A Symmetric Van ’t Hoff Equation and Equilibrium Temperature Gradients

D. P. Sheehan

doi:10.1515/jnet-2017-0007

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A Symmetric Van ’t Hoff Equation and Equilibrium Temperature Gradients

Veröffentlicht/Copyright: 21. Juni 2018

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Aus der Zeitschrift Journal of Non-Equilibrium Thermodynamics Band 43 Heft 4

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.

Keywords: non-equilibrium thermodynamics; Van ’t Hoff equation; epicatalysis; catalysis; second law of thermodynamics

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=T2−T1) 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, λrx≳xg. 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=T2−T1) maintained by thermally driven chemical cycle.

The device’s areal power density should scale as P≃σ∗ΔEτc, 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., τtrans≃xg/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.

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Received: 2018-01-17

Revised: 2018-05-25

Accepted: 2018-05-28

Published Online: 2018-06-21

Published in Print: 2018-10-25

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https://doi.org/10.1515/jnet-2017-0007

Schlagwörter für diesen Artikel

non-equilibrium thermodynamics; Van ’t Hoff equation; epicatalysis; catalysis; second law of thermodynamics