# Nernst Equation

In electrochemistry , the Nernst equation is an equation that relates the reduction potential of a reaction (of a half-cell or full-cell reaction) to the standard electrode potential , temperature , and activities (often estimated at concentrations) of chemical species undergoing reduction and oxidation . It was named after the German physical chemist Walther Nernst , who formulated the equation.

##### expression of Nernst Equation

A quantitative relationship between cell potential and anion concentration

Ox + z  e− → Red

Standard thermodynamics says that the real Gibbs free energy G is related to the free energy change under the standard state G.
By relationship:

{\displaystyle \Delta G=\Delta G^{\ominus }+RT\ln Q_{r}}

where q r is the reaction quotient . The cell potential E associated with an electrochemical reaction is defined as a decrease in the Coulomb Gibbs free energy per charge transferred, which leads to the relation . The constant f ( Faraday’s constant ) is a unit conversion factor f = q, where n is the Avogadro constant and q is the fundamental electron charge. This immediately leads to the Nernst equation, which for an electrochemical half-cell

{\displaystyle \Delta G=-zFE}
{\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-{\frac {RT}{zF}}\ln Q_{r}=E_{\text{red}}^{\ominus }-{\frac {RT}{zF}}\ln {\frac {a_{\text{Red}}}{a_{\text{Ox}}}}}.

For a complete electrochemical reaction (complete cell), the equation can be written as:

{\displaystyle E_{\text{cell}}=E_{\text{cell}}^{\ominus }-{\frac {RT}{zF}}\ln Q_{r}}
E_{red}~ is~ the~ half-cell~ reduction ~potential ~at ~the ~temperature ~of ~interest,\\
E^o_{red}~ is~ the~ standard~ half-cell ~reduction~ potential,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
E_{cell}~ is ~the~ cell ~potential ~(electromotive~ force) ~at~ the~ temperature~ of~ interest,
E^o_{cell} ~is ~the ~standard ~cell ~potential,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

R is the universal gas constant: R = 8.31446261815324 J K−1 mol−1,

T is the temperature in kelvins,
z is the number of electrons transferred in the cell reaction or half-reaction,
F is the Faraday constant, the number of coulombs per mole of electrons: F = 96485.3321233100184 C mol−1,
Qr is the reaction quotient of the cell reaction, and
a is the chemical activity for the relevant species, where aRed is the activity of the reduced form and aOx is the activity of the oxidized form.

Similarly for the equilibrium constant, activities are always measured with respect to the standard state (1 mol/L for solutes, 1 atm for gases). The activity of species X, A X , can be related to physiological concentrations of C X through A X = X C X , where X is the activity coefficient of the species X , due to the activity coefficient, tending to unity in small amounts The Nernst equation is often replaced with simple concentrations. Alternatively, defining a formal capacity as:

{\displaystyle E^{\ominus '}=E^{\ominus }+{\frac {RT}{zF}}\ln {\frac {\gamma _{Ox}}{\gamma _{Red}}}}

The half-cell Nernst equation can be written as the concentration:

{\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus '}-{\frac {RT}{zF}}\ln {\frac {C_{\text{Red}}}{C_{\text{Ox}}}}}

and similarly for full cell expression.

At room temperature (25 °C), the thermal voltage is approximately 25,693 mV. The Nernst equation is often expressed in terms of the base-10 logarithm ( ie , the common logarithm ) rather than the natural logarithm , in which case it is written:

{\displaystyle V_{T}={\frac {RT}{F}}}
{\displaystyle E=E^{0}+{\frac {V_{T}}{z}}\ln {\frac {a_{\text{Ox}}}{a_{\text{Red}}}}=E^{0}+{\frac {\lambda V_{T}}{z}}\log _{10}{\frac {a_{\text{Ox}}}{a_{\text{Red}}}}}.

where = ln(10) and V t = 0.05916 … V. The Nernst equation is used in physiology to find the electrical potential of a cell membrane with respect to a type of ion . This can be linked to the acid dissociation constant.

## applications in biology

### Nernst Capacity

The Nernst equation has a physical application when used to calculate the potential of an ion of charge Z in a membrane. This potential is determined using the concentration of the ion inside and outside the cell:

E={\frac {RT}{zF}}\ln {\frac {[{\text{ion outside cell}}]}{[{\text{ion inside cell}}]}}=2.3026{\frac {RT}{zF}}\log _{10}{\frac {[{\text{ion outside cell}}]}{[{\text{ion inside cell}}]}}.

When the membrane is in thermodynamic equilibrium (ie, no net flow of ions), and if the cell is permeable to only one ion, the membrane potential must be equal to the Nernst potential for that ion.

### goldman equation

When the membrane is permeable to more than one ion, as is essentially the case, the resting potential can be determined from the Goldmann equation, which is a solution to the GHK flow equation under constraints that by electrochemical force. The driven total current density is zero:

{\displaystyle E_{\mathrm {m} }={\frac {RT}{F}}\ln {\left({\frac {\displaystyle \sum _{i}^{N}P_{\mathrm {M} _{i}^{+}}\left[\mathrm {M} _{i}^{+}\right]_{\mathrm {out} }+\displaystyle \sum _{j}^{M}P_{\mathrm {A} _{j}^{-}}\left[\mathrm {A} _{j}^{-}\right]_{\mathrm {in} }}{\displaystyle \sum _{i}^{N}P_{\mathrm {M} _{i}^{+}}\left[\mathrm {M} _{i}^{+}\right]_{\mathrm {in} }+\displaystyle \sum _{j}^{M}P_{\mathrm {A} _{j}^{-}}\left[\mathrm {A} _{j}^{-}\right]_{\mathrm {out} }}}\right)},}
• Em is the membrane potential (in volts, equivalent to joules per coulomb),
• Pion is the permeability for that ion (in meters per second),
• [ion]out is the extracellular concentration of that ion (in moles per cubic meter, to match the other SI units, though the units strictly don’t matter, as the ion concentration terms become a dimensionless ratio),
• [ion]in is the intracellular concentration of that ion (in moles per cubic meter),
• R is the ideal gas constant (joules per kelvin per mole),
• T is the temperature in kelvins,
• F is the Faraday’s constant (coulombs per mole).

The potential in the cell membrane which is exactly opposite to the net diffusion of a particular ion through the membrane is called the Nernst potential for that ion. As seen above, the magnitude of the Nernst potential is determined by the ratio of the concentration of that specific ion on both sides of the membrane. The higher this ratio, the greater the tendency for the ion to diffuse in one direction, and therefore the greater the Nernst potential required to prevent diffusion. A similar expression exists that includes r (absolute value of transport ratio). It takes into account transporters with uneven exchange. See: sodium-potassium pump where the transport ratio will be 2/3, so in the formula below r is equal to 1.5. The reason we put a factor r = 1.5 here is because the electrochemical force J eccurrent density. (Na + ) + J E. C. (K + ) is no longer zero, but J ec (Na + )+1.5J ec . (K + )=0 (as the electrochemical force is compensated by the pump for the flow of both ions, i.e. J ec = -J pump ), changing the constraints to apply the GHK equation. Other variables are the same as above. The following example involves two ions: potassium (K + ) and sodium (Na + ). The chloride is assumed to be in equilibrium.

{\displaystyle E_{m}={\frac {RT}{F}}\ln {\left({\frac {rP_{\mathrm {K} ^{+}}\left[\mathrm {K} ^{+}\right]_{\mathrm {out} }+P_{\mathrm {Na} ^{+}}\left[\mathrm {Na} ^{+}\right]_{\mathrm {out} }}{rP_{\mathrm {K} ^{+}}\left[\mathrm {K} ^{+}\right]_{\mathrm {in} }+P_{\mathrm {Na} ^{+}}\left[\mathrm {Na} ^{+}\right]_{\mathrm {in} }}}\right)}.}

When chloride (Cl  ) is taken into account,

{\displaystyle E_{m}={\frac {RT}{F}}\ln {\left({\frac {rP_{\mathrm {K} ^{+}}\left[\mathrm {K} ^{+}\right]_{\mathrm {out} }+P_{\mathrm {Na} ^{+}}\left[\mathrm {Na} ^{+}\right]_{\mathrm {out} }+P_{\mathrm {Cl} ^{-}}\left[\mathrm {Cl} ^{-}\right]_{\mathrm {in} }}{rP_{\mathrm {K} ^{+}}\left[\mathrm {K} ^{+}\right]_{\mathrm {in} }+P_{\mathrm {Na} ^{+}}\left[\mathrm {Na} ^{+}\right]_{\mathrm {in} }+P_{\mathrm {Cl} ^{-}}\left[\mathrm {Cl} ^{-}\right]_{\mathrm {out} }}}\right)}.}

## etymology

### using the boltzmann factor

For the sake of simplicity, we will consider a solution of redox-active molecules that undergo a one-electron reversible reactionaux + e  red

and which have a standard potential of zero, and in which the activities are well represented by concentrations (ie unit activity coefficients). The chemical potential μc of this solution is the difference between the energy barriers to take electrons and give electrons to the working electrode, determining the electrochemical potential of the solution. ratio of molecules less oxidized,[Bull]/[Red], is equal to the probability of being oxidized (giving an electron) over the probability of being reduced (taking an electron), which we can write in terms of the Boltzmann factor for these processes:

{\displaystyle {\begin{aligned}{\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}&={\frac {\exp \left(-[{\text{barrier for gaining an electron}}]/kT\right)}{\exp \left(-[{\text{barrier for losing an electron}}]/kT\right)}}\\[6px]&=\exp \left({\frac {\mu _{\mathrm {c} }}{kT}}\right).\end{aligned}}}

Taking the natural logarithm of both the sides, we get

{\displaystyle \mu _{\mathrm {c} }=kT\ln {\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}.}

If μc ≠ 0 at

\frac{[OX]}{[Red]}=1 ,

{\displaystyle \mu _{\mathrm {c} }=\mu _{\mathrm {c} }^{0}+kT\ln {\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}.}

To convert from chemical potential to electrode potential, divide the equation by E , and remember that K / I = R / F We obtain the Nernst equation for the one-electron process Ox + E  → Red :

{\displaystyle {\begin{aligned}E&=E^{0}-{\frac {kT}{e}}\ln {\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}\\&=E^{0}-{\frac {RT}{F}}\ln {\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}.\end{aligned}}}

### Using Thermodynamics (Chemical Potential)

Here the quantities are given per molecule, not per mole, and therefore the Boltzmann constant k and the electron charge e are used instead of the gas constant R and Faraday’s constant F. To convert to the molar volume given in most chemistry textbooks, it is necessary to multiply by the Avogadro constant: R = kN a and F = eN a . The entropy of a molecule is defined as

{\displaystyle S\ {\stackrel {\mathrm {def} }{=}}\ k\ln \Omega ,}

where is the number of states available to the molecule . The number of states should vary linearly with the volume V of the system (here an ideal system is assumed for better understanding, so that the activities are very close to the actual concentrations. The fundamental statistical proof of the linearity mentioned is beyond its scope. section, but to see that this is true, it is easy to consider the general isothermal process for an ideal gas where the change of entropy s = nr ln (V2 / V1) happens. It follows from the definition of entropy and the condition of constant temperature and gas volume n such that the change in the number of states must be proportional to the relative change in volume V2 / V1 In this sense, there is no difference in the statistical properties of ideal gas atoms compared to dissolved species of a solution with an activity coefficient equal to one: the particles freely “hang around” filling the volume provided), which is the concentration is inversely proportional to c , so we can also write entropy as

S=k\ln \ (\mathrm {constant} \times V)=-k\ln \ (\mathrm {constant} \times c).

The change in entropy from some state 1 to another state 2 is because

\Delta S=S_{2}-S_{1}=-k\ln {\frac {c_{2}}{c_{1}}},

so that the entropy of state 2 is

S_{2}=S_{1}-k\ln {\frac {c_{2}}{c_{1}}}.

If state 1 is in standard conditions in which 1 is unity (eg, 1 atm or 1 M), it will only cancel units of 2 . Therefore, we can write the entropy of an arbitrary molecule A as

{\displaystyle S(\mathrm {A} )=S^{0}(\mathrm {A} )-k\ln[\mathrm {A} ],}

where S0 is the entropy at standard conditions and [A] denotes the concentration of A. change in entropy for a reaction

a  A + b  B → y  Y + z  Zis then . is given by

{\displaystyle \Delta S_{\mathrm {rxn} }={\big (}yS(\mathrm {Y} )+zS(\mathrm {Z} ){\big )}-{\big (}aS(\mathrm {A} )+bS(\mathrm {B} ){\big )}=\Delta S_{\mathrm {rxn} }^{0}-k\ln {\frac {[\mathrm {Y} ]^{y}[\mathrm {Z} ]^{z}}{[\mathrm {A} ]^{a}[\mathrm {B} ]^{b}}}.}

We define the ratio in the last term as the reaction quotient:

{\displaystyle 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}}},}

where the counter-reaction product is a product of activities, j , each stoichiometric coefficient of the power of a, j , and every reactant is an equal product of activities. All activities refer to a time t . Under certain circumstances (see chemical equilibrium) each activity term such as aJJ _
_
Can be replaced by a concentration word, [a]. In an electrochemical cell, the cell potential E is the chemical potential available from redox reactions ( E =μc/Ie is related to the Gibbs energy change g : only by a constant g  = zFE , where n is the number of electrons transferred and f is the Faraday constant. has a negative sign because a spontaneous reaction has a negative free energy G and a positive potential E. The Gibbs energy is related to the entropy by G = H – Ts , where H is the enthalpy and T is the temperature of the system. Using these relations, we can now write the change in Gibbs energy,

{\displaystyle \Delta G=\Delta H-T\Delta S=\Delta G^{0}+kT\ln Q_{r},}

and cell potential,

{\displaystyle E=E^{0}-{\frac {kT}{ze}}\ln Q_{r}.}

This is the more general form of the Nernst equation. for the redox reaction ox + n e  → red ,

{\displaystyle Q_{r}={\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}},}

and we have:

{\displaystyle {\begin{aligned}E&=E^{0}-{\frac {kT}{ne}}\ln {\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}\\&=E^{0}-{\frac {RT}{nF}}\ln {\frac {[\mathrm {Red} ]}{[\mathrm {Ox} ]}}\\&=E^{0}-{\frac {RT}{nF}}\ln Q_{r}.\end{aligned}}}

At standard conditions the cell potential E0 is often replaced by the formal potential E0 ‘ , which includes some small corrections to the logarithm and is the potential that is actually measured in an electrochemical cell .

## relation to balance

At equilibrium, the electrochemical potential ( E ) = 0 and therefore the reaction quotient acquires a special value known as the equilibrium constant: Q = eq . Hence,

{\displaystyle {\begin{aligned}0&=E^{0}-{\frac {RT}{zF}}\ln K\\\ln K&={\frac {zFE^{0}}{RT}}.\end{aligned}}}

or at standard temperature,

{\displaystyle \log _{10}K={\frac {zE^{0}}{\lambda V_{T}}}={\frac {zE^{0}}{0.05916{\text{ V}}}}\quad {\text{at }}T=298.15~{\text{K}}.}

Thus we have related the standard electrode potential and equilibrium constant of a redox reaction.

## borders

In dilute solutions, the Nernst equation can be expressed directly as concentrations (since the activity coefficients are close to unity). But at higher concentrations, the actual activities of the ions must be used. This complicates the use of the Nernst equation, as estimating the non-ideal activities of ions usually requires experimental measurements. The Nernst equation also applies when there is no net current flowing through the electrodes. The activity of ions at the electrode surface changes when there is current flowing, and there are additional overpotential and resistive loss conditions, thereby contributing to the measured potential.

At very low concentrations of potential-determining ions, the potential estimated by the Nernst equation approaches ± . This is physically meaningless, because under such conditions, the exchange current density is greatly reduced, and there can be no thermodynamic equilibrium required for the Nernst equation to hold. In such a case the electrode is said to be unpolarised. Other effects tend to control the electrochemical behavior of the system, such as the participation of the solvated electron in power transfer and electrode balance, as observed by Alexander Frumkin and B. Damskin, [4] Sergio Trasatti, et al.

### time dependence of capacity

The expression of dependence on time has been established by Karoglanoff. [5] [6] [7] [8]

## Importance to the respective scientific domain

The equation has been involved in the scientific controversy surrounding cold fusion. The discoverers of cold fusion, Fleischmann and Ponce, calculated that a palladium cathode immersed in a heavy water electrolysis cell could reach 1027 atmospheres of pressure at the surface of the cathode , which is a pressure high enough to cause spontaneous nuclear fusion. In fact, only 10,000–20,000 atmospheres were obtained. John R. Huizenga claimed that his original calculations were affected by a misinterpretation of the Nernst equation. [9] He cited a paper about Pd-Zr alloys. [10]The equation allows the extent of the reaction between two redox systems to be calculated and can be used, for example, to decide whether a particular reaction will be completed. At equilibrium the emfs of the two half cells are equal. This enables one to calculate hence the range of the reaction.

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