Salt-free electrolysis of water facilitated by hydrogen gas

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Salt-free electrolysis of water facilitated by hydrogen gas ( salt-free-electrolysis-water-facilitated-by-hydrogen-gas )

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Li et al Dovepress Supplementary materials Expressions for the limiting current are derived in these supple- mentary materials. Since a solution of the steady state diffusion equation in the domain 0 ,x ,w is a linear variation of x, concentrations of [H+] = c+ and [H2] = cH2 are expressed by c+=p+sx, cH2=q−rx (S1) where p, q, r, and s are positive constants. Fluxes at x = 0 by both species are given by D+ (dc+ /dx)x=0 = −2DH2 (dcH2 /dx)x=0 = j/F or we have D+s = 2DH2r = j/F  c* + jw / 2FD  2 + + + jw / 4FD H2 − jw/4FD We define the dimensionless current density as H2 H2 c* ∆E=ln H2 2F RT Then Equation S6 becomes (S6) c* − jw/2FD  f = jw/2Fc* D (S7) c* H2 H2+ + 2F  2+ f b+ f2 exp RT ∆E = 2 − f  b − f  (S8) c* D b= + + (S9) The kinetics-involved diffusion equations for [H+] = c+ and [OH−] = c– at a planar electrode under the steady state is given by where D+ and DH2 are diffusion coefficients of H+ and H2, respectively. Letting the electrode potentials at the cathode and the anode be EC and EA, respectively, we can express the Nernst equations at both electrodes as RT (c+ ) RT E−E= ln x=0= ln p2 C 2F (c ) 2F q 2 dc 2 d2c+ =−(k[HO]−kcc)/D dx2 d2 r+− (S10) (S11) where (S2) c* D H2 H2 H2 x=0 EA −E =RTln(p+sw) (S3) The voltage controlled by a two-electrode potentiostat is ∆E = EA – EC. Then we have − =−(k[HO]−kcc)/D dx2 d2 r+− 2 2F q−rw where kd is the dissociation rate constant and kr is the recom- bination rate constant. These rate constants can be related with the equilibrium constant, K, through (S12) The boundary conditions for the concentrations are given by the Nernst equations under the assumption of the invariance of cH2 (S13) 2 ∆E=RTln(p+sw) q 2F q−rw p2 k c*c* K= d = + − , K[HO]=c*c* =K =10−14M2 (S4) kr [H2O] 2 +− w Since the amount of the species does not change by elec- trolysis within the cell, it should keep the loaded amount, regardless of concentration profiles. By letting loaded con- centrations be c * and c *, the trapezoidal profiles yield 22 H2 + (2p+sw)w/2=c*w, (2q−rw)w/2=c* w (c ) (c ) E−E=RTln + x=0,E−E=RTln + x=w + H2 Extracting p and q and replacing s and r by j/F in Hydrogen ion participates in the electrode reaction, whereas hydronium ion does not. Then the fluxes at the electrodes define the current density: Equation S2, we obtain p=c* −sw/2=c* − jw/2FD +++(S5) j = D  dc+  = D  dc+  , q=c* +rw/2=c* + jw/4FD H2 H2 H2 F  d x   dc−   dx x=0  d x   dc−  x=w = 0 C 2F c* A 2F c* H2 H2 Eliminating p, q, s, and r from Equation S4 by use of Equations S2 and S5 yields x=0 = 0, (S14) Powered by TCPDF (www.tcpdf.org) 14 submit your manuscript | www.dovepress.com Dovepress  dx x=w Reports in Electrochemistry 2013:3

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