Editing Nernst equation (section)

===Using thermodynamics (chemical potential)===
Quantities here are given per molecule, not per [[Mole (unit)|mole]], and so [[Boltzmann constant]] {{math|''k''}} and the [[Elementary charge|electron charge]] {{math|''e''}} are used instead of the [[gas constant]] {{math|''R''}} and [[Faraday constant|Faraday's constant]] {{math|''F''}}. To convert to the molar quantities given in most chemistry textbooks, it is simply necessary to multiply by the [[Avogadro constant]]: {{math|1 = ''R'' = ''kN''<sub>A</sub>}} and {{math|1 = ''F'' = ''eN''<sub>A</sub>}}. The [[entropy]] of a molecule is defined as

<math display="block">S \ \stackrel{\mathrm{def}}{=}\ k \ln \Omega,</math>

where {{math|Ω}} is the number of states available to the molecule. The number of states must vary linearly with the volume {{math|''V''}} of the system (here an idealized system is considered for better understanding, so that activities are posited very close to the true concentrations). Fundamental statistical proof of the mentioned linearity goes beyond the scope of this section, but to see this is true it is simpler to consider usual [[isothermal process]] for an [[ideal gas]] where the change of entropy {{math|1=Δ''S'' = ''nR'' ln({{sfrac|''V''<sub>2</sub>|''V''<sub>1</sub>}})}} takes place. It follows from the definition of entropy and from the condition of constant temperature and quantity of gas {{mvar|n}} that the change in the number of states must be proportional to the relative change in volume {{math|{{sfrac|''V''<sub>2</sub>|''V''<sub>1</sub>}}}}. In this sense there is no difference in statistical properties of ideal gas atoms compared with the dissolved species of a solution with [[activity coefficient]]s equaling one: particles freely "hang around" filling the provided volume), which is inversely proportional to the [[Molar concentration|concentration]] {{mvar|c}}, so we can also write the entropy as
<math display="block">S = k\ln \ (\mathrm{constant}\times V) = -k\ln \ (\mathrm{constant}\times c).</math>

The change in entropy from some state 1 to another state 2 is therefore
<math display="block">\Delta S = S_2 - S_1 = - k \ln \frac{c_2}{c_1},</math>
so that the entropy of state 2 is
<math display="block">S_2 = S_1 - k \ln \frac{c_2}{c_1}.</math>

If state 1 is at standard conditions, in which {{math|''c''<sub>1</sub>}} is unity (e.g., 1&nbsp;atm or 1&nbsp;M), it will merely cancel the units of {{math|''c''<sub>2</sub>}}. We can, therefore, write the entropy of an arbitrary molecule A as
<math display="block">S(\mathrm{A}) = S^\ominus(\mathrm{A}) - k \ln [\mathrm{A}],</math>
where <math>S^\ominus</math> is the entropy at [[Standard temperature and pressure|standard conditions]] and [A] denotes the concentration of A. The change in entropy for a reaction

{{block indent|em=1.5|text={{mvar|a}}A + {{mvar|b}}B → {{mvar|y}}Y + {{mvar|z}}Z}}
is then given by
<math display="block">
\Delta S_\mathrm{rxn} = \big(yS(\mathrm{Y}) + zS(\mathrm{Z})\big) - \big(aS(\mathrm{A}) + bS(\mathrm{B})\big) 
= \Delta S^\ominus_\mathrm{rxn} - k \ln \frac{[\mathrm{Y}]^y [\mathrm{Z}]^z}{[\mathrm{A}]^a [\mathrm{B}]^b}.
</math>

We define the ratio in the last term as the [[reaction quotient]]:
<math display="block">Q_r = \frac{\displaystyle\prod_j a_j^{\nu_j}}{\displaystyle\prod_i a_i^{\nu_i}} \approx \frac{[\mathrm{Z}]^z [\mathrm{Y}]^y}{[\mathrm{A}]^a [\mathrm{B}]^b},</math>
where the numerator is a product of reaction product [[Thermodynamic activity|activities]], {{math|''a<sub>j</sub>''}}, each raised to the power of a [[stoichiometric coefficient]], {{math|''ν<sub>j</sub>''}}, and the denominator is a similar product of reactant activities. All activities refer to a time {{math|''t''}}. Under certain circumstances (see [[chemical equilibrium]]) each activity term such as {{math|''a{{su|b=j|p=ν<sub>j</sub>}}''}} may be replaced by a concentration term, [A].In an electrochemical cell, the cell potential {{math|''E''}} is the [[chemical potential]] available from [[redox]] reactions ({{math|1=''E'' = {{sfrac|''μ''<sub>c</sub>|''e''}}}}). {{math|''E''}} is related to the [[Gibbs free energy]] change {{math|Δ''G''}} only by a constant: 
{{math|1=Δ''G'' = −''zFE''}}, where {{math|''n''}} is the number of electrons transferred and {{math|''F''}} is the [[Faraday constant]]. There is a negative sign because a spontaneous reaction has a negative [[Gibbs free energy]] {{math|Δ''G''}} and a positive potential {{math|''E''}}. The Gibbs free energy is related to the entropy by {{math|1=''G'' = ''H'' − ''TS''}}, where {{math|''H''}} is the [[enthalpy]] and {{math|''T''}} is the temperature of the system. Using these relations, we can now write the change in Gibbs free energy,
<math display="block">\Delta G = \Delta H - T \Delta S = \Delta G^\ominus + kT \ln Q_r,</math>
and the cell potential,
<math display="block">E = E^\ominus - \frac{kT}{ze} \ln Q_r.</math>

This is the more general form of the Nernst equation. 

For the redox reaction {{nowrap|Ox + {{mvar|z}} e<sup>−</sup> → Red}}, 
<math display="block">Q_r = \frac{[\mathrm{Red}]}{[\mathrm{Ox}]},</math>
and we have:
<math display="block">\begin{align}
E &= E^\ominus - \frac{kT}{ze} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
&= E^\ominus - \frac{RT}{zF} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
&= E^\ominus - \frac{RT}{zF} \ln Q_r.
\end{align}</math>

The cell potential at [[standard temperature and pressure]] (STP) <math>E^\ominus</math> is often replaced by the formal potential <math>E^{\ominus'}</math>, which includes the [[activity coefficient]]s of the dissolved species under given experimental conditions (T, P, [[ionic strength]], [[pH]], and complexing agents) and is the potential that is actually measured in an electrochemical cell.