Chap. 4 electrolysis cell

當E>>E₀

at x=0, E=E₀-(RT/nF)ln(c_R/cₒ)|ₓ₌₀ c_R/cₒ→0 as c_R<<cₒ ⸫R→O

limiting current, iᴅₐ=-nFAc_R*D_R/Xᴅ_R....(1), iᴅ_c=-nFAcₒ*Dₒ/Xᴅₒ...(2)

general case:

i= -nFAJ₀|ₓ₌₀=nFAJ_R|ₓ₌₀

Jₒ|ₓ₌₀= -Dₒ[cₒ*-cₒ(0,t)]/Xᴅₒ=-i/nFA → cₒ(0,t)=cₒ*-(i/nFA)(Xᴅₒ/Dₒ)....(3)

(2)代入(3),  ....(5)

同理, c_R(0,t)=c_R*+(i/nFA)(Xᴅ_R/D_R)....(4)

(1)代入(4),  .….(6)

E=E°+(RT/nF)ln[cₒ(0,t)/c_R(0,t)]....(4.16)

(5), (6)代入(4.16), 

  ⸪ Xᴅᵢ=(πDᵢt)^½

⁽⁷⁾

^i.e. E_½= E°+(RT/2nF)ln(D_R/Dₒ)=E°-(RT/2nF)ln(Dₒ/D_R)

實驗上把時間固定

1. dme: dropping mercury electrode

2. rotation disk → steady state electrode

applcation:

⸪t=constant, Xᴅₒ=(πDₒt)^½=constant

iᴅ_c=δₒcₒ* ….(8)→ δₒ= nFADₒ/Xᴅₒ

iᴅₐ=-δ_Rc_R*....(9)

O+ne⁻→R, x: mole fraction of O cₒ*=xc* and c_R*=(1-x)c*

(8), (9)代入(7)→ E=E_½+(RT/nF)ln[(δₒxc*-i)/(i+δ_R(1-x)c*)]

可作i-E-x的圖, 如Fig. 4.12

1. 當x固定, E-i圖

2. i=0, E-x圖: potentiometric titration curve, Fig. 4.13(b)

⸪ Xᴅᵢ=(πDᵢt)^½, and E_½=E°-(RT/2nF)ln(Dₒ/D_R)

 → Nerst eq., +號是因為O+ne⁻→R(還原電位)

1. E=constant, i-xdiagram

constant composition, iᴅ_c=-nFAcₒ*Dₒ/Xᴅₒ≈k_ct⁻^½ Xᴅₒ=(πDₒt)^½

       2. E=constant,  i-t圖

Dropping mercury electrode(d.m.e.)

當V↑, Xᴅ↓   ↑       D₀_dme≈7D₀/3

iᴅ=nFAcₒ*Dₒ/Xᴅₒ=nFAcₒ*(Dₒ/πt)^½=nF4π(3ut/4πd)^⅔cₒ*(7Dₒ/3πt)^½=706nDₒ^½cₒ*u^⅔t^⅙