The article information
 Tiejun Xiao, Yun Zhou
 肖铁军, 周匀
 Stochastic Thermodynamics of Mesoscopic Electrochemical Reactions
 介观电化学反应的随机热力学
 Chinese Journal of Chemical Physics, 2018, 31(1): 6165
 化学物理学报, 2018, 31(1): 6165
 http://dx.doi.org/10.1063/16740068/31/cjcp1705110

Article history
 Received on: May 27, 2017
 Accepted on: June 26, 2017
Stochastic thermodynamics (ST) has been a very active field in the modern statistic physics [14]. The fluctuations could be significant in mesoscopic systems, and hence various thermodynamic properties such as work, heat, and system entropy become stochastic variables [58]. One can introduce the system entropy
As a special case of the birthdeath stochastic processes, chemical reactions have gained considerable attentions. In general, the state of a reaction system is described by the number vector
In this study, we focus on the electrochemical reactions where the electrons are transferred between ions and electrodes, and hence the electrode potential
In this work, a model electrochemical reaction system is first introduced. ST of electrochemical reactions is shown in a thermodynamic relationship between the reaction rate constant and the electrode potential is presented and the fluctuation theorem is discussed.
Ⅱ. MODEL OF THE MESOSCOPIC ELECTROCHEMICAL SYSTEMSConsider a model electrochemical reaction system with fixed volume
$ \begin{eqnarray} {Y} \rightleftharpoons {X} + e^(\textrm{electrode}) \end{eqnarray} $  (1) 
where
$ \begin{array}{*{20}c} W_{+}( \textbf{X} ) = k_{+} \textbf{Y}\\ W_{}( \textbf{X} ) = k_{} \textbf{X}\\ \end{array} $  (2) 
where
The
$ \begin{eqnarray} \partial_t p(\textbf{X};t) &=& \sum\limits_{\rho = \pm } [W_{\rho}(\textbf{X}v_{\rho}) p(\textbf{X}v_{\rho};t)\nonumber\\ &&{}W_{\rho}(\textbf{X}) p(\textbf{X};t)] \end{eqnarray} $  (3) 
where
For the reaction system, the reaction events change the state variable of the system and hence generate a stochastic trajectory. Denote
$ \begin{eqnarray} \chi(t)&=&\{ \textbf{X}_0(t''=t_0)\rightarrow\textbf{X}_1(t''=t_1)\rightarrow... \rightarrow\nonumber\\[2ex] &&{}\textbf{X}_{n1}(t''=t_{n1})\rightarrow\textbf{X}_n(t''=t_n) \rightarrow\nonumber\\[2ex]&&{}\textbf{X}_n(t''=t)\}\nonumber \end{eqnarray} $ 
its conjugate backward trajectory
$ \begin{eqnarray} \chi^R(t)&=&\{\textbf{X}_n(t''={t'}_0)\rightarrow\textbf{X}_{n1}(t''={t'}_1)\rightarrow...\rightarrow\nonumber\\[2ex] &&{} \textbf{X}_{1}(t''={t'}_{n1})\rightarrow\textbf{X}_0(t''={t'}_n)\rightarrow\textbf{X}_0(t''=t)\}\nonumber \end{eqnarray} $ 
$ \begin{eqnarray} P[\chi(t) \textbf{X}_0]&=& \Pi_{j=1}^n \textrm{e}^{ r( \textbf{X}_{j1}) \tau_j } \displaystyle{W_{\rho_j}( \textbf{X}_{j1}) \over r( \textbf{X}_{j1})}\nonumber\\ &&{} \textrm{e}^{r( \textbf{X}_n)(t t_n)} \end{eqnarray} $  (4) 
$ \begin{eqnarray} P[\chi^R(t) \textbf{X}_t]&=& \Pi_{j=1}^n \textrm{e}^{ r( \textbf{X}_{j1}) \tau_j } \displaystyle {W_{\rho_j}( \textbf{X}_j) \over r( \textbf{X}_{j1})}\nonumber\\ &&{} \textrm{e}^{r( \textbf{X}_n)(t t_n)} \end{eqnarray} $  (5) 
where
$ \begin{eqnarray} \label{Sigm} \Sigma &\equiv& \ln {P[\chi(t) \textbf{X}_0] \over P[{\chi^R(t)} \textbf{X}_t]}\nonumber\\ &=& \sum\limits_{j=1}^n \ln {W_{\rho_j}( \textbf{X}_{j1}) \over W_{\rho_j}( \textbf{X}_j)} \end{eqnarray} $  (6) 
For the simple reaction systems, the dynamic irreversibility
In order to study the energetics of the reaction system, we consider the thermodynamic properties. The chemical potential
Denote
$ \beta \Delta A^j = v_{\rho_j} [\beta \mu_{X}^* + \ln {(X_{j1} + v_{\rho_j})}] $  (7) 
$ \Delta S_0^j = v_{\rho_j} s_X $  (8) 
$ \beta \Delta U^j = \beta \Delta A^j + \Delta S_0^j $  (9) 
In order to keep the particle number of
$ \begin{eqnarray} \beta w_{\textrm{chem}}^j &=&  u_{\rho_j} \beta \mu_{Y}\nonumber\\ &=&  u_{\rho_j} (\beta \mu_{Y}^* + \ln \textbf{Y}) \end{eqnarray} $  (10) 
where
$ \begin{eqnarray} \beta w_{\textrm{ele}}^j= v_{\rho_j} e_0 \psi \end{eqnarray} $  (11) 
where
Since the temperature
$ \begin{eqnarray} \beta q^j = \beta w_{\textrm{chem}}^j +\beta w_{\textrm{ele}}^j \beta \Delta U^j \end{eqnarray} $  (12) 
Along a single trajectory which consists of
$ \begin{eqnarray} \beta q &=&  \sum\limits_{j=1}^n [u_{\rho_j} \beta \mu_{Y}^* + v_{\rho_j} \beta \mu_{X}^* + v_{\rho_j} \textrm{e}_0 \psi]\nonumber \\ &&{}\hspace{0.2cm}\sum\limits_{j=1}^n [u_{\rho_j} \ln \textbf{Y} + v_{\rho_j} \ln {(\textbf{X}_{j1} + v_{\rho_j}) }]\nonumber \\ &&{}\hspace{0.2cm} \sum\limits_{j=1}^n \Delta S_0^j \end{eqnarray} $  (13) 
which implies that the heat dissipation has a linear dependence on the electrode potential.
C. Dynamicthermodynamic relationHerein we demonstrate that the heat dissipation
$ \textbf{x}_s\equiv\lim\limits_{V\rightarrow \infty} \frac{\textbf{X}}{V} $  (14) 
$ \textbf{y}\equiv\lim\limits_{V\rightarrow \infty} \frac{\textbf{Y}}{V} $  (15) 
The zero current condition in the equilibrium system is:
$ \begin{eqnarray} k_{+}\textbf{y}k_{} \textbf{x}_s=0\end{eqnarray} $  (16) 
or equivalently:
$ \begin{eqnarray} \ln K= \ln\frac{k_{+}}{ k_{}}= \ln\frac{\textbf{x}_s}{\textbf{y}} \end{eqnarray} $  (17) 
The electric potential equilibrium condition reads:
$ \begin{eqnarray} \beta \mu_X(\textbf{x}_s V)\beta \mu_Y(\textbf{y} V)+e_0 \psi=0\end{eqnarray} $  (18) 
or equivalently:
$ \begin{eqnarray} u_{+} \beta \mu_Y^*+v_{+} \beta \mu_X^*+v_{+} e_0\psi=\ln{\frac{\textbf{x}_s}{\textbf{y}}} \end{eqnarray} $  (19) 
By comparing these two constraints, one can find that
$ \begin{eqnarray} \ln K &\equiv& \ln \frac{k_{+ }}{k_{ }} \nonumber\\ &=& ( u_{+} \beta \mu_{Y}^* + v_{+} \beta \mu_X^* + v_{+}e_0 \psi) \end{eqnarray} $  (20) 
One may note that it is possible to derive this equation in a different way, e.g., one can use the reaction rate constant
$ \begin{eqnarray} \beta q&=&\displaystyle \sum\limits_{j=1}^n \ln\displaystyle \frac{W_{\rho_j}( \textbf{X}_{j1})}{ W_{\rho_j}( \textbf{X}_j)}  \sum\limits_{j=1}^n \Delta S_0^j\\ &=& \Sigma  \Delta S \end{eqnarray} $  (21) 
$ \Delta S_0\equiv\displaystyle\sum\limits_j \Delta S_0^j $  (22) 
where
For the reaction system along the single trajectory
$ \begin{eqnarray} \label{entropy} s = S_0  \ln [p(\textbf{X}_t;t)] \end{eqnarray} $  (23) 
The system entropy change can then be evaluated as
$ \begin{eqnarray} \label{entropychange} \Delta s = \Delta S_0 + \ln {p(\textbf{X}_0, t_0)\over p(\textbf{X}_t;t)} \end{eqnarray} $  (24) 
When the internal entropy term is evaluated analytically from the grand canonical ensemble, one can find that
Using Eq.(21) and Eq.(24), the total entropy change
$\begin{eqnarray} \label{1Lawc} \Delta s_\textrm{t} &=& \Sigma + \ln{ p(\textbf{X}_0, t_0) \over p(\textbf{X}_t, t)}\nonumber\\ &=& \ln\frac{P(\chi)}{P(\chi^R)} \end{eqnarray} $  (25) 
According to Eq.(25), the total entropy change is equal to the log ratio of the probability of a forward trajectory to that of a conjugate backward trajectory. Then following the standard procedure in the stochastic thermodynamics [3, 22], it is straightforward to prove that
$ \begin{eqnarray} \langle e^{\Delta s_\textrm{t}}\rangle =1 \end{eqnarray} $  (26) 
and then the second law of thermodynamics follows as
As one can see, the propensity function
In this study, we discussed the stochastic thermodynamics of electrochemical reactions, where the electrode potential plays a role. We find a relationship between the reaction constant
This work was supported by the National Natural Science Foundation of China (No.21403041 and No.21503048) and the Startup Packages from Guizhou Education University.
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