 18 degrees Celsius, with RgasR_{gas} the gas constant. He explicitly left open whether this is only a convenient mathematical identity of the ideal-dilute model or reflects a genuine physical transport of heat, recalling F. Kohlrausch's hypothesis of heat bound to moving electricity, and stated that experiments on the question were unknown to him. In his Section 9 he drew the experimental corollary: at a diffusion front the dilute side must warm and the concentrated side cool — in Planck's own analogy, just as when gas streams out of a vessel and the vessel cools while the region receiving the gas warms, the solution losing ions cools and the solution receiving them warms — and for an ideal-dilute normal (one gram-equivalent per litre) hydrochloric acid solution diffusing into pure water at 18 degrees Celsius he computed a maximum temperature rise of about 0.0077 degrees Celsius on the dilute side and an equal fall on the concentrated side, about 0.015 degrees Celsius between the two extrema, which separate diffusively (about 0.6 cm apart after one hour). He judged a detection, perhaps by thermoelectric means, at least qualitatively not impossible, noted that a quantitative test would have to reckon with the incomplete dissociation of real solutions (normal hydrochloric acid only about 80 percent dissociated in his estimate), and recorded that convection was not taken into account. His Section 10 added an estimate of how quickly the electrostatic charge settles at the start of the process (the charging transient).

In modern language, the heat carried by each ion became the heat of transport Qi∗Q_i^{*} of irreversible thermodynamics: the heat transported per mole of species ii migrating under its electrochemical-potential gradient in an isothermal mixture, an Onsager cross-coefficient coupling heat flow to matter flow. Its observable consequences are the Soret effect (an imposed temperature gradient drives a concentration gradient) and, reciprocally, the diffusion thermoeffect or Dufour effect (an imposed concentration gradient produces a temperature change); Planck's Section 9 prediction is precisely the Dufour-type signal of an electrolyte, masked in his treatment by the assumption of local electroneutrality outside the space-charge layers. Because single ionic heats of transport cannot all be separated by purely thermodynamic measurements (only charge-neutral combinations are directly observable, and a convention must fix the single-ion split), they are measured indirectly through thermal liquid junction potentials, the potentials of thermocells (electrochemical cells whose identical electrodes are held at different temperatures), and through the electrolyte Seebeck coefficient (the potential gradient produced by a unit temperature gradient). The junction dynamics that Planck treated by a two-stage idealization belongs today to the Poisson-Nernst-Planck (PNP) equations, the Nernst-Planck transport equations coupled to Poisson's equation for the electrostatic potential, whose charging regime is the classical Debye-Falkenhagen transient problem (the relaxation of ionic space charge after a disturbance). For a real, partially dissociated, non-ideal solution, the relevant modern ingredients of such an account are the measured ion-specific heats of transport (or their measurable charge-neutral combinations), enthalpies of dilution, and activity or dissociation corrections.

The electrostatic side of this program is mature: exact and numerical liquid-junction solutions exist for arbitrary composition and concentration (including non-ideal, Onsager-Stefan-Maxwell-based descriptions), for two- and three-dimensional geometries, and for the full formation dynamics of the junction on nanosecond to second timescales. The thermal side is active but incomplete: molecular-dynamics and hydrodynamic theories of ionic heats of transport exist, thermocell and electrolyte-Seebeck measurements are an active field (connected to thermoelectric energy harvesting and thermally driven nanofluidics), and the single-ion heats of transport at finite concentration are acknowledged to remain unsettled, with recent measurements in mutual tension. Absent from the literature, however, is the one experiment the paper itself proposed: direct observation of the temperature changes at an electrolyte diffusion front in the concentration-gradient geometry, and a quantitative account of what is seen for real solutions.

Problem Statement

Observe directly, with spatial and temporal resolution, the temperature changes produced by diffusion at a junction between two electrolyte solutions of different concentration, and account for them quantitatively for a real, partially dissociated, non-ideal solution. In Planck's ideal-dilute model the prediction is concrete: for normal (one gram-equivalent per litre) hydrochloric acid diffusing into pure water at 18 degrees Celsius, the dilute side warms and the concentrated side cools by about 0.0077 degrees Celsius each (about 0.015 degrees Celsius peak-to-peak), with the extrema separating diffusively, roughly 0.6 cm apart after one hour. The observation is to resolve the temperature as a function of position and time at a characterized diffusion front between specified solutions, with the sensor calibration, uncertainty budget, and control of convection, evaporation, radiative pickup, and sensor self-heating that a quantitative measurement of this signal requires. The resulting transient profile is to be accounted for quantitatively, within the measurement uncertainty, for the stated solutions.

References

  1. Max Planck. Ueber die Erregung von Electricität und Wärme in Electrolyten. Annalen der Physik 275(2), 161-186 (1890). DOI: 10.1002/andp.18902750202. URL: https://doi.org/10.1002/andp.18902750202 Open access: https://zenodo.org/records/1748444
  2. Max Planck. Ueber die Potentialdifferenz zwischen zwei verdünnten Lösungen binärer Electrolyte. Annalen der Physik 276(8), 561-576 (1890). DOI: 10.1002/andp.18902760802. URL: https://doi.org/10.1002/andp.18902760802
  3. William Duane. Ueber electrolytische Thermoketten. Annalen der Physik 301(6), 374-402 (1898). DOI: 10.1002/andp.18983010614. URL: https://doi.org/10.1002/andp.18983010614
  4. E. D. Eastman. Thermodynamics of non-isothermal systems. Journal of the American Chemical Society 48, 1482-1493 (1926). DOI: 10.1021/ja01417a004. URL: https://doi.org/10.1021/ja01417a004
  5. E. D. Eastman. Theory of the Soret effect. Journal of the American Chemical Society 50, 283-291 (1928). DOI: 10.1021/ja01389a007. URL: https://doi.org/10.1021/ja01389a007
  6. Richard J. Bearman and V. S. Vaidhyanathan. Theory of the single-ion heat of transport in nonisothermal electrolytic solutions. The Journal of Chemical Physics 39, 3411-3419 (1963). DOI: 10.1063/1.1734208. URL: https://doi.org/10.1063/1.1734208
  7. William H. Smyrl and John Newman. Potentials of cells with liquid junctions. The Journal of Physical Chemistry 72, 4660-4671 (1968). DOI: 10.1021/j100859a051. URL: https://doi.org/10.1021/j100859a051
  8. Digby D. Macdonald, Arthur C. Scott, and Paul Wentrcek. Silver-silver chloride thermocells and thermal liquid junction potentials for potassium chloride solutions at elevated temperatures. Journal of The Electrochemical Society 126, 1618-1624 (1979). DOI: 10.1149/1.2129342. URL: https://doi.org/10.1149/1.2129342
  9. R. L. Rowley and M. D. Hall. Heats of transport from the diffusion thermoeffect in binary liquid mixtures of toluene, chlorobenzene, and bromobenzene. The Journal of Chemical Physics 85, 3550-3555 (1986). DOI: 10.1063/1.450927. URL: https://doi.org/10.1063/1.450927
  10. J. N. Agar, C. Y. Mou, and Jeong Long Lin. Single-ion heat of transport in electrolyte solutions: a hydrodynamic theory. The Journal of Physical Chemistry 93, 2079-2082 (1989). DOI: 10.1021/j100342a073. URL: https://doi.org/10.1021/j100342a073
  11. John W. Perram and Peter J. Stiles. On the nature of liquid junction and membrane potentials. Physical Chemistry Chemical Physics 8, 4200 (2006). DOI: 10.1039/b601668e. URL: https://doi.org/10.1039/b601668e
  12. Edmund J. F. Dickinson, Leon Freitag, and Richard G. Compton. Dynamic theory of liquid junction potentials. The Journal of Physical Chemistry B 114, 187-197 (2010). DOI: 10.1021/jp908024s. URL: https://doi.org/10.1021/jp908024s
  13. Massimo Marino, Lorenza Misuri, and Doriano Brogioli. A new open source software for the calculation of the liquid junction potential between two solutions according to the stationary Nernst-Planck equation. arXiv:1403.3640 (2014). URL: https://arxiv.org/abs/1403.3640
  14. Silvia Di Lecce, Tim Albrecht, and Fernando Bresme. A computational approach to calculate the heat of transport of aqueous solutions. Scientific Reports 7, 44833 (2017). DOI: 10.1038/srep44833. URL: https://doi.org/10.1038/srep44833
  15. Mathijs Janssen and Markus Bier. Transient dynamics of electric double-layer capacitors: exact expressions within the Debye-Falkenhagen approximation. Physical Review E 97, 052616 (2018). DOI: 10.1103/PhysRevE.97.052616. URL: https://doi.org/10.1103/PhysRevE.97.052616
  16. Dmitri P. Zarubin. A closer look at the Nernst-Planck model for liquid junctions in galvanic cells and the concept of EMF. Journal of Electroanalytical Chemistry 833, 275-280 (2019). DOI: 10.1016/j.jelechem.2018.11.049. URL: https://doi.org/10.1016/j.jelechem.2018.11.049
  17. Patrick B. Warren. Non-Faradaic electric currents in the Nernst-Planck equations and nonlocal diffusiophoresis of suspended colloids in crossed salt gradients. Physical Review Letters 124, 248004 (2020). DOI: 10.1103/PhysRevLett.124.248004. URL: https://doi.org/10.1103/PhysRevLett.124.248004
  18. Astrid Fagertun Gunnarshaug, Signe Kjelstrup, and Dick Bedeaux. The heat of transfer and the Peltier coefficient of electrolytes. Chemical Physics Letters 738, 100040 (2020). DOI: 10.1016/j.cpletx.2019.100040. URL: https://doi.org/10.1016/j.cpletx.2019.100040
  19. André Luiz Sehnem and Mathijs Janssen. On the time-dependent electrolyte Seebeck effect. The Journal of Chemical Physics 154, 164511 (2021). DOI: 10.1063/5.0045137. URL: https://doi.org/10.1063/5.0045137
  20. Rohit Rungta and Charles W. Monroe. Nernst-Planck laws, distilled from Onsager-Stefan-Maxwell theory. The Journal of Chemical Physics 164, 144907 (2026). DOI: 10.1063/5.0299400. URL: https://doi.org/10.1063/5.0299400
  21. M. Bonetti, S. Nakamae, M. Roger, and P. Guenoun. Huge Seebeck coefficients in nonaqueous electrolytes. The Journal of Chemical Physics 134, 114513 (2011). DOI: 10.1063/1.3561735. URL: https://doi.org/10.1063/1.3561735 (preprint: arXiv:1102.2425, https://arxiv.org/abs/1102.2425)