3. Relationship of Chemical Energy to Electrochemical Potential
- Page ID
- 81853
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Electrochemical reactions have a similar drive toward the lowest possible energy. Instead of referring to G for electrochemical reactions, we refer to the electrochemical potential (E). Similar to chemical energy, Eo refers to the difference in electrochemical potential between the standard state and equilibrium. E refers to the difference in electrochemical potential between non-standard state conditions and equilibrium. An interesting facet of electrochemical reactions is that with proper design the change in chemical energy can be converted to electrical energy in the form of an electrical current.
The standard state electrochemical potential (Eo) can be related to ΔGo by the following equation:
\[\mathrm{ΔG^o = -nFE^o}\]
In this equation, n is the number of electrons transferred in the overall balanced electrochemical reaction and F is Faraday’s Constant. Faraday’s Constant relates the total charge in Coulombs (C) of a reaction to the amount of product that forms. For a reaction in which n = 1, there are 96,485 C/mole.
The non-standard state electrochemical potential (E) can be related to ΔG by the following equation:
\[\mathrm{ΔG = -nFE}\]
There are several important outcomes of the relationship between chemical energy and electrochemical potential. Setting equal and rearranging the two expressions for ΔGo leads to the following:
\[\mathrm{\Delta G^o= -RT\ln K\: and\: \Delta G^o= -nFE^o}\]
\[\mathrm{E^o= \dfrac{(RT)}{(nF)} \ln K}\]
Substituting in the standard state temperature, gas constant and Faraday’s constant and converting the natural log to a base ten log leads to the following expression:
\[\mathrm{E^o= \dfrac{(0.059)}{(n)} \log K}\]
Setting equal and rearranging the two expressions for ΔG leads to the following:
\[\mathrm{\Delta G= -RT\ln K + RT\ln Q\: and\: \Delta G= -nFE}\]
\[\mathrm{E= \dfrac{(RT)}{(nF)} \ln K - \dfrac{(RT)}{(nF)}\ln Q}\]
Again, substituting in the standard state temperature, gas constant and Faraday’s constant and converting the natural log to a base ten log leads to the following expression:
\[\mathrm{E= \dfrac{(0.059)}{(n)} \log K - \dfrac{(0.059)}{(n)}\log Q}\]
Substituting in for the first expression in this provides:
\[\mathrm{E= E^0 - \dfrac{(0.059)}{(n)} \log Q}\]
This last equation is especially important in electrochemistry and is known as the Nernst Equation. In the Nernst Equation:
\(\mathrm{\dfrac{0.059}{n} \log K}\) (denoted as Eo) is the difference in electrochemical potential between the standard state and equilibrium.
\(\mathrm{\dfrac{0.059}{n}\log Q}\) is the difference in electrochemical potential between the non-standard state starting conditions and the standard state.
One last thing worth pointing out is the sign convention of electrochemical potentials and how they relate to whether the reaction favors products or reactants. An examination of the Nernst Equation shows that electrochemical reactions that favor products will have positive values of electrochemical potential. Electrochemical reactions that favor reactants will have negative values of electrochemical potential.