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LECTURE NOTE ON & NERNST EQUATION

-Dr S P Singh

Historical Background: The connection between chemistry and electricity is a very old one, going back to Allesandro Volta’s discovery, in 1793, that electricity could be produced by placing two dissimilar metals on opposite sides of a moistened paper. In 1800, Nicholson and Carlisle, using Volta’s primitive battery as a source, showed that an electric current could decompose water into oxygen and hydrogen. This was surely one of the most significant experiments in the history of chemistry, for it implied that the atoms of hydrogen and oxygen were associated with positive and negative electric charges, which must be the source of the bonding forces between them. By 1812, the Swedish chemist Berzelius could propose that all atoms are electrified, hydrogen and the metals being positive, the nonmetals negative. In , the applied voltage was thought to overpower the attraction between these opposite charges, pulling the electrified atoms apart in the form of (named by Berzelius from the Greek for “travellers”). It would be almost exactly a hundred years later before the shared pair theory of G.N. Lewis could offer a significant improvement over this view of chemical bonding. Meanwhile, the use of electricity as a means of bringing about chemical change continued to play a central role in the development of chemistry. Humphrey Davey prepared the first elemental sodium by electrolysis of a sodium hydroxide melt. It was left to Davey’s former assistant, Michael Faraday, to show that there is a quantitative relation between the amount of electric charge and the quantity of electrolysis product. James Clerk Maxwell immediately saw this as evidence for the “molecule of electricity”, but the world would not be receptive to the concept of the electron until the end of the century.

Walther Hermann Nernst, a German physical chemist advanced a theory to explain the phenomena regarding electrode potential in 1889. According to this theory; 1. All the metallic elements and hydrogen have a tendency to pass into solution in the form of (+)ve ions. This tendency is known as solution tension or electrolytic solution pressure which is constant at a given temperature. Due to migration of (+)ve ions, the metallic electrode is left (-)vely charged and thus an electrical double layer is set up at the electrode which prevents the further expulsion of ions from the metal and thus there is a state of equilibrium rapidly established with a definite potential deference termed as electrode potential.

2. If the metallic electrode is dipped in a solution of one of its salts, the tendency of the ions is to be deposited back on the electrode due to the osmotic pressure. The difference between solution pressure of metal and the osmotic pressure of metals is known as standard electrode potential. Thus three possibilities arise; Case-1 Solution pressure > Osmotic pressure: In this case the tendency of the metal to lose ions predominates. The (+)ve ions will enter the liquid and leave the metal (-)vely charged. Example: Zn, Cd, Mn and Alkali metals. Case-2 When Solution pressure < Osmotic pressure: The ions have a greater tendency to leave the solution and get deposited on the metal. The metal in this case acquires (+)ve charge w.r.t the solution. Example: Cu, Ag, Hg, Au etc. Case-3 When Solution pressure = Osmotic pressure:

In this case no relative charge is developed and hence no potential difference exists such a system is known as null electrode. The above mentioned theory of Nernst is thus applicable to such metals which yield (+)ve ions and also to materials capable of producing (-)ve ions.

The solution pressure of an electrode material depends;

(i) on its physical condition e.g. fine crystals of a metal having a higher solution pressure than that of large crystals due to more (-)ve potential of the former in the same solution. (ii) on the allotropic forms of a metal (metastable has a higher value).

The solution pressure of an element also varies with the Single electrode potential arises in a nature of the solvent medium containing the ions. The half-cell. An solution pressure of a gas depends upon the pressure of consists of two half cells. With an the gas. open circuit, the metal electrode in each half cell transfers its ions into From the Nernst theory, the potential difference of the solution. double layer formed at an electrode is termed as single electrode potential.

Derivation of Nernst Equation for electrode potential

The Nernst equation is an equation that relates the of an electrochemical reaction (half-cell or full cell reaction) to the standard electrode potential, temperature, and activities (often approximated by concentrations) of the chemical species undergoing reduction and oxidation. Even under non-standard conditions, the cell potentials of electrochemical cells can be determined with the help of the Nernst equation.

Let us consider a metal M in contact with aqueous solution of its own salt. A Metal looses n electron to become a (+)vely charged metal ion and the ion gains n electron to return to its atomic state. The forward and backward reactions are equally feasible and reach to the state of equilibrium;

푛푀(푠) ⇌ 푀푛+(푎푞) + 푛푒 (i)

Let P1 be the osmotic pressure of metal ion and P2 be the solution pressure of the metal and E be the difference of potential between the metal and its ion i.e. electrode potential. A current of electricity is passed reversibly through the electrode until 1g ion of the metal is dissolved.

Therefore, electricity required for the dissolution of 1g ion of the metal will be nF .

The electrical work done (w1) = nF E -

i.e., w1= nFE volt coulomb (ii) where F is Faraday = 96487 coulomb = electrical charge carried by one of

The solution is now diluted, so that osmotic pressure is reduced from P1 to P1 – dP1, the difference of potential between the metal and its ion is now changed from E to E – dE.

Work done to cause the dissolution of 1gm ion of the metal (w2) = (E – dE)nF volt-coulombs.

i.e., w2 = (E – dE)nF volt-coulomb (iii) The difference in electrical energy = nFE – (E – dE) nF = nFdE volt-coulombs (iv) The difference in electrical energy = Osmotic work done in transforming 1gm ion of the metal from P1 to P1 – dP1

푛퐹푑퐸 = 푉푑푃1 (v) where V is the volume of the solvent containing 1gm ion of the metal. For an ideal solution, Osmotic pressures ∝ Activities of the ions 푅푇 and 푉 = 푃1 On putting the value of V in equation (v), we have

푅푇 푛퐹 푑퐸 = 푑푃1 (vi) 푃1 On integrating the equation (vi), we get 푑푃 푛퐹 ∫ 푑퐸 = 푅푇 ∫ 1 푃1

∴ nFE= RT log P1 + integrating constant (vii)

When P1= P2, then E= 0 , Integrating constant = RT log P2 Now the equation (vii) becomes nFE = RT log P1 – RT log P2 푃 = 푅푇 log 1 푃2

푅푇 푃 ∴ 퐸 = log 1 (viii) 푛퐹 푃2 This is known as Nernst’s equation for electrode potential. If two similar electrodes are dipped in two different solutions (electrolytes), the potential difference between the two electrodes (two half cells) will be given by;

푅푇 푃1퐴 푙표푔푃1퐵 푅푇 푃1퐴 퐸1 − 퐸2 = (log − ) = log (ix) 푛퐹 푃2 푃2 푛퐹 푃2퐵

where P1A and P1B are the osmotic pressures of the metal ions in two solutions A and B respectively.

Since Osmotic pressure ∝ concentration, we have

푅푇 퐶1퐴 퐸1 − 퐸2 (∆E)= log (x) 푛퐹 퐶2퐵 The standard cell potentials we have been discussing that refers to cells in which all dissolved substances are at unit activity, which essentially means an “effective concentration” of 1 M. Similarly, any gases that take part in an electrode reaction are at an effective pressure (known as the fugacity) of 1 atm. If these concentrations or pressures have other values, the cell potential will change in a manner that can be predicted from the principles. During the reaction, concentration keeps changing and the potential also will decrease with the rate of reaction.

The Gibbs function (∆G = −nFE or ∆G◦ = −nFE◦ ) is more than a criterion for spontaneity. G◦ & E◦ refer to & Electrode Potential under standard conditions. The value of ∆G◦ expresses the maximum useful work that a system can do on the surroundings. “Useful” here means work other than P-V work that is simply a consequence of volume change, which cannot be channelled to some practical use. This maximum work can only be extracted from the system under the limiting conditions of a reversible change, which for an electrochemical cell, implies zero current. The more rapidly the cell operates, the less electrical work it can supply.

For a reversible equilibrium reaction, vant Hoff isotherm says:

∆G = ∆G° + RT lnK (xi) where,

• K is the • K = Product/Reactant = [M]n/[M]n+ • R is the =8 .314J/K mole • T is the temperature in scale. Substituting for free energy changes in vant Hoff equation, – nFE = – nFE° + RT ln [M]/[Mn+] = – nFE° + 2.303 RT log [M]n/[Mn+] Dividing both sides by – nF, E = E° – (2.303RT/nF) log[M]n+/[M]n (xii)

This is termed as Nernst’s equation.

Applications The Nernst equation can be used to:

• calculate single electrode reduction / oxidation potential at any conditions. • measure standard electrode potentials (reduction / oxidation). • compare the relative ability as a reductant or oxidant. • find the feasibility of the combination of such single electrodes to produce . • measure Emf of an electrochemical cell. • find unknown ionic concentrations. • know the pH of solutions. • measure solubility of sparingly soluble salts.

Limitations

1. In order to use the Nernst equation in case of solutions having very high concentrations where the ion concentration is not equal to the ion activity (unlike dilute solution, experimental measurements must be conducted to obtain the true activity of the ion. Hence the use of Nernst equation for the purpose is complicated. 2. It cannot be used to measure cell potential when there is a current flowing through the electrode because the flow of current affects the activity of the ions on the surface of the electrode and additional factors such as resistive loss and over potential must be considered when there is a current flowing through the electrode.

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