HELMHOLTZ PERRIN MODEL OR THE PARALLEL PLANE CONDENSER MODEL

According to this theory in order to develop potential differences one phase certain positive charge and outer phase a negative charge of the phase boundary is with the plates of a charged condenser. Thus at the interface, there are two sheets of charge on the electrode and the other in solution the charge densities on the two sheets are equal in magnitude and opposite in sign as in the case of ∣∣ electrode plate condenser. These charges are at a distance from each other and together from electrical "double layer" which is compact and rigid.

                         He postulated that the electrical equivalent of the double layer is a ∣∣ electrode plate condenser. Therefore the electrostatic theory capacitors can be applied for double layer.

The potential difference across a condenser is, 

         V =  4πdqm   dqm...................... (1)

                   ∈i

Where d - distance between the plates 

∈i - dielectric constant of the medium.

Based on ∣∣ electrode plate model of double layer,  

                 dv =  4πd    dqm............. (2)

                          ∈i

This is the functional relationship required for the integration of the Lippmann equation.

    ∫dʋ = − ∫qm dv............. (3)

=− ∫qm   4πd    dqm

                ∈i

∫dʋ = −    4πd  ∫ qm  dqm ≫ −   4πd  

                ∈i                                 ∈i

1/2 qm² = ʋ is constant.

When qm = 0 (i.e) at pʐe , υ = ʋ max

∴ constant =− ʋ max

Hence, ʋ − ʋmax = −  4πd        qm²   

                                       ∈i           2

ʋ = −   4πd          q²m    + ʋ max   ..........(4)

             ∈i              2

W.K.T,  v =   4πd   q   (or) q =   V∈i     ....(5)

                      ∈i                         4πd

Equation (5) in (4)

     ʋ = ʋ max −   v²∈i       ...........(6)

                           4πd

This is equation for parallel, symmetrical about υ max is for electrocapillary curve which are perfect parabola.

Limitations :-

1) This model predicts perfect parabolic electrocapillary curves. But curves are not parabola always. There's always a slight asymmetric.

2) it's also found that the electrocapillary curves show a marked sensitivity to the nature of the present electrolytic. Large organic cations seems to affect significantly the electrocapillary curves.

3) The electrode plate condenser model predicts a constant capacity which show a differential capacity which is not constant with all potential consider a typical electrocapillary curve which is not exactly a parabola.

The differential capacity a capacitor, 

       C = ( ∂qm )

                 ∂v

is constant composition.

A plot of C vs V can be obtained by differential of q vs v curve, 

     (∂ʋ )     = − qm

      ∂v

∴  C = −∂ (   ∂v   )

                  ∂v.∂v

  C = − (   ∂²ʋ    )

                ∂v²

The equation of theoretical predictions and observation and Helmholtz model of the satisfactory curves that are perfect parabola.

Helmholtz model of the interface is not satisfactory is explaining the structure of the interface.

The electrified interface doesn't behave like a simple double layer. 

                                    When the electrocapillary ʋ vs E curve is a perfect parabola, the electrode charge density varies linearly with potential independent. However, experimental electrocapillary curves are not perfect parabola.