Interface-Lippmann equation

 Thermodynamics explaination of electrified interface-Lippmann equation

1.Consider electrode/ electrolyte interface as closed system where no matter leaves or enter in it, then

   dU =TdS-W

TdS=Q= Heat produced/supplied reversibly

W= Work carried out reversibly.

 2.Again consider it as an open system, matter can exchange between and surrounding .Let us introduce a mol of i species , the chemical potential on the system is μi

 dni is the change of the number of moles, then work done = - μidni

dμ= TdS-W-∑μidni

∑μidni= work done required to expel dni of species i

The work done in the metal/solution interface (MI/S) are due to (a) volume expansion PdV (b) increase of surface area dA , the corresponding increase of interfacial tension ν. dA

3. Changing the external source of electricity for altering the charge on the metal by dq' M is

 MI∆S φ.dq'M

MI∆S φ=V, MI∆S φ= interfacial tension, V=applied potential

dU=TdS-PdV- νdA - MI∆S φ.dq'M - ∑μidni…….(1)

Each term in the RHS of the equation is the product of two factors

i.e, T,P, ν, MI∆S φ, μi with Ds, Dv,dA, dq'M and dni

Each of the first factors is known as intensive factor(which does not depend on the amount of matter in the system) and each of the second factor is known as extensive factor(which depend on the amount of matter in the system).

Hence, du= ∑Intensive factor X Extensive factor 

increase the extensive factor to absolute value i.e integrate the equation

We have

U=TS –PV- ν.A- MI∆S φ.q'M - ∑μ,

 Again differentiate this equation by changing both the intrinsic and extensive factors, we have

du = (TdS-PdV- νdA - MI∆S φ.dq'M - ∑μidni)- (SdT-VdP- Adν- q'M. d MI∆S φ- ∑ni. dμi)………(2)

Here equation 1 and 2 are equal 

0= SdT-VdP- A dν- q'M. d MI∆S φ- ∑ni. dμi, 

at constant temperature and pressure we have 

0=- A dν- q'M. d MI∆S φ- ∑ni. dμi

dν= - q'M/A . d(MI∆S φ)- ∑ni/A. dμi……….(3)

Surface tension is related to the 

  • Potential difference across electrode/electrolyte.
  • the change in chemical potential of species.

        Here we introduce another quantity, τi ,

              τi = (ni /A) - (ni0/A)

 ni =number of moles at the interface, when double layer is formed

ni0 = number of moles at the interface, when no double layer is formed

A= Area

(n i /A) = τi + (ni0/A)

(n i /A) dμi = τ i dμi + (n i 0/A) dμi

∑ (n i /A) dμi = ∑τ i dμi + ∑ (ni0/A) dμi,………(4)

We know from Gibbs Duhem relation 

∑ n i 0. dμi =0

Then equation 4 becomes,

 ∑ (ni /A) dμi = ∑τ i dμi.

Substituting this relation in equation 3 we have,

dν= - q'M. d(MI∆S φ)- ∑τ i dμi…….(5)

Equation 5 contains the quantity d(MI∆S φ), i.e the change of inner potential difference across electrode/electrolyte surface.

d(MI∆S φ) can be determined if 

  1. MI/S is a polarizable one
  2. MI/S is connected to non polarizable interface MII/S (as in electrochemical cell),when MI/S and MII/S are connected , the total potential of the electrochemical cell is

           V= (MI∆S φ)+ (S ∆MII φ)+ (MI∆MII φ),

          (MI∆MII φ) = Inner potential which does                   not depend on external potential, So, the                 change of external potential V , i.e 

           dV = d (MI∆S φ)+ d(S ∆MII φ), put the value             of d (MI∆S φ) in equation 5,

we have 

dν= - q'M d V + q'M. d(S ∆ MII φ)- ∑τ i dμi……(6)

At equilibrium , d (S ∆MII φ) = - 1/(Zj.F) . dμj , j is the leakage of charge across the non polarizable interface.  Put this value in equation 6 we have ,

dν= - q'M d V - q'M /(ZjF). dμj -∑τ i dμi

This is the thermodynamics treatment of polarizable interface. 

From this we can study electrocapillary curve for a fixed composition, dμi = 0, dμj =0

(dν/dv) at fixed composition = - q'M, This is known as Lippmann equation.