Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 21, 2018

A Symmetric Van ’t Hoff Equation and Equilibrium Temperature Gradients

  • D. P. Sheehan EMAIL logo

Abstract

Thermodynamically isolated systems normally relax to equilibria characterized by single temperatures; however, in recent years several systems have been identified that challenge this presumption, demonstrating stationary temperature gradients at equilibrium. These temperature gradients, most pronounced in systems involving epicatalysis, can be explained via an underappreciated symmetry in the Van ’t Hoff equation.

Acknowledgment

The author thanks M. W. Anderson, T. Herrinton, G. Levy, D. Miller, and W. F. Sheehan for their comments and insights. Figures were prepared by S. Grubb and C. Ibarra. The author is indebted to the two anonymous reviewers whose critical comments greatly improved the paper.

Appendix

This appendix describes a device designed to accentuate STGs. Its underlying effects have been corroborated by experiments, and it is currently the subject of laboratory research and development [48].

The STG device consists of two narrowly spaced parallel epicatalytic surfaces (S1 and S2) that are differentially active with respect to a dimeric gas AB (Fig. 7). S1 preferentially dissociates the dimer endothermically (AB + ΔE A + B), while S2 preferentially recombines the monomers exothermically (A + B ⟶ AB + ΔE). Because ΔE is provided by the surfaces, it follows that S1 cools and S2 heats. (In the Duncan experiments [7] [Section 3.2], AB was H2, S1 rhenium, and S2 tungsten.) KT-reciprocity is satisfied in that S1 and S2 foster distinct equilibria for their gas-surface reaction; thus, a temperature differential (ΔT=T2T1) forms between them. This chemical cycle is driven solely by internal thermal energy, and none of the reactants or products are net consumed; thus, in theory, the cycle can persist indefinitely in a sealed chamber.

Surfaces S1 and S2 are separated by a small gap (width xg). This permits the gas number density to be high so as to generate a large ΔT, while still satisfying the cardinal condition for epicatalysis that the mean free path for gas-phase reactions be appreciable compared with the distance between gas-surface reactions; that is, λrxxg. For a gap spacing of 200 nm, the gas number density can approach the Loschmidt number. A small xg also helps reduce the cycling time for gas particles between S1 and S2, thereby increasing the heat transfer rate between them. The total number of gas molecules inside the cavity should not significantly exceed that corresponding to monolayer coverage of the active surface sites, otherwise S1 and S2 might become so loaded they would be rendered effectively the same chemically, canceling the effect.

Figure 7 STG device. Stationary temperature difference between S1 and S2 (ΔT=T2−T1\Delta T={T_{2}}-{T_{1}}) maintained by thermally driven chemical cycle.
Figure 7

STG device. Stationary temperature difference between S1 and S2 (ΔT=T2T1) maintained by thermally driven chemical cycle.

The device’s areal power density should scale as PσΔEτc, where σ is the areal number density of epicatalytic reaction sites, and τc is the average cycling time for molecules between the surfaces. The cycling time will be largely set by species adsorption or desorption times, but at a minimum it must be at least the transit time for thermal particles to cross the surface-to-surface gap; i. e., τtransxg/vth, where vth is the gas thermal speed. The power density P does not represent chemical free energy; rather, it represents the device’s capacity to transfer heat (W/m2) up the temperature gradient from S1 to S2, using the reaction gases as the working fluid [49]. Ultimately, P is not derived from the device itself; rather, it must be obtained from the surrounding heat bath (e. g., air or water).

The temperature differential ΔT can be increased by minimizing the thermal inertia of the S1 and S2 substrates, perhaps by utilizing one of a growing number of 2-D surfaces, like graphene and its analogs (e. g., Si, Ge, Sn, hexagonal boron nitride); transition-metal dichalcogenides (e. g., MoS2, WSe2, ReS2, TiS2); III–IV compounds (e. g., GaS, InSe); and black phosphorus and its analogs (e. g., SnS) [50].

Thermal shorting between S1 and S2 should be minimized. Radiative heat backflow from S2 to S1 is inevitable if T2>T1, but it can be reduced by choosing mechanical substrates that have poor radiative coupling, perhaps by mirroring the inward-facing surfaces. The active layers of S1 and S2 can be thin films, in principle just nanometers thick, as for self-assembling monolayers, in which case the underlying mechanical substrate can factor strongly into (or even dominate) the device’s optical behavior. The gap distance xg probably should not be reduced much below 100–200 nm so as to avoid super-blackbody radiative coupling between S1 and S2 [51], [52]. To reduce thermal conduction, the pillars separating S1 and S2 (Fig. 7) should minimize the cross-sectional area of contact between S1 and S2; also, they should be made from materials with low thermal conductivity.

Radiative and conductive heat backflows appear manageable, but gas convection is unavoidable because it is precisely gas flow that cycles chemical energy between S1 and S2. Detailed analysis and numerical simulations [53] indicate that, for most room-temperature STG designs, convection dominates heat transfer (and thermal shorting), whereas at high temperatures radiation does. Analysis and simulations also indicate that sizable stationary temperature differences should be achievable. For example, it is predicted that a 1-micron thick cavity containing hydrogen-bonded dimers (e. g., methanol, formic acid) might attain temperature differences of ΔT 10–100 K. The STG for this design would be up to about 108 K/m, roughly three orders of magnitude greater than those seen thus far.

References

[1] S. H. Van ‘t Hoff, Études de Dynamique Chimique, Frederik Muller, Amsterdam, 1884.10.1002/recl.18840031003Search in Google Scholar

[2] D. P. Sheehan, Phys. Rev. E 88 (2013), 032125.10.1103/PhysRevE.88.032125Search in Google Scholar PubMed

[3] The term epicatalysis was coined in 2013, but examples of it appear in the literature earlier, e. g., (1998) [4].Search in Google Scholar

[4] D. P. Sheehan, Phys. Rev. E 57 (1998), 6660.10.1103/PhysRevE.57.6660Search in Google Scholar

[5] T. L. Duncan, Phys. Rev. E 61 (2000), 4661.10.1103/PhysRevE.61.4661Search in Google Scholar

[6] D. P. Sheehan, J. T. Garamella, D. J. Mallin and W. F. Sheehan, Phys. Scr. T151 (2012), 014030.10.1088/0031-8949/2012/T151/014030Search in Google Scholar

[7] D. P. Sheehan, D. J. Mallin, J. T. Garamella and W. F. Sheehan, Found. Phys. 44, (2014) 235.10.1007/s10701-014-9781-5Search in Google Scholar

[8] This section opens following the development from F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Co, New York, 1965, pp. 319–25.Search in Google Scholar

[9] H. Le Chatelier and O. Boudouard, Bulletin de la Société Chimique de France (Paris) 19 (1898), 483.Search in Google Scholar

[10] P. W. Atkins, The Elements of Physical Chemistry, 3rd Ed., Oxford University Press, Oxford, 1993.Search in Google Scholar

[11] L. Onsager, Phys. Rev. 37 (1931), 405.10.1103/PhysRev.37.405Search in Google Scholar

[12] It might be wondered whether surface type could be added to K as another independent variable like T or V, e. g., K(T,V,SY), where SY is surface type. In practice, given the complexities of surface interactions, it is unclear how meaningful including SY would be; moreover, it is unnecessary because surface type is automatically incorporated into K via the states included in the partition function Z. Variation in surface type is already implicit in equations (3), (4); however, if desired it could be made formally explicit by writing (ln(K)x)(ln(K)SY)(SYx).Search in Google Scholar

[13] For the sake of visualization, consider solutions of NaOH and HCl poured into a beaker, forming NaCl solution. The reaction (NaOH + HCl ⟶ NaCl + H2O) occurs in the thin turbulent interfaces where the two solutions mix.Search in Google Scholar

[14] V. Smil, Enriching the Earth: Fritz Haber, Carl Bosch and the Transformation of World Food Production, MIT Press, Cambridge, 2001.10.7551/mitpress/2767.001.0001Search in Google Scholar

[15] B. Strömgren, Astrophys. J. 89 (1939), 526.10.1086/144074Search in Google Scholar

[16] Even under optimal conditions, depletion regions are unlikely to exceed about 100 μm.Search in Google Scholar

[17] L. Dufour, Poggend. Ann. Physik 148 (1873), 490.10.1002/andp.18732240311Search in Google Scholar

[18] S. E. Ingle and F. H. Horne, J. Chem. Phys. 59 (1973), 5882.10.1063/1.1679957Search in Google Scholar

[19] R. G. Mortimer and H. Eyring, Proc. Natl. Acad. Sci. USA 77 (1980), 1728.10.1073/pnas.77.4.1728Search in Google Scholar

[20] M. N. Saha, Proc. R. Soc. A, Math. Phys. Eng. Sci. 99 (1921), 135.10.1098/rspa.1921.0029Search in Google Scholar

[21] R. W. Motley, Q-Machines, Academic Press, New York, 1975.10.1016/B978-0-12-508650-9.50006-0Search in Google Scholar

[22] N. Rynn and N. D’Angelo, Rev. Sci. Instrum. 31 (1960), 1326.10.1063/1.1716884Search in Google Scholar

[23] Q-plasmas do not require magnetic fields; they can also be created inside unmagnetized, three-dimensional blackbody cavities.Search in Google Scholar

[24] K. Kingdon and I. Langmuir, Phys. Rev. 22 (1923), 148.10.1103/PhysRev.22.148Search in Google Scholar

[25] M. J. Dresser, J. Appl. Phys. 39 (1968), 338–339.10.1063/1.1655755Search in Google Scholar

[26] The Q-plasma’s electron number density, set by Richardson emission, also depends on the hotplate work function.Search in Google Scholar

[27] It has been said that the sheath is the first thing one learns about plasmas, but it is the last thing one understands.Search in Google Scholar

[28] K. W. Kolasinski, Surface Science: Foundations of Catalysis and Nanoscience, 2nd ed., John Wiley, England, 2009.10.1016/S1351-4180(09)70251-XSearch in Google Scholar

[29] G. Rothenberg, Catalysis: Concepts and Green Applications, Wiley-VCH, Weinheim, 2008.10.1002/9783527621866Search in Google Scholar

[30] V. Parmon, Thermodynamics of Non-Equilibrium Processes for Chemists with a Particular Application to Catalysis, Elsevier, Amsterdam, 2010.10.1016/B978-0-444-53028-8.00001-0Search in Google Scholar

[31] D. P. Sheehan, Phys. Plasmas 3 (1996), 104.10.1063/1.871834Search in Google Scholar

[32] L. Schäfer, C.-P. Klages, U. Meier and K. Kohse-Höinghaus, Appl. Phys. Lett. 58 (1991), 571.10.1063/1.104590Search in Google Scholar

[33] T. Otsuka, M. Ihara and H. Komiyama, J. Appl. Phys. 77 (1995), 893.10.1063/1.359015Search in Google Scholar

[34] D. P. Sheehan, Phys. Lett. A 280 (2001), 185.10.1016/S0375-9601(01)00060-3Search in Google Scholar

[35] X. Qi, Z. Chen and G. Wang, J. Mater. Sci. Technol. 19 (2003), 235.Search in Google Scholar

[36] F. Jansen, I. Chen and M. A. Machonkin, J. Appl. Phys. 66 (1989), 5749.10.1063/1.343643Search in Google Scholar

[37] D. P. Sheehan and T. A. Zawlacki, Rev. Sci. Instrum. 87 (2016), 074101.10.1063/1.4954971Search in Google Scholar PubMed

[38] D. M. Rowe, ed., Thermoelectrics Handbook: Macro to Nano, CRC Press, 2005.Search in Google Scholar

[39] S. Iwanaga, E. S. Toberer, A. LaLonde and G. J. Snyder, Rev. Sci. Instrum. 82 (2011), 063905.10.1063/1.3601358Search in Google Scholar PubMed

[40] G. Levy, Entropy 15 (2013), 4700.10.3390/e15114700Search in Google Scholar

[41] G. W. Neudeck, Volume II: The PN Junction Diode, 2nd ed., Modular Series on Solid State Devices, G. W. Neudeck and R. F. Pierret, eds, Addison-Wesley Publishing Co., Reading, MA, 1989.Search in Google Scholar

[42] B. Gaveau and L. S. Schulman, Phys. Rev. E 79 (2009), 021112.10.1103/PhysRevE.79.021112Search in Google Scholar PubMed

[43] S. R. de Groot and P. Mazur Non-equilibrium Thermodynamics, Dover, New York, 1984.Search in Google Scholar

[44] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, 1993.10.1007/978-3-642-97430-4Search in Google Scholar

[45] R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics, Wiley-Interscience, New York, 1975.Search in Google Scholar

[46] D. P. Sheehan, J. Sci. Explor. 12 (1998), 303.10.1076/clin.12.2.303.2007Search in Google Scholar

[47] Radiation dominates the energy budget in these high-temperature systems, however, it couples weakly to the particles (electrons, ions, and atoms), such that they can maintain temperature disparities and gradients.Search in Google Scholar

[48] D. P. Sheehan, Epicatalytic Thermal Diode; U.S. patent No. 9,212,828, (2015).Search in Google Scholar

[49] For the sake of illustration, let the number of epicatalytically active surface sites on S1 and S2 be σ=1016m2 and the reaction energy be ΔE=0.5eV=8×1019J, typical of hydrogen-bonded dimers like methanol and formic acid [37]. Let the average cycling time τc be the thermal transit time for room-temperature 100 amu molecules (typical for volatile species) to cross the S1–S2 gap of thickness 10−6 m, that is, τcxgvth108s. With these, the areal power density for the STG device is estimated to be P106W/m2. This, of course, will be reduced by unavoidable convective, radiative, and conductive backflows.Search in Google Scholar

[50] P. Ajayan, P. Kim and K. Banerjee, Phys. Today 69(9) (2016), 39.10.1063/PT.3.3297Search in Google Scholar

[51] E. Rousseau, et al., Nat. Photonics 3 (2009), 514.10.1038/nphoton.2009.144Search in Google Scholar

[52] K. Kim, et al., Nat. Nanotechnol. 10 (2015), 253.10.1038/nnano.2015.6Search in Google Scholar PubMed

[53] D. P. Sheehan and T. Welsh, in review (2018).Search in Google Scholar

Received: 2018-01-17
Revised: 2018-05-25
Accepted: 2018-05-28
Published Online: 2018-06-21
Published in Print: 2018-10-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/jnet-2017-0007/html
Scroll to top button