# Nernst Equation

The electrical potential disparity across the cell membrane of all living cells is called the membrane potential, the inner part of the cell being negative compared to the outside. The magnitude of the membrane potential varies from cell to cell and in an exceptional cell following its functional state. For example, a nerve cell has a membrane potential of -70mv at rest, but the membrane potential drops to about +30mv when excited. The membrane potential at rest is called the **resting potential. **RMP is basically due to-

- Uneven distribution of ions across the cell membrane due to its selective permeability.
- Due to the combined effect of forces acting onions. This origin of RMP is dependent

**Selective Permeability of Cell Membrane**

The cell membrane is selectively permeable, i.e. it is freely permeable to K^{+} and Cl^{–}, from the medium to Na^{+}, and is impermeable to proteins and organic phosphates which are negatively charged ions. Usually, the intracellular cation is K^{+} and major intracellular anions are proteins and organic phosphate. General extracellular cation is Na^{+} and anion is Cl^{–}.

The presence of gated protein channels in the cell membrane is responsible for the variable permeability of ions. The forces executive on the ions across the cell membrane Production of variations in the membrane potential. The magnitude of forces acting thenceforward the cell membrane on each ion can be analyzed by the Nernst equation.

**Concentration gradient: **The Donnan effect results in an uneven distribution of diffuse ions across the cell membrane, in the form of additional diffuse cations, resulting in a concentration gradient.

**Electrical gradient: **As a consequence of concentration gradient cation K+, will try to disseminate back into ECF from ICF. But it is counteracted by an electrical gradient which will be created due to the impendence of nondiffusible anions within the cell. The membrane potential at which the electric force is equal in magnitude but opposite in the direction of the concentration force is called the equilibrium potential for that ion. The magnitude of the equilibrium potential is determined by the Nernst equation.

**E _{m } = -RT/ZF In C_{in}/C_{out}**

37 Â°C at normal body temperature, substituting for the constants (R, T, and F) and converting to the normal logarithm,

**E _{m}= -61.5 log C_{in}/C_{out}**

## Nernst equation

The Standard electrode potentials are measured at their standard states when the concentration of the electrolyte solution is fixed as 1M and the temperature is 298 K. Despite this, in actual practice, electrochemical cells do not always have a fixed concentration on the electrolyte solution. The electrode potential depends on the concentration of the electrolyte solution. The Nernst equation gave a relationship between the electrode potential and the concentration of electrolyte solutions, known as the Nernst equation. For a general electrode:

M^{n+}+ ne^{– }â†’ M

E_{cell }= E^{Â°}_{cell}â€“(RT/nF)lnQ

E_{cell }= EÂ°_{cell}â€“(RT/nF)ln([C]^{c}[D]^{d}/ [A]^{a}[B]^{b)}where,

- R = The gas constant (8.314JK
^{-1}mol^{-1})- F = Faraday constant (96,500Cmol
^{-1})- T = Temperature in kelvin
- Q = Reaction quotient
- n = Total number of moles of electrons translocate

It should be remembered that when writing the Nernst equation for the overall cell reaction, the log term is the same as the expression for the equilibrium constant for the reaction. The relation of both is similar.

E_{cell} = EÂ°_{cell} – (2.303 RT/nf) ln([C]^{c}[D]^{d}/ [A]^{a}[B]^{b})

E_{cell} = EÂ°_{cell} + 2.303 RT [A] [B]ln([A]^{a}[B]^{b}/ [C]^{c}[D]^{d})

Similarly, for the electrode reaction:

M^{n+ }+ ne^{– } â†’ M

The Nernst equation is-

**E _{cell = }E_{Â°cell – }(2.303 RT/nf)log[1/M^{n+}]**

**or E _{cell = }E_{Â°cell + }(2.303RT / nF)log[1/M^{n+}]**

when T = 273K, F=96500 Cmol^{-1}, R=8.314JK^{-1}mol^{-1 }and concentration of solid M is taken as unit

E_{cell} = EÂ°_{cell} – (0.059/n)log[1/M^{n+}]

Relationship between equilibrium constant and standard potential of a cell

E_{cell} = E_{cell}Â° – (2.303 RT/nF)log[K_{c}], [K_{c }= equilibrium constant]

At equilibrium, E_{cell }=0

E^{Â°}_{cell} = (2.303 RT/nF)log[K_{c}]

**K _{c = }antilog[nE_{cell}^{Â°}/0.0591]**

### Limitations of Nernst Equation

- The Nernst equation applies exclusively because no current flows through the electrodes. When current flows, the movement of ions at the electrode surface changes, and the conditions of excess potential and resistance loss contribute to the measured potential.
- At very low concentrations of commutation potential-determining ions, the potential approach Â± is found using the Nernst equation. This is corporeally useless, because, underneath such a situation, the commutation current density is reduced and tends to control the electrochemical behaviour of the system more than other effects.
- Since the active coefficients are close to unity in dilute solutions, the Nernst equation can be expressed in the directly implicit form of the concentration. But in the case of higher concentrations, the actual activities of the ions must be used. This creates complexity for the use of the Nernst equation because estimating the non-ideal activities of ions usually requires experimental measurements.

**Sample Questions**

**Question 1: Will the EÂş value change when the coefficients in the chemical equation change?**

**Answer:**

The EÂş value does not depend on the coefficient in the chemical equation i.e. when we double or triple the coefficient, the EÂ° value does not change.

For example:

- ZnÂ˛+ 2e
^{–}â†’ Zn; EÂ° =-0.76 V- 2ZnÂ˛+ 4e
^{–}â†’ 2Zn; EÂ° =-0.76 V- 3ZnÂ˛+ 6e
^{–}â†’ 3Zn; EÂ° =-0.76 VIn the half-reaction, if the coefficients change, the number of electrons will change to cancel out the effect of the change in n coefficients.

**Question 2: Which reference electrode is used to measure the electrode potential of other electrodes?**

**Answer:**

The standard hydrogen electrode is used as a reference electrode whose electrode potential is assumed to be zero. The electrode potential of the other electrode is measured concerning it.

**Question 3: Zinc rod is dipped in 0.1M solution of ZnSO4. The salt is 95% dissociated at this dilution at 298 K. Calculate the electrode potential given that E (ZnÂ˛+ | Zn) = -0.76 V.**

**Answer:**

The electrode reaction is :

Zn

^{2+}+ 2e^{– }â‡† Zn(s)According to Nernst equation, at 298 K

E(Zn

^{2+}|Zn)=E^{Â°}(Zn^{2+}|Zn)- (0.059 /n) log [ angle n] [Zn]/[Zn^{2+}(aq)]E

^{Â°}(Zn^{2+ }|Zn)=-0.76 V, [Zn] = 1,[Zn

^{2+}(aq)]=0.1*95/100=0.095 ME(Zn

^{2+}|Zn)=-0.76- (0.009/n )log1/-0.095=-0.76-0.03=

-0.79V

**Question 4: What advantage do the fuel cells have over primary and secondary batteries?**

**Answer:**

Primary batteries restrain delimited congeries of reactants and are destroyed when the reactants have been consumed. Secondary batteries can be recharged but retract a longer to recharge. The fuel cell is conducted consecutive as long as reactants are recoupment to it and products are continuously removed.

**Question 5: How will the pH of the brine (aq. NaCl solution) be affected when it is electrolyzed?**

**Answer:**

Since NaOH is formed during electrolysis, the pH of the brine solution will increase

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