Concentration of Overpotential :
When the cell is reversible, the electrolyte concentration near the electrode and in bulk will be the same and is represented as Co. Under these conditions,
(ie) Current is not passing through the solution. For reversible cell, emf is given by the following equation ( emf is governed by Co)
M⁺ + e⁻ →M
E rev = E⁰ + RT ln Co......... (1)
nF
...... (2)When the applied potential corresponds to the decomposition (or) deposition potential of metal ion.
At this potential metal ions presents near the electrode get deposited immediately, lowering the metal ion concentration near the electrode compared to the bulk (ie) Cx = 0 ( near the electrode) is less than Co. Hence in solutions concentration gradient is set up.
When the cell becomes irreversible the potential is governed by ions present near the electrode namely. Cx = 0
E irrev = E⁰ + RT ln Cx = 0
nF
η arises due to the difference in concentration or concentration gradient and hence it's known as concentration Overpotential.
ηc = = RT ln [ Co ] ............(3)
nF Cx = 0
(or) ηc = ( RT ) ln ( Cx =0)
nF Co
Always, the velocity of electrode reaction (Electronation) is faster than the velocity of Diffusion of ions. The during the course of experiments, there will be a concentration gradient and hence Diffusion will occur continuously.
The current carried by ions diffusing is known as called "Diffusion current"
To relate Diffusion current with concentration fick's first law is used. Fick's first law is derived from phenomenological equation or macroscopic equation.
J = A+ BF + CF²+..................
When F C (driving force) is small, it get simplified into Fick's law which is given by
JD = - D ( dc )
dx
JD - Diffusion flux.
(D-Diffusion coefficient of ion concerned or ion diffusing)
No. of ions diffusing per second per cm².
dc - concentration gradient
dx
Assuming the curve in the above diagram to a line in the slope (dc)
dx
Is given as
dc = C₀ −Cx = 0
dx δ
( δ - thickness of the layer)
JD = - D [ Co−Cx = 0 ]
δ
id = Ζ F JD = − ZFD [ Co−Cx =0] .....(4)
δ
With increase applied potential, the velocity of electronation increase. Hence concentration gradient increases.Hence the velocity of Diffusion will also increase. At particular applied potential the ῡ becomes so fast, the Cx=0 become zero near the electrode. When Cx =0,
id = iL = - ζ F D ( Co) .................... (5)
δ
Hence the concentration gradient attains a limit, resulting in limiting υ Diffusion and the corresponding id is known as limiting current (iL)
iL = Co
id Co - Cx = o
Inverting, we get
id /iL = Co-Cx = o = 1 - Cx = o
Co Co
Cx =o = iL - id ................... (6)
Co iL
Substituting Equation (6) in (3)we get,
ηc = RT ln ( iL- id )
nF iL
E irrev - Erev = ηc = RT ln ( iL - id)
nF. iL
The above equation relates the potential with Diffusion current at a stationery electrode.
In photography instead of a stationery cathode, a mobile Hg cathode is used. Hence the expression for id has to be modified.
id = - Z F D ( Co- Cx = o) ............. (7)
δ
1) Modification :
The SA of dropping Hg cathode is sharing continuously with time. Hence in the expression for Diffusion current, the SA of mobile Hg cathode At introduced.
Assumption :
The Hg drop is assumed to be spherical.
At = 4πrₜ² rₜ - unknown
The velocity of Hg flow through the capillary gives the volume of Hg flowing in 1 second.
Velocity = ʋ ml/sec.
Experimentally this velocity is measured
t = time taken for formation of successive drops (sec)
∴ The volume of 1 Hg drop = ( velocity) xt
= V ml/sec xt (sec)
Volume of 1 Hg drop = Vt = 4/3 π rₜ³
rₜ³ = 3Vt ≫ rₜ = ( 3Vt )¹/₃
4π 4π
∴ Area of cathode at time ' t ' = 4π (3Vt)²/₃
4π
2) Modification :
With time, there is a continuous charge SA of cathode. This area is larger than the area of the thickness of electrified interface.
With charging cathodic SA, there will be a corresponding charge in thickness of electrified interface δ.
To arive at δ, it's assumed that planer Diffusion occurs rather than spherical Diffusion to evaluate the charge in δ with time. Fick's (2) law of Diffusion is used namely.
∂c = D ∂²c
∂t ∂x²
By solving this equation, we get δ as
δ = ( 3 πDt)¹/₂ ( ∂δ) δ α t¹/₂
7
These two concentration are introduced in the expression for Diffusion current (Equation 1) get modified as follows :
id =. - ZFD ( Co - Cx =o)
At. ( 3/7 π Dt) ¹/₂
id = - ZFD ( Co - Cx =o)
4π ( 3Vt)²/₃ ( 3 π Dt) ¹/₂
4π 7
This is Ilkovic equation is applicable to a dropping Hg cathode. When Cx = 0 = 0,
id = iL = - ZFD Co.
4π ( 3Vt)²/₃ ( 3 π Dt) ¹/₂
4π 7
This is also Ilkovic equation for limiting current, The measurement of such id as a fn. of applied potential is known as polygraphy.
Half wave potential ( E1/2) :
E 1/2 is the potential corresponding to in flexion Pt on the pologram and the in flexion Pt ∂E = 0
∂i
∂²E = 0 .
∂²i
By solving the twob Diffusion current, the potential value obtained is E¹/₂.
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