Transport Equation. For constant ε, the force per unit fluid volume due to electric field becomes,

Transcription

1 Trasport Eqato For ostat ε, the fore per t fld volme de to eletr feld beomes, - ρ f E N/m 3 or ρ f ψ Mometm Eq. (trodg the eletr fore term as body fore term) ρ + ρ = p + µ d t Steady state, reep flow d + ρ g f ρ ψ.1 0 = p f + µ + ρ g ρ ψ Free harge desty s gve by, f ρ = z e = F z. F s Faraday ost. C s molar o. Mass oservato: = 0 s defed as followg

2 Coveto-Dffso-Mgrato Eq. The movemet of o spees a eletrolyt solto take plae de to blk moto, dffso bease of dfferee oetrato ad mgrato der eletr potetal. Loal mass average veloty, ρ ρ = 1 = 1 = = = 1 ρ ρ...3 s the veloty vetor of the th spees w.r.t. statoary observer s the blk fld veloty w.r.t statoary observer Molar average veloty = 1 = 1 = 1 = = spees (mol/m 3 ).3 s the molar o. of the th Molar o. (mol/m 3 ) ad mass desty (kg/m 3 ) are related by,

3 = 1000 (ρ / M ) M s the molar mass (kg/kmol) of the th spees. M = 1000 ρ Mass of a mole of a sbstae s alled molar mass ( M ) kg/kmol. It has magtde as the formerly sed moleler weght. The fator 100 appears order to reole the s gve terms of mol/m 3 ot as kmol/m 3 ot mol/l. Mass frato of the th spees s the desty of th spees dvded by the desty of the mxtre W = ρ I / ρ Smlarly molar frato, x = / The flx gve by Fk s law s prely de to dffso as a reslt of molelar moto. The molar flx of the th spees wth respet to the mass average veloty,, s gve by Fk s Law as D mol/m s

4 The flx of th spees wth respet to the statoary observer s a ombato of the ovetve flx de to blk movemet ad the dffsoal flx. Coseqetly, = ρ = ρ D ρ mass bass..4 = = D molar bass..5 For dlte solto = = D molar bass.5 The frst term the RHS s flx de to blk moto. The seod term o RHS s the flx of the th spees de to dffso resltg from the oetrato gradet,. I a eletrolyt solto sbeted to eletr feld, mgrato of o wll or. The flx de to mgrato a eletr feld s proportoal to the fore atg o the partle tmes the os oetrato,.e. The fore atg o the th spees s

5 -Z F ψ N/mol where, Z F s the harge per t mole ad F s the Faraday ostat. Ths a be expressed terms of per t volme -Z F ψ N/m 3 Addg the proportoalty ostat betwee flx ad the fore -υ Z F ψ mol/m s where, -υ I s the moblty of the o ( th spees). The t of -υ I s m mol/(j s) or mol m/(n s) The flx of the th spees s gve by, = D υ Z FC ψ molarbass...6 = ρ D ρ υ Z F ρ ψ massbass..7 I eletroket problems t s better to work wth molar bass.

6 Makg se of Nerst-Este Eq. D = R T υ The molar flx eq beomes, = D ZF D RT ψ molarbass..8 The above eqatos are kow as Nerst- Plak eq. Also, F = N a e ; k N a = R; F / RT = e / k T = N a C Makg se of the above relatos, oe a wrte flx based o mber o. = N a t: 1/m s = D Z F RT D ψ...9 Defg moblty as ω I = D / (k T) = ktω Ze ω ψ..3.0

7 Boltzma Dstrbto: Boltzma dstrbto was prevosly derved by herst argmets. Use Nerst-Plak eq (.8) to derve the expresso for Boltzma dstrbto for o mber o. Near harged srfae. Cosder a srfae wth ormal dreto as x x = x D d dx Ze k T D dψ dx Assmg at eqbm. zero fld veloty ad zero flx, d z e + d x kt or d (l d x ) z e + kt dψ =0 d x dψ = 0 d x Let = ad ψ = ψ Solto of above eq. wth b.. z = e ψ exp k T

8 Coservato of o mass Materal balae over a statoary volme elemet wth eletrolyt solto, d dt =. + R 3.1 R s the prodto rate de to hemal reato gve as mol/m 3 s, mpt term.. s the et For steady state wth o hemal reato,. = Crret Desty: The flow of rret reslt from flx of dvdal flx of all o spees preset the eletrolyt solto, = F z = e Makg se of expresso for (Nmber o. bass) z..3.3 (.9)

9 e ψ = e z edz z D kt.3.4 or (molar bass) = F z F D Z F ψ z υ 3.5 For a eletrally etral solto the frst term RHS drops ot, z = 0 Ths s eqvalet to sayg that the blk moto of zero volme harge desty a ot otrbte aythg to the rret desty. Whe there s o o. gradet, egletg the seod term to gve, = F ψ z υ or, terms of odtvty (σ) of the solto = σ ψ (Ohm s law) e σ = F where, z υ = z D kt S/m S semes = Amp/volt Molar odtvty (odtvty of oe mole of sbstae oe b meter of solto) of o spees: σ λ = = F z υ S m / mol

10 Eqvalet odtvty = λ I / z Codtvty of solto σ = λ For a sgle salt eletrolyte, σ = λ+ + + λ S/m (wrte molar odtvty of Na SO 4 oe mole oe m 3 ) Coservato of Charge: Mltplyg (3.1) by z F z = F. z F + t F z R Smmg over all spees: F z = F. z + t F z R 0 0 steady state homogeeos rx whh s or eletrally etral eletrally balaed. = 0 Coservato of Charge