THE GOUY CHAPMANN MODEL - POTENTIAL DROP IN THE DIFFUSE LAYER

 dψ  is the potential gradient

dx

 dψχ  = - [ 32πn⁰kT ]¹/₂ sinh   ʐe0ψχ  

 dχ            ∈i                             2kT

Assume,

Sinh  Ze0ψχ  ≈   Ze0ψχ   

           2kT           2kT

So,

   dψχ   = - [32πn⁰kT]¹/₂     ʐe0ψχ    

    dχ              ∈i                   2kT

= - [ 32πn⁰kT ʐ² e₀² ] ¹/₂ ψχ

         ∈i 4k² T²

 dψχ   = - [ 8πn⁰ ʐ² e₀²]¹/₂ ψχ

dχ               ∈i kT

 dψχ   = - k ψχ

 dχ

Where k =   ( 8πn⁰ʐ²e₀²) ¹/₂

                       ∈i kT

K is the ∂max distance from diffussed electrode and it depends on concentration (n⁰) 

  dψχ         = - k dχ

  Ψχ

On integrating, 

lnψχ   = - kχ + constant............. (1)

Where x = 0, ψχ = ψb

∴ hn ψ₀ = constant .........(2)

Equation (2) in (1)

ln ψχ = − kχ + ln ψ₀

ψχ = ψ₀ e⁻ᵏˣ   ........(3)

i.e The potential decays experimentally as the distance from the equation. 

When the solution concentration (n⁰) ↑ˢᵉˢ ᴷ ↑ˢᵉˢ and ψχ falls more and more sharply. 

                        The potential distance relations is on important and simple result from Gouy - Chapmann Model. 

The potential differences ∆v, The potential differences a llᵉ plate condenser is, 

∆v =  4π    dqm ..... (4)

           ∈

qm - charge of metal 

qm =-qd - [²(ɛin⁰kT)¹/₂ sinh  Ze₀ψχ ] ......(5)

                      2π                       2kT

d = k⁻¹ ( ɛi kT          )¹/₂ ........(6)

              8πn⁰z²e₀²         

Equation (5)and (6) in (4)

∆v =  4π  ( ɛikT        )¹/₂  2 (  ɛin⁰kT  )¹/₂

          ɛi   8πn⁰z² e₀²                2π

sinh   Ze₀ψχ     

          2kT

∆v = ψχ

∆v = ψ₀ when the distance x =0. When the electrode is considered as a point charge. 

Differential capacity according to Gouy - Chapmann Model :

C= ∂qm  = − ∂qd     = 

      ∂ψm       ∂ψm

= ∂ ( ɛiχ²e₀²n⁰ )¹/₂ sinh  Ze₀ψ₀      

        2πκT                         2kT.                    

                    ∂ψm

 ∂qd  = ²(ɛin⁰kT) ¹/₂   Ze₀   cosh (Ze₀ψm  )

∂ψm        2π               2kT               2kT

c = ( 4ɛn⁰kT Z² e₀²)¹/₂  cosh   Ze₀ψm   

         2π 4 k² T²                          2kT

= ( ɛn⁰Z²e₀²) ¹/₂ cosh  ( Ze₀ψm  ).... (7)

     2πkT                            2kT

Differential capacity ∴ given by the equation :-

C =  ∂q    = ( ɛiZ²e²n⁰ )¹/₂ cosh  Zeψ    

       ∂ψ         2πkT                          kT

This proves that cis not a constant and c is varies as a hyperbolic each function of potential showing inverted parabola. This is a good result because the major weakness of the Helmholtz and perrin model is that doesn't predict any variation of capacity with potential although such a variation is found experimentally. 

Draw back :- 

It's found that only in very dilute solutions (20.001 mol/dm³)² potential near pze there are portion of the experimental curves which suggests that the interface is behaving in a Gouy - Chapmann way. 

But at potential farther away from that of zero charge and in concentrated solutions this model bears no relation to reality. 

Application :

This model finds are in the understanding of the stability of collide and hence the stability of system and in certain areas of electrodes.