Understanding the principles of thermodynamics is of interest for multiple scientific disciplines. Thermodynamic analysis is the bread and butter of many physicists and engineers. For them, it is usually unproblematic to understand the basics of thermodynamics because of their strong background in mathematics. On the other end of the spectrum, chemists also frequently have to analyze thermodynamics in order to predict and optimize chemical reactions, and biologists also come into contact with the topic in the context of toxycology, enzymology or metabolic flux analysis. For this audience it is often not easy to thoroughly understand the concepts of thermodynamics. To make things worse, in chemistry text books like “Physical Chemistry” (Atkins 2006), the considered thermodynamic systems are not always defined rigorously enough. The following unreviewed article is a summary of the authors understanding of the topic as a graduate student in biochemistry. This text is provided without guarantees of completeness or accuracy.

Thermodynamics in reactive open systems

A scientific law is a statement that describes what phenomena happen (observed or predicted). Overlap between scientific theory and scientific laws allows for predictions. By repeated successful predictions, the appropriate field of application for the law can be extended. If we consider a system which together with its surroundings constitutes a larger, isolated system (the universe), one important law is the law of conversion of energy which states that the energy of the universe is constant. Thus

$$\begin{array}{}\text{(1)}& \mathrm{d}{E}_{universe}=\mathrm{d}{E}_{system}+\mathrm{d}{E}_{surroundings}=0\end{array}$$

where $E_{system}$ the energy of the system, which is the sum of the system’s kinetic energy $E_{kin}$, the system’s potential Energy $E_{pot}$ and the system’s internal energy $U$. Thus

$$\begin{array}{}\text{(2)}& \mathrm{d}{E}_{system}=\mathrm{d}{E}_{kin}+\mathrm{d}{E}_{pot}+\mathrm{d}U.\end{array}$$

If the kinetic energy of the system as a whole and potential Energy of the system as a whole (e.g. due to interaction with an external gravitational field) is not considered, i.e. for $\mathrm{d}{E}_{kin}=0$, $\mathrm{d}{E}_{pot}=0$, it follows that $\mathrm{d}{E}_{system}=\mathrm{d}U$.

The zeroth law of thermodynamics states that if two thermodynamic systems are each in thermal equilibrium with a third one, then they are in thermal equilibrium with each other. Maxwell expressed it in the words “All heat is of the same kind” (Maxwell, Clerk 1871). It can also expressed by saying that there is precisely one kind of non-mechanical, non-matter-transferring contact equilibrium between thermodynamic systems.

The most general case that can be considered is a reactive, open system that can exchange energy and matter with its surroundings. For an open, reaktive system where elementary chemical reactions k take place the first principle of thermodynamics can be expressed as

$$\begin{array}{}\text{(3)}& \mathrm{d}U(S,V,N)=\mathrm{d}\mathrm{\Phi }+\delta w\end{array}$$

with $\mathrm{d}\mathrm{\Phi }$: General heat, and

$\delta w$: infinitesimally small change in work done by the system (inexact derivative)

The second principle of thermodynamics can then be expressed as (Petrucci et al. 2016, p.600):

$$\begin{array}{}\text{(4)}& \mathrm{d}{S}_{universe}=\mathrm{d}{S}_{system}+\mathrm{d}{S}_{surroundings}\ge 0.\end{array}$$

For reversible processes,

$$\begin{array}{}\text{(5)}& \mathrm{d}{S}_{universe}=\mathrm{d}{S}_{system}+\mathrm{d}{S}_{surroundings}=0\end{array}$$

For irreversible processes,

$$\begin{array}{}\text{(6)}& \mathrm{d}{S}_{universe}=\mathrm{d}{S}_{system}+\mathrm{d}{S}_{surroundings}>0\end{array}$$

According to the second principle of thermodynamics (De Decker 2015),

$$\begin{array}{}\text{(7)}& \mathrm{d}S=\mathrm{d}{S}_{int}+\sum _i(\mathrm{d}{S}_{e,i});\mathrm{d}{S}_{int}\ge 0,\end{array}$$

with $\mathrm{d}{S}_{int}$: infinitesimally small change in entropy due to internal processes (caused by chemical reactions), and

$\mathrm{d}{S}_{e,i}$: infinitesimally small entropy, tranfered from the connected system i through mass exchange of compound i.

$$\begin{array}{}\text{(8)}& \sum _i\mathrm{d}{S}_{e,i}=\frac{\mathrm{d}\mathrm{\Phi }}{T}-\frac{1}{T}\sum _k({\mu }_k\mathrm{d}{n}_{e,k})\end{array}$$

$$\begin{array}{}\text{(9)}& \mathrm{d}{S}_{int}=-\frac{1}{T}\sum _k{\mu }_k\mathrm{d}{n}_{int,k}\end{array}$$

Thus

$$\begin{array}{}\text{(10)}& \mathrm{d}S=\frac{\delta \mathrm{\Phi }}{T}-\frac{1}{T}\sum _k({\mu }_k\mathrm{d}{n}_{e,k})-\frac{1}{T}\sum _k({\mu }_k\mathrm{d}{n}_{int,k})\end{array}$$

If the heat $\delta q=T\mathrm{d}S+\sum _k({\mu }_k\mathrm{d}{n}_{int,k})$ is introduced, it follows (De Decker 2015) that

$$\begin{array}{}\text{(11)}& \mathrm{d}\mathrm{\Phi }=\delta q+\sum _k({\mu }_k\mathrm{d}{n}_{e,k})\end{array}$$

Here, $\delta q$: infinitesimally small change in heat (inexact derivative).

Inserting $\text{(11)}$ into $\text{(3)}$ yields

$$\begin{array}{}\text{(12)}& \mathrm{d}U(S,V,N)=\delta q+\delta w+\sum _k{\mu }_k\mathrm{d}{n}_{e,k}\end{array}$$

Different approaches towards a thermodynamic description of systems far from equilibrium (Jazinsky 2011; Esposito and Van den Broeck 2011; Prokovskii 2013) and description of chemically reactive systems were presented in recent literature (De Decker 2015; Rao and Esposito 2016).

Prigogine introduced the concept of internal variables ${\xi }_i$ for discussion of non-equilibrium thermodynamics (Prigogine 1955; Prigogine 1967). The energy dissipation is caused by entropy production due to relaxation of internal variables which upon relaxation to discuss thermodynamics of chemical reactions. Prigogine showed that in the case of chemical reactions, the internal parameter $\xi$ is a measure of the extend of reaction (Prigogine 1955; Prigogine 1967). The infinitesimally small change in the extend of reaction is defined as

$$\begin{array}{}\text{(13)}& \mathrm{d}\xi =\frac{\mathrm{d}{n}_i}{{\nu }_i}\end{array}$$

with ${\nu }_i$: stochiometric number (stochiometric coefficient) of reactand i.

Prigogine showed that for chemically reactive systems, the extend of reaction $\xi$ From $\mathrm{\Delta }\xi =\frac{\mathrm{\Delta }{n}_i}{{\nu }_i}$ and from the fact that at the beginning of the reaction by definition ${\nu }_i=0$ follows that $\mathrm{\Delta }\xi =\xi$. Therefore:

$$\begin{array}{}\text{(14)}& \xi =\frac{\mathrm{\Delta }{n}_i}{{\nu }_i}\end{array}$$

The rate of reaction is (Davis and Davis 2003)

$$\begin{array}{}\text{(15)}& r=\frac{\mathrm{d}\xi }{\mathrm{d}t}\end{array}$$

The vector $J$ whose components are all reaction rates $r_i$ relevant for the system under consideration (De Decker 2015; Pekar 2018) looks as if it could be the reaction flux.

Unreactive open systems

For an unreactive open system that is connected to other systems via different walls that are only permeable to either heat (diathermic wall), or work (piston for adiabatic work) or matter of a specific compound i (semipermeable membrane that is permeable only by compound i) the internal energy of the open system is (Knuiman et al. 2012):

$$\begin{array}{}\text{(16)}& \mathrm{d}U(S,V,N{)}_o=\mathrm{d}{U}_c+\sum _i\mathrm{d}{U}_{em,i}\end{array}$$

with $\mathrm{d}U(S,V,N{)}_{em,i}$: infinitesimally small internal energy, transfered from the connected system i through mass exchange of compound i and

$\mathrm{d}U(S,V,N{)}_c$: inner engery of the whole closed system.

According to the first law of thermodynamics,

$$\begin{array}{}\text{(17)}& \mathrm{d}U(S,V,N{)}_o=\delta q+\delta w+\sum _i\mathrm{d}{U}_{em,i}\end{array}$$

The mass exchange is quantized, therefore the last equation becomes

$$\begin{array}{}\text{(18)}& \mathrm{d}U(S,V,N{)}_o=\delta q+\delta w+\sum _i{u}_i\mathrm{d}{n}_i,\end{array}$$

with $n_i$: amount of substance of type i

$u_i$: internal energy per unity of amount of substance i.

Equation $\text{(18)}$ makes clear that change in internal energy cannot always be uniquely resolved into heat and work components Knuiman et al. 2012.

Quasistatic processes and reversible processes

If the process of mass exchange is quasistatic or even reversible (quasistatic, no dissipation), the expression for the work is

$$\begin{array}{}\text{(19)}& \delta w=\delta w_o+\delta w_{flow}.\end{array}$$

Since for the given process $\delta w_{flow}=\sum _i{p}_{I,i}{v}_i\mathrm{d}{n}_i$ (Knuiman et al. 2012) the expression for the work becomes

$$\begin{array}{}\text{(20)}& \delta w=\delta w_o+\sum _i{p}_{I,i}{v}_i\mathrm{d}{n}_i.\end{array}$$

When the mass exchange is a quasistatic (or even reversible) transfer, the second law of thermodynamics holds:

$$\begin{array}{}\text{(21)}& \mathrm{d}S=\frac{\delta q}{T}+\sum _i\mathrm{d}{S}_{em,i}\end{array}$$

The mass exchange is quantized, hence

$$\begin{array}{}\text{(22)}& \mathrm{d}S=\frac{\delta q}{T}+\sum _i{s}_i\mathrm{d}{n}_i\end{array}$$

with $s_i$: Entropy per unity of amount of substance i

Inserting $\text{(20)}$ and $\text{(22)}$ into $\text{(18)}$ yields

$$\begin{array}{}\text{(23)}& \mathrm{d}U(S,V,N{)}_o=T\mathrm{d}{S}_o+\sum _i{s}_i\mathrm{d}{n}_i+\delta w_o+\sum _i{p}_{I,i}{v}_i\mathrm{d}{n}_i+\sum _i{u}_i\mathrm{d}{n}_i\end{array}$$

In equation $\text{(23)}$ the contributions of exchanged heat and work are still seperate.

That seperation vanishes when the equation is simplified to

$$\begin{array}{}\text{(24)}& \mathrm{d}U(S,V,N{)}_o=T\mathrm{d}{S}_o+\delta w_o+\sum _i{\mu }_i\mathrm{d}{n}_i\end{array}$$

with ${\mu }_i$: chemical potential of component i, ${\mu }_i=(\frac{\mathrm{d}G}{\mathrm{d}{n}_i}{)}_{p,T,n^{\prime }}$

and $\delta w_o=\sum _j{X}_j\mathrm{d}{a}_{o,j}$.

Notably, in equation $\text{(24)}$, the first term does not exactly (uniquely) correspond to the exchanged heat (see $\text{(21)}$). Similarly, the term involving the chemical potential is neither a pure work term nor $\sum _i\mathrm{d}{U}_{em,i}$ (see $\text{(23)}$). Rather, there are mixed contributions of heat and work in all terms of equation $\text{(24)}$.

For any system and any general process the Gibbs equation is:

$$\begin{array}{}\text{(25)}& \mathrm{d}U(S,V,N)=T\mathrm{d}S+\sum _j{X}_j\mathrm{d}{a}_j+\sum _i{\mu }_i\mathrm{d}{n}_i,\end{array}$$

Equation $\text{(25)}$ still holds, if the transfer of energy is not quasistatic, but in that case the equation cannot easily be derived from the first and second law of thermodynamics, because those laws are only defined for quasistatic processes in certain systems or for any process in a closed system. Thus the first term does not necessarily correspond to pure heat ($\delta q\ne T\mathrm{d}S$) and the later terms do not necessarily correspond to pure work ($\delta w\ne \sum _j{X}_j\mathrm{d}{a}_j$) (Callen 1985, p97).

Fundamental equations of thermodynamics

$$\begin{array}{}\text{(26)}& H(S,p,N)=U(S,V,N)+pV\end{array}$$

$$\begin{array}{}\text{(27)}& F(T,V,N)=U(S,V,N)–TS\end{array}$$

$$\begin{array}{}\text{(28)}& G(p,T,N)=H–TS=U+pV–TS\end{array}$$

For any system and any general process the Gibbs equation is given by $\text{(25)}$. With $X_{Vol}=-p$ and $a_{Vol}=V$, the work can be seperated into volume-work and non-volume-work: $\delta w=\delta w_{Vol}+\delta w^{\prime }=-p\mathrm{d}V+\sum _l{Y}_l\mathrm{d}{y}_l$. Thus

$$\begin{array}{}\text{(29)}& \mathrm{d}U(S,V,N)=T\mathrm{d}S-p\mathrm{d}V+\sum _l{Y}_l\mathrm{d}{y}_l+\sum _i({\mu }_i\mathrm{d}{n}_i),\end{array}$$

Insertion of $\text{(26)}$ into $\text{(29)}$ yields

$$\begin{array}{}\text{(30)}& \mathrm{d}H(S,p,N)=T\mathrm{d}S+V\mathrm{d}p+\sum _l{Y}_l\mathrm{d}{y}_l+\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

Insertion of $\text{(27)}$ into $\text{(29)}$ yields

$$\begin{array}{}\text{(31)}& \mathrm{d}F(T,V,N)=-S\mathrm{d}T-p\mathrm{d}V+\sum _l{Y}_l\mathrm{d}{y}_l+\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

Insertion of $\text{(28)}$ into $\text{(29)}$ yields

$$\begin{array}{}\text{(32)}& \mathrm{d}G(p,T,N)=-S\mathrm{d}T+V\mathrm{d}p+\sum _l{Y}_l\mathrm{d}{y}_l+\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

From $\text{(32)}$ follows with the second principle of thermodynamics, $\text{(7)}$, that $\mathrm{d}G\le \delta w^{\prime }$. If the system is at equilibrium (which requires that all processes under consideration are reversible), $dG=\delta w^{\prime }$, else $dG<\delta w^{\prime }$.

For homogenous macroscopic systems, it follows from equation $\text{(25)}$ according to Eulers homogeneous function theorem that

$$\begin{array}{}\text{(33)}& U(S,V,N)=TS–pV+\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

Insertion of $\text{(33)}$ into $\text{(26)}$ yields

$$\begin{array}{}\text{(34)}& H(S,p,N)=TS+\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

Insertion of $\text{(33)}$ into $\text{(28)}$ yields

$$\begin{array}{}\text{(35)}& G(p,T,N)=\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

Spontaneous irreversible processes and reversible processes

From the second law of thermodynamics $\text{(4)}$ follows the inequality for spontaneous change (Petrucci et al. 2016; Nelson et al. 2008):

$$\begin{array}{}\text{(36)}& \mathrm{d}S\ge \frac{\delta q}{T}\end{array}$$

For reversible processes,

$$\begin{array}{}\text{(37)}& \mathrm{d}S=\frac{\delta q}{T}.\end{array}$$

Irreversible processes are spontaneous if

$$\begin{array}{}\text{(38)}& \mathrm{d}S>\frac{\delta q}{T}\end{array}$$

Reorganizing $\text{(38)}$ and inserting it into $\text{(12)}$ yields the following inequality that holds for spontaneous changes (Nelson et al. 2008):

$$\begin{array}{}\text{(39)}& \mathrm{d}U(S,V,N)-T\mathrm{d}S-\delta w-\sum _k({\mu }_k\mathrm{d}{n}_{e,k})<0\end{array}$$

with $\delta w=\delta w_{Vol}\delta w^{\prime }=-p\mathrm{d}V+\sum _l(Y_l\mathrm{d}{y}_l)$. Seperation of volume work $\variation{w_{Vol}} and non-volume-work \variation{w’} yields

$$\begin{array}{}\text{(40)}& \mathrm{d}U(S,V,N)-T\mathrm{d}S+p\mathrm{d}V-\delta w^{\prime }-\sum _k({\mu }_k\mathrm{d}{n}_{e,k})<0.\end{array}$$

For an isolated system ($\delta q=0$, $\delta w=0$, $\mathrm{d}V=0$, $\mathrm{d}{n}_{e,k}=0$, $\mathrm{d}U=0$):

$$\begin{array}{}\text{(41)}& (\mathrm{d}S{)}_{U,V}\ge 0.\end{array}$$

For closed systems with varying constant parameters, the inequality for spontaneous change is given below. Indices indicate constant parameters.

$$\begin{array}{}\text{(42)}& (\mathrm{d}U(S,V,N){)}_{S,V}\le 0.\end{array}$$

$$\begin{array}{}\text{(43)}& (\mathrm{d}H(S,p,N){)}_{S,p}\le \delta w^{\prime }\end{array}$$

$$\begin{array}{}\text{(44)}& {\mathrm{\Delta }}_{\mathrm{r}}H={\left(\frac{\partial H(S,p,\xi )}{\partial \xi }\right)}_{S,p}\le \delta w^{\prime }\end{array}$$

$$\begin{array}{}\text{(45)}& (\mathrm{d}S{)}_{H,p}\ge \delta w^{\prime }\end{array}$$

$$\begin{array}{}\text{(46)}& {\mathrm{\Delta }}_{\mathrm{r}}S={\left(\frac{\partial S}{\partial \xi }\right)}_{H,p}\ge \delta w^{\prime }\end{array}$$

$$\begin{array}{}\text{(47)}& (\mathrm{d}F(T,V,N){)}_{V,T}\le \sum _l(Y_l\mathrm{d}{y}_l)\end{array}$$

$$\begin{array}{}\text{(48)}& {\mathrm{\Delta }}_{\mathrm{r}}F={\left(\frac{\partial F(T,V,\xi )}{\partial \xi }\right)}_{T,V}\le \delta w^{\prime }\end{array}$$

$$\begin{array}{}\text{(49)}& (\mathrm{d}G(p,T,N){)}_{p,T}\le \delta w^{\prime }\end{array}$$

$$\begin{array}{}\text{(50)}& {\mathrm{\Delta }}_{\mathrm{r}}G={\left(\frac{\partial G(p,T,\xi )}{\partial \xi }\right)}_{p,T}\le \delta w^{\prime }\end{array}$$

The changes in the potentials of thermodynamics above are equal to $\delta w^{\prime }$ if, and only if the system is at thermodynamic equilibrium, which requires that all processes under consideration are reversible.

Confusingly, in the equations above, the symbol ${\mathrm{\Delta }}_{\mathrm{r}}$ stands for the partial derivative against the extend of reaction, not for a difference.

Generalized work

In any case, the relevant generalized work terms relevant for the system and process under consideration are required:

$$\begin{array}{}\text{(51)}& \mathrm{d}w=\sum _j{X}_j\mathrm{d}{a}_j,\end{array}$$

where $X_j$: Generalized force (intrinsic variable) conjugated with the extrinsic parameter $a_j$ .

The most relevant intrinsic and extrinsic variables are given in the table below. For details, see (Balmer 2011).

Type of work	Symbol	Generalized force	extrinsic parameter
volume-pressure	$\delta w_{vol}$	pressure ($p$)	volume ($V$)
electrochemical	$\delta w_e$	electromotive force ($\epsilon$)	charge ($Q$)

In the context of electrochemical cells, the electromotive force is called source voltage. It is equal to the electric potential difference for zero current through the cell IUPAC 2019.

Closed systems and unreactive systems

For closed systems $\mathrm{d}{n}_{e,i}=0$ and therefore also $\sum _i({\mu }_i\mathrm{d}{n}_i)=0$.

Thus, from $\text{(25)}$ follows for closed systems:

$$\begin{array}{}\text{(52)}& \mathrm{d}U=T\mathrm{d}S–\sum _j(X_j\mathrm{d}{a}_j)\end{array}$$

For an isochoric system ($\mathrm{d}V=0$) where no no work other than volume work is performed ($\delta w=-p\mathrm{d}V$):

$$\begin{array}{}\text{(53)}& \mathrm{d}H=\delta q\end{array}$$

For an isobaric system ($\mathrm{d}p=0$), where no no work other than volume work is performed ($\delta w=-p\mathrm{d}V$):

$$\begin{array}{}\text{(54)}& \mathrm{d}H=dU+p\mathrm{d}V\end{array}$$

For an isobaric system, if temperature is held constant ($\mathrm{d}p=0$, $\mathrm{d}T=0$), the free reaction enthalpy is defined as

$$\begin{array}{}\text{(55)}& {\mathrm{\Delta }}_{\mathrm{r}}G={\left(\frac{\partial G(p,T,\xi )}{\partial \xi }\right)}_{p,T}\end{array}$$

Confusingly, the symbol ${\mathrm{\Delta }}_{\mathrm{r}}$ stands for the partial derivative against the extend of reaction, not for a difference.

Inserting $\text{(35)}$ into $\text{(55)}$ yields

$$\begin{array}{}\text{(56)}& {\mathrm{\Delta }}_{r}G={\left(\frac{\partial G(p,T,\xi )}{\partial \xi }\right)}_{p,T}={\left(\frac{\partial \sum _i{\mu }_i\mathrm{d}{n}_i}{\partial \xi }\right)}_{p,T}.\end{array}$$

With $\text{(13)}$, which after reorganizing reads $\mathrm{d}{n}_i={\nu }_i\mathrm{d}\xi$, $\text{(56)}$ becomes

$$\begin{array}{}\text{(57)}& {\mathrm{\Delta }}_{\mathrm{r}}G={\left(\frac{\partial G(p,T,\xi )}{\partial \xi }\right)}_{p,T}=\sum _i({\nu }_i{\mu }_i)\end{array}$$

The chemical potential of substance i can be calculated from two seperate terms:

$$\begin{array}{}\text{(58)}& {\mu }_i={\mu }_i^o+RT\ln A\end{array}$$

with ${\mu }_i^o$: standard chemical potential, the chemical potential of substance i at standard conditions (IUPAC Standard Atmospheric Temperature and Pressure, SATP: pressure $p=100\mathrm{kPa}$, temperature $T=298.15\mathrm{K}$, concentration $c=1.0\cdot {10}^{-3}\mathrm{mol}{\mathrm{m}}^{-3}$. NIST uses Normal Temperature and Pressure, NTP: pressure $p=101.325\mathrm{kPa}$, temperature $T=293.15\mathrm{K}$, concentration $c=1.0\cdot {10}^{-3}\mathrm{mol}{\mathrm{m}}^{-3}$).

Inserting $\text{(58)}$ into $\text{(57)}$ yields

$$\begin{array}{}\text{(59)}& {\mathrm{\Delta }}_{\mathrm{r}}G=\mathrm{d}{G}^o+RT\ln \left(\prod _i{A}_i^{{\nu }_i}\right)\end{array}$$

with $\mathrm{d}{G}^o$: Standard free reaction enthalpy, $\mathrm{d}{G}^o=\sum _i({\nu }_i{\mu }_i^o)$

For a system at thermodynamic equilibrium, $\mathrm{d}G=0$. Hence, in equilibrium

$$\begin{array}{}\text{(60)}& \mathrm{d}{G}^o=-RT\ln K\end{array}$$

with $K$: Equilibrium constant, $K=\prod _i(A_{eq,i}^{{\nu }_i})$ , and

$A_{eq,i}$: Equilibrium activity of component i.

Electrochemical thermodynamics - Nernst equation

Equation $\text{(32)}$ reads

$$\begin{array}{}\text{(61)}& \mathrm{d}G(p,T,N)=-S\mathrm{d}T+V\mathrm{d}p+\sum _l{Y}_l\mathrm{d}{y}_l+\sum _i({\mu }_i\mathrm{d}{n}_i)\end{array}$$

For an isochoric system ($\mathrm{d}V=0$) at constant temperature ($\mathrm{d}T=0$), if $\delta w^{\prime }=\delta w_e$ and no other non-volume-pressure work is performed, $\text{(47)}$ yields

$$\begin{array}{}\text{(62)}& \mathrm{d}F(T,V,N)\le \delta w_e\end{array}$$

For an isobaric system ($\mathrm{d}p=0$) at constant temperature ($\mathrm{d}T=0$), if $\delta w^{\prime }=\delta w_e$ and no other non-volume-pressure work, $\text{(49)}$ yields

$$\begin{array}{}\text{(63)}& \mathrm{d}G(p,T,N)\le \delta w_e\end{array}$$

When the reaction proceeds for $\mathrm{d}\xi$,

$$\begin{array}{}\text{(64)}& \mathrm{d}Q=-e\mathrm{d}N(\mathrm{electrons})=-N_{A}e\mathrm{d}n(\mathrm{electrons})\end{array}$$

Inserting $\text{(13)}$ yields

$$\begin{array}{}\text{(65)}& \mathrm{d}Q=-N_{A}e\nu (\mathrm{electrons})\mathrm{d}\xi =-F\nu (\mathrm{electrons})\mathrm{d}\xi \end{array}$$

with $F$: Faraday constant, $F=N_{A}e$.

Inserting $\text{(13)}$ into $\text{(63)}$

$$\begin{array}{}\text{(66)}& \mathrm{d}G(p,T,\xi )\le -\nu (\mathrm{electrons})F\epsilon \mathrm{d}\xi \end{array}$$

and

$$\begin{array}{}\text{(67)}& {\mathrm{\Delta }}_{\mathrm{r}}G={\left(\frac{\partial G(p,T,\xi )}{\partial \xi }\right)}_{p,T}\le {\left(\frac{\partial (-\nu (\mathrm{electrons})F\epsilon \mathrm{d}\xi )}{\partial \xi }\right)}_{p,T}=-\nu (\mathrm{electrons})F\epsilon \end{array}$$

For reversible processes,

$$\begin{array}{}\text{(68)}& {\mathrm{\Delta }}_{\mathrm{r}}G={\left(\frac{\partial G(p,T,\xi )}{\partial \xi }\right)}_{p,T}=-\nu (\mathrm{electrons})F\epsilon \end{array}$$

Reorganizing yields:

$$\begin{array}{}\text{(69)}& \epsilon =-\frac{{\mathrm{\Delta }}_{\mathrm{r}}G}{\nu (\mathrm{electrons})F}\end{array}$$

From $\text{(59)}$ follows

$$\begin{array}{}\text{(70)}& \epsilon =-\frac{{\mathrm{\Delta }}_{\mathrm{r}}{G}^o}{\nu (\mathrm{electrons})F}-\frac{RT}{\nu (\mathrm{electrons})F}\ln \left(\prod _i\left(A_i^{{\nu }_i}\right)\right)\end{array}$$

Thus

$$\begin{array}{}\text{(71)}& \epsilon ={\epsilon }^o-\frac{RT}{\nu (\mathrm{electrons})F}\ln \left(\prod _i\left(A_i^{{\nu }_i}\right)\right)\end{array}$$

See Balmer 2011.

The cell potential can be calculated from the electrode potentials of the two balanced half-reactions ($\nu {(\mathrm{electrons})}_{cathode}=\nu {(\mathrm{electrons})}_{anode}$).

For a electrochemical cell with the cell formula

Anode (oxidation) $\mathrm{X}⟶z\mathrm{X}{\phantom{A}}^{+}+\mathrm{z}\mathrm{e}{\phantom{A}}^{-}$ | electrolyte | $z\mathrm{Y}{\phantom{A}}^{+}+\mathrm{z}\mathrm{e}{\phantom{A}}^{-}⟶\mathrm{Y}$ (reduction) Cathode

$$\begin{array}{}\text{(72)}& \epsilon ={\epsilon }_{reduction,cathode}-{\epsilon }_{oxidation,anode}\end{array}$$

See here

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