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Paper PCCP View Article Online Let us first consider this process of transformation of chemical energy into an emf from the point of view of thermo- dynamics. The total energy of the LEC can be expressed as E 1⁄4 1MX_ 2 þ UðX; QÞ; f ðX; QÞ 1⁄4 @ UðX; QÞ; (19) 2 @X with time derivative E ̇ = (gdiss + gload)MX: 2 + Vd(Q,X)Q: . (20) Eqn (20) can be interpreted in terms of the first law of thermo- dynamics. Taking the system of interest to be the battery’s half- cell, we have that dE = dQ dW + mdN, (21) with internal energy E, heat flow to the environment Q = gdissMX: 2, and electric power output W: = gloadMX: 2. We identify the electrochemical potential m in eqn (21) with qVd(Q,X), and the quantity of matter N with Q/q. It should be stressed that the two damping coefficients gdiss and gload, which appear in eqn (8) and (20), are qualitatively different from a thermodynamic point of view: gdiss represents the dissipation of mechanical energy into the disordered motion of the microscopic components of the environment in which the double layer is immersed, while gload describes the transfer of energy from the self-oscillating double layer into the pumping of a coherent, macroscopic current that carries no entropy. The fluctuation-dissipation theorem therefore applies only to gdiss, a point that should be born in mind if the presentmodel were extended to incorporate thermal noise via a Fokker–Planck equation. The second law of thermodynamics is satisfied because heat is dissipated into the: environment (dQ o 0) at constant temperature T, so that S = Q/T 4 0. Integrating eqn (21) over a complete cycle of the system’s thermodynamic state we obtain that Fig. 4 Chemical-engine cycle represented in the (m,N)-plane. W0 III dW o VdðQ; XÞdQ 1⁄4 mdN; (22) to the generation of an emf E 1⁄4 W0: (23) Q Note that the modulation of m with respect to N, required by the Rayleigh-Eddington criterion, is possible only if the thermo- dynamic cycle (which in this case corresponds to the self- oscillation of the double layer) is slow compared to the time-scale of the chemical reactions that control the value of m at each point within the cycle. For instance, an automobile can run because it takes in pristine fuel at high chemical potential and expels burnt fuel at low chemical potential.† Combustion must therefore proceed quickly compared to the period of the motion of engine’s pistons. If the time taken by the combustion were comparable to the period of the piston, m would not vary effectively with dN in eqn (22) and little or no net work could be done by the engine. The self-oscillation of the double layer is also the pumping cycle, which converts mechanical into electrical work, as we shall discuss in more detail in Section III E. This pumping must therefore be slow with respect to the redox reactions from which the battery ultimately takes the energy to generate the electrical work. This conceptual distinction between the fast chemical reaction and the slower pumping is, in our view, the key missing ingredient in all previous theoretical treatments of the battery. E Current pumping Having described the extraction of mechanical work by the double layer’s self-oscillation, we proceed to consider how that work can be used to pump an electrical current, thereby generating the battery’s emf. As shown in ref. 13, the instanta- neous power that an irrotational electric field E(t,r) delivers to a current density J(t,r) contained in a volume V is ð 3 ð@r3 P1⁄4 EJdr1⁄4 f@tdr; (24) VV where f(t,r) is the potential (such that E = =f) and r(t,r) is the charge density (such that =J = qr/qt). † The internal combustion engine is usually conceptualized as a heat engine, with the air in the cylinder as working substance. But it is also possible to consider it as a chemical engine, with the fuel-air mixture as working substance. Of course, in the latter analysis, eqn (22) gives only a loose upper bound on the extracted work because of the large amount of heat (and therefore entropy) that the engine dumps into the environment when it expels the burnt fuel. where W0 is the useful work generated by one limit cycle of the self-oscillation. The right-hand side of eqn (22) is the area contained within a closed trajectory on the (m,N)-plane, as shown in Fig. 4. This curve is the thermodynamic cycle of the half-cell considered as a chemical engine. Eqn (22) implies that sustained work extraction (W0 4 0) by an open system coupled to an external chemical disequilibrium requires the system to change its state in time in such a way that m varies in phase with N. This is a particular instance of what was called the generalized ‘‘Rayleigh-Eddington criterion’’ (after the physicists who clearly formulated this principle for heat engines) in ref. 38. A clear example of this principle for a microscopic chemical engine is provided by the electron shuttle.29–31 The work W0 extracted by the cycle can then be used to to pump electric charge Q% against the time-averaged electric potential difference at the half-cell, which corresponds This journal is © the Owner Societies 2021 Phys. Chem. Chem. Phys., 2021, 23, 9428–9439 | 9433 Open Access Article. Published on 23 March 2021. Downloaded on 6/26/2022 1:50:45 PM. 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