7/28/2019 Termo - Activities Debye-Hackel Theory

1/4

Electrolyte Solutions: Activity CoefficientsTheory

Chemistry 243David Ronis

McGill University

and Debye-Huckel

To understand the thermodynamics of electrolyte solutions, we have to understand hownature tries to keep systems electrically neutral. (WHY?). In particular, this leads to some addi-tional definitions for the chemical potentials and activities. Consider the electrolyte dissociation

e1":A'r_ + v +Az* * v -8"-, (l)where v; is the stochiometric coefficient of the ith ion which has charge Zi, i = + or -, where eis the magnitude of the charge on an electron.

There is some ambiguity in horv to think about the solution, i.e., as comprised of dissoci-ated ions or of undissociated molecules. Of course, in reality an equilibrium for Eq. (l) will beestablished, and thus,

llA,.B,_ = V+llA^ * V-Pg"-. (2)We've seen that chemical potentials can be written in terms of activities ds ltr = pi + RT h(o,),which in our case gives

F A,.8,- = FoA,*o,- + RT ln(a 4P,-),P A,. = lt'A,. + RT ln(a 4.),

andltB,_ = ltoB"_+ RT ln(ag,_).

It tunrs out that the activities of the positive and negative ions always appear. together,the equation for the dissociation equilibrium:oui,-oti, -K,

v: v++v_,Winter Term 2010

(3a)(3b)

(3c)e.g., as in

o Ar.Br_cf. Eqs. (2) and (3). Because of this it is convenient to define meatr ionic activities as:

p*=v*FL.*v-Pe'- = pl+ RTlna+, (4a)Vwith

(4b)

7/28/2019 Termo - Activities Debye-Hackel Theory

2/4

= v+ - v- s,_,and

a!*= au1,.d4.Hence, the dissociation equilibrium becomes:

au-L _L'%-^Finally, if we define a mean mole fraction as x! = x?x!- it follows from Eq. (4d) that the meanactivity coefficient is given by fl = fi.f!-.

We next consider a model for the activities of dilute electrolyte solutions. Since the undis-sociated molecule is neutral, v+z++v-z-=O.If pi is the concentration of the ith ion, then thefact that we start with a electrically neutral compound implies that

lp;ezi =0. (s)Does electro-neutrality hold locally; i.e-, very near to one of the ions? Consider a statisticalmodel for the local charge density. Based on our work on the kinetic theory of gases, we knowthat

c

(4d)

Pro b ab il ity n exp(- Energt I k 6T),and for electrical problems it is reasonable to assume that

Energt = ezi$(r)

(6)

(7)

(e)

(10)

(l l)

where p(r) is the electrical potential around one of the ions in the solution, here assumed to be atthe origin of our coordinate system, Using these probabilities to moderate the local density, wesee that local charge density, o(r) is

o(r) = L p, "r t exp(-u ig@)l k BT)i (8)For a positive ion at the origin, we expect that Q@) > 0 and hence, Eq, (8) predicts a net excessnegative charge density in the local vicinity of our ion. This is the physical basis of the phenom-ena of screening. If the elechostatic energies are small compared with thermal energies, i.e.,e0@)

7/28/2019 Termo - Activities Debye-Hackel Theory

3/4

Activities & Debye-Huckel Theory Chemistry 243cf. Eq. (5), was used to eliminate the leading order terms.

How much charge will be induced by having an ion (with charge Q) at the origin? Gauss'slaw strongly suggests that the induced charge, cf. Eqs. (8) or (10), exactly balance the charge atthe origin; i.e., local electro-neutrality will hold, and thus,e=- !dro(r)=4n !* arto{i=2# li *toe) (r2)

where the second to last equalrty follows by switching to polar coordinates, and the final one byusing Eq. (10). In the absence of screening, the potential is simply Coulomb's law; namely,

-3-

where e is the electric pennittivity of the solvent. How will this change if the charge isscreened? To answer this requires some physics that is a bit beyond this course* and insteadwe'll simply guess the form of the screened potential, i.e.,0U)= +Eer

0?) = I {*o,,+fiEr

(l 3)

(14)

(16)

where the exponential accormts for the screening and rp is lnown as the Debye screening wave-vector. The screening sets in on a length scale of 22 = llro.What is the screening length? To answer this we'll impose local electro-neutrality, cf. thelast equality in Eq. (12); hence,

2te2Q I (ls)ekBT *o='# !; *r{*il =Thus, by solving for rp,we see that

where the second equality is obtained when fte ionic sfiengtr and electron charge are expressedas mollm3 and the Faraday, I', respectively. For a l:l electrolyte in water at25C, this gives ascreening length of,,= *=# (io,i),

where c is the molar salt concentration; hence, the counter-ion cloud around any given ion in a0.01M salt solution is about SO i.in radius (1o =30.44 but the cloud extends beyond .12 some-whaQ.Specifically, as you will see in PHYS 242,yort necd to know Poisson's equation for the electricpotential due to a charge distribution in a dielectric medium; i.e.,

Y'oe)= -o(r)le.When Eq. (8) is used for the charge density, we obtain what is known as the nonlinear Poisson-Boltmrann equation, which cannot be solved analytically for all but l-dimensional problems.When Eq. (10) is used the linearized Poisson-Bolhnann or Debye-Huckel equation is obtained;this can be solved in closed form and gives the result presented in the text.

Winter Term 2010

7/28/2019 Termo - Activities Debye-Hackel Theory

4/4

Activities & Debye-Huckel Theory Chernistry 243How much work was done on the system nplacng the ion and the screening ions at theorigin? Clearly,

4-

where we've again switched to polar coordinates and used Eqs. (10), (14), and (16). Thus,mole of either of the ions, the work is just done is

(17)for a

Narpdz!8reOf course, this is just another contribution to the non-PV work or chemical potential associatedwith dissolving the ith ionic species. Hence,

w -- [ dr g{r)o(r) = o" li &124Q)o(') = - 8rle2 f *]s2b\=-*rAksT to w ' Y \"- 8ne '

, rod @*f,+v-*-1---s61lEst E VrDe2= 67;'*'-,

where the second relation follows by noting that the undissociated molecule is neutral.

or(18a)

(18b)

(1ea)

(leb)

This is known as the Debye-Huckel limiting law and is valid at low concentrations (typically, itbreals doum in the 0.005 to 0.01 M range). Note that the sign of the ionic charge is irrelwantand that lnyrnlt/2.

With Eq. (18) it is easy to work out the mean activity coefficient; specffically,

Winter Term 2010