Notes on Debye-Hückel Theory

Transcription

1 GEOS 0300/30300 Autumn 004 Thermodynamcs and Phase Change MS Ghorso Notes on Debye-Hückel Theory We seek: µ µ o + RT ln X + µ ex where µ ex RT ln λ or the devaton from deal behavor due to long range nteracton of ons n soluton µ ex s electrcal n nature From a theorem of electrostatcs: that the mutual energy of a system of charges s onehalf the sum of the products of the charges of each and the potentals due to the others we have for the th on: µ ex z eψ where z e s the charge n coulombs and ψ s the potental on the surface of the th on due to all the other ons n soluton We want a smple expresson for ψ One of the fundamental laws of electrostatcs (Maxwell s frst equaton) s: E 4πϕ where E s the electrc feld vector ϕ s the charge densty per unt volume and s the delectrc constant of the medum Now E s ust ψ (ust lke the force s the gradent of the potental energy) so: whch s the Posson equaton ψ 4πϕ or ψ 4πϕ For the th on: ψ 4πϕ where we assume that s the same everywhere and s equal to the value for pure water ϕ s a functon of radus about the th on Clearly the charge on the th on added to the charge of every other on n soluton must sum to zero e z e 0 snce the soluton s neutral Thus

2 Notes on Debye-Hückel Theory con t 4πr ϕ dr + z e 0 a where 4πr s an element of surface area and a s the radus of the th on What does the charge densty look lke? We assume the Boltzmann dstrbuton C C exp( E kt ) where C s the number of ons per unt volume and E z eψ the energy of the on s equal to ts charge tmes the potental Then ( ) ϕ z ec exp z eψ kt Expandng n a Taylor seres (and keepng only the frst two terms): Note that the frst of these sums s zero! ϕ z ec z ec z eψ kt ( ) So ψ 4πϕ becomes Settng κ 4πe C z ψ kt ψ 4πe C z we have n Cartesan coordnates kt or n Polar coordnates ψ κ ψ ψ + r ψ κ ψ 0 A soluton s: ψ A r e κ r + B r eκ r As r ψ must be fnte so B s zero Thus ψ A r e κ r so

3 Notes on Debye-Hückel Theory con t 3 ϕ C z e kt A r e κ r How do we determne the constant A? From a 4πr ρ dr + z e 0 we have e C 4πr z A kt r e κ r dr z e a and A z e e κ a 1 +κa At last: ψ z e e κ a 1+κa e κ r r f we are only nterested n the potental feld of all ons except the th on we must subtract ts potental z e r from ψ yeldng ψ * z e r e κ a e κ r 1+κa 1 Therefore the potental at the surface of the th on (r a) due to the presence of all other ons n soluton s gven by: * ψ a z e κ 1+κa From µ ex z eψ we get µ ex z e κ 1 +κa N where N s Avogadro s number to put the potental on a molar bass So:

4 Notes on Debye-Hückel Theory con t 4 ln λ z e kt κ 1 + κa 4πe C Now we must modfy κ z kt to convert C to M unts We dvde by N and convert cc to lters (1000 cc/lter): 4π Ne M κ z 1000kT m ρ * To go from Molarty to molalty we use the relaton M where W 1+ k 0001m k W k k s the molecular weght of the k th component and ρ * s the soluton densty For geologcal (natural) fluds ths expresson may be smplfed to M m ρ where ρ s the densty of pure water snce ρ * 1+ k 0001m k W k ρ These manpulatons gve us κ 4πe Nρ m z 1000kT f we defne 1 m z and call t the onc strength then κ 8πe Nρ 1000kT Hence: ln λ z e kt 8πe Nρ 1000kT 1+ a 8πe Nρ 1000kT

5 Notes on Debye-Hückel Theory con t 5 Defne and and we get A γ B γ 8πe Nρ 1000kT π Ne 3 ρ ln( 10) 1000 ( kt ) 3 logλ A γ z 1+ ab γ Now we must convert ratonal to molal actvty coeffcents Frst note that X m m and So: As m 0 so Thus Defnng µ µ o + RT ln( λ X ) µ o* + RT ln( γ m ) µ o* µ o RT 1 ln 5551 ln ln X + ln λ m γ µ o* o µ 1 ln RT m ln( λ ) ln( γ ) + ln 1+ m + ln λ γ 5551 Γ γ log1+ m 5551

6 Notes on Debye-Hückel Theory con t 6 we have fnally: logγ A γ z 1+ ab γ + Γ γ Ths s the Debye-Hückel law At low concentratons logγ A γ z Ths s the Debye-Hückel Lmtng Law Note that a s an on parameter for the soluton A γ and B γ are f(tp) and are tabulated by Helgeson and Krkham (1974)