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Dovepress Furthermore, electric neutrality holds within the cell: ∫w c dx = ∫w c−dx (S15) 0+0 Taking the subtraction of Equation S10 from S11 yields d2(c+ –c−)/dx2 =0orc+ –c− =k1 +k2xforconstantsk1 andk2. Applying conditions S14 and Equation S15, we have jw j c+−c−=−2FD+FDx (S16) It is assumed that the reaction rate is so fast that the reac- tion zone is restricted to domain near the electrode surface without effects of diffusion. Then the interesting value of x within the reaction layer is much smaller than w. On the other hand, c+ near the cathode is almost zero, leading to (c+)x ≈ 0 ,, jw/FD. Then Equation S16 for x ≈ 0 becomes Salt-free electrolysis of water A solution of Equation S19 is given by c+ =2−Bexp−λ gx (S21) S21 yields j=BDλ g Fw Subtracting B from the above equation and Equation S21 for x = 0, we have c* g w where B is a constant. Applying condition S14 to Equation (c+)x=0 2 jw 2 g c* =g−Fc*Dλ g=g−λ (S22) jw jw c− =FD2−x+c+ ≈2FD and Equation S10 can be reduced to d2c+ =−kd[H2O]+ kr jw c On the other hand, we assume that the reaction at the anode (H2 → 2H+ + 2e−) is in equilibrium, H+ + OH− ↔ H2O, because the produced amount of H+ at the anode is smaller than the consumed amount of H+ at the cathode. Then Equa- tion S16 at x = w is given by or (c+)x = 0 and (c+)x = w by Equation S19 and S23, respectively, we have (S19) dx2 It is further written as D 2FD2 + 2k [H O]FD (S17) (S18) (c+)x=w =Kw/(c+)x=w =jw/2FD (c+)x=w/c*=g/4+ g2/16+1 d2(c /c*) k jw c + =r +−d2 = +− 2 g (S23) Subtracting the two Nernst equations S13 and replacing d x 2 where 2 F D 2 c * c * k r j w w 2 * c * λ2gc F∆E≡F(EA−EC)=ln(c+)x=w =lnλg(g+ g2+16) g=jw,λ=wkrc (S20) Fc*D 2D RT RT (c ) 4(2λ−g3/2) + x=0 (S24) Reports in Electrochemistry Publish your work in this journal Reports in Electrochemistry is an international, peer-reviewed, open access journal publishing original research, reports, reviews and com- mentaries on all areas of electrochemistry. The manuscript manage- ment system is completely online and includes a very quick and fair Dovepress peer-review system. Visit http://www.dovepress.com/testimonials.php to read real quotes from published authors. Submit your manuscript here: http://www.dovepress.com/reports-in-electrochemistry-journal Powered by TCPDF (www.tcpdf.org) Reports in Electrochemistry 2013:3 submit your manuscript | www.dovepress.com Dovepress 15PDF Image | Salt-free electrolysis of water facilitated by hydrogen gas
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