Electrolytes. Chapter Basics = = 131 2[ ]. (c) From both of the above = = 120 8[

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1 Chapter 1 Electrolytes 1.1 Basics Here we consider species that dissociate into positively and negatively charged species in solution. 1. Consider: 1 H (g) + 1 Cl (g) + ()+ () = { } = (+ )+ ( ) = 167[ ]. The ve heats of formation for the ions is principally due to the association (solvation) of the polar water molecules to the ions. The (aq) is present to remind you of this association. 3. The separate enthalpy of formation of IONs cannot be determined directly as the solutions are always neutral, i.e. there are always contributions from +ve and ve ions to the heat evolved on solution. 4. This problem is overcome by a convention for H + at an activity of 1. (+ ) =1 for all T. As a consequence of this convention, we get a non- 3 law entropy. (+ ) =1 { (+ ) } = and (+ ) =1 (+ )+ (+ )= 5. Thus, returning to our numerical example: (a) the constant pressure calorimetry + convention ( )= 167[ ] (b) from the equilibrium constant, in this case, 1 (see section 3 for the ionic strength complication) the standard reaction free energy change can be determined. 1 The conductivity of the solution is often used to determine the extent of dissolution. The conductvity is roughly proportional to = = 131[ ]. (c) From both of the above = = 18[ ] = 167 ( 131) 98 18[ ]= ( )+ 1 { ( )+ ( )} (d) From the standard entropies (from integration of heat capacity/t) one gets the values inside {...}. Thus 18[ ] = ( ) 1 { } ( )= {3538} =565[ ] 6. With these values for Cl, one can now bootstrap ones way to an entire table, see Table 1.1. For example, one might move next to consider () + ()+ (); (a) : 411? 167[ ] With a measured value of =39[ ] (+ ) = 41 [ ] can be deduced. (b) With another equilibrium constant measurement (K from conductivity), one can deduce (+ ), and (+ ) Voilà, Table 1.1. ANALYSIS: As the reference is to ( + ) ve values of (aq) mean that the formation of the solvated ions is more exothermic than for the formation of (+ aq). Generally, the higher the charge and the smaller the ion, the concentration of ions in solution, but all ions do not conduct equally. Ions that move through solution easily conduct better. For example, small, fast moving ions like H + impart greater conductivity to solutions than do bulky ions like bromide ion (Br ), or heavily hydrated ions like sulfate (SO 4 ). 76

2 themorenegativethe (aq). Why does the size matter? The smaller the ion the closer the polar ions can get to the charge and thus the greater the stabilization of the hydrated cluster. As energy is conserved 1 more energy released on solvation. Also note that (aq) can be +ve or ve because they reflect the loss of quantum states relative to loss from the solvation of H +. States are lost as water is associated with the ion, rather than doing its own thing. 1. : Born model A theoretical model for can help firm up our idea of we are talking about. 1. The non-expansion for a reversible process is. As the solvation energy should not include the ionization energy () =( ) ( ) 3. Lets calculate the reversible work required to charge up a sphere of radius r. Each added charge, repels the preexisting charge and thus adding charge gets harder and harder. When doing this in a medium, as compared to vacuum, a correction factor, which reduces the cost must be considered. (a) ( ) = R (b) ( ) = R 4 = 8 4 = 8 Where (= [ ]) and are the permittivity of free space and the relative permittivity (or dielectric constant )ofthematerial, respectively. (c) Low-frequency relative permittivities at 5. material ² (5 ) Dipole? No 4.74 No No No 9.6 Yes Yes Yes Yes H O Yes d 1 = 39pm 4. Dialing up a mole of charge, with = in soln. () = 8 ( 1 1) The dielectric constant (at sufficiently low frequency) is the relative capacitance of a condenser when the material between the plates is the medium rather than a vacuum. = 5. If 1 () Vsolvation is spontaneous. 6. But what is the relevant mean distance? It is not the ion-ion distance in a crystal. One uses an effective distance to account for the facts that: (a) the center (H O)-to-center (ion) distance is generally further than the ion-ion distance in crystals and (b) that +ve ions can get closer to the center of H Othan ve ions [] ( ) = a line with negative slope and zero intercept, i.e. validating the underlying assumptions. (See plot 1. b.) 8. While this theoretical model is reasonable, the experimental ( bootstrap derived) values are far more accurate. 1.3 Mean activities ( ± )&activities coefficients ( ± ) 1. The normal form of Henry s Law is not followed (even in dilution) for electrolytes.. Molality is the concentration unit of choice (with the reference state at 1) because it is invariant with temperature. 3. While one can define the single-ion (charge) CPs, + ( + ) = + + ln( + + ) THEY ARE INDETERMINATE and thus, except for mental gymnastic exercises, useless. 4. Neutrality requires: =where ± are the charge (subscript) ion multiplicity. (a) :(1 1 + )+(1 1 )= (b) :(1 + )+( 1 )= = Using a geometric-mean ionic activity, ± ( + + ) 1, a collective CP of the electrolyte is defined, () + ln{ ±} (a) = + ln{ + } = + ln{ ±} with ± = { + } 77

3 (b) = + ln{ ++ } = + ln{ 3 ±} with 3 ± = { ++ } 6. The geometric-mean activity coefficient and molality are defined by: ± ( + + )1 ± ( + + ) 1 =[( + ) + ( ) ] 1 =[( + ) + ( ) ] 1 7. Employing these we can write an expression for the CP of the electrolyte that conforms to what we are used to. () = + ln{ ±} = + ln( ± ± ) =( + ln[ + + ]) + ln + ln ± However, this begs the questions: what is ± and how do you measure it? 8. The ionic strength of solutions is defined as 1/ the sum of the concentrations times the charge of each species. 1 X (1.1). ln =ln ln ± or in general, =ln ln ± 3. If the solubility is small and (i.e. no noncommon ions) ± 1 (e.g. () 1 5 ) 4. However, % with %. An effect sometimes called salting in. When is plotted as a function of the dependence is universal. That is, is a function of but not of the specific ionsthatgive its numerical value. 5. To get ± : (a) Measure () (b) Intercept, ± 1 and thus = lim ± (c) The extracted activity coefficients ± () = ( ) 1 i. depend on and on the ionic strength but NOT (or only weakly) on the specific ions. ii. decrease (below 1) with % and, ± & with % and and () & with % and for very low (This nonideal effect is called salting-in.) iii. Ultimately, ± () values increase above 1 at high 1 (An effect called saltingout.) 6. WHY should activity coefficients depend on I? That is: why should the chemical potential DECREASE with ionic strength (at low I)? To understand this one must think about how charges (noncommon ions) affect the liquid environment. 1.4 Debye-Hückel Theory ± from solubility measurements Consider: () + ()+ () OBSERVATION: (), i.e. the solubility depends on the ionic environment as well as This is a noncommon ion effect. 1. ( )= ( )=( +)( ) =( + )( + ) = ()± extrapole = meas. * deduce. This is an elaboration on the Born model for solvation which allows one to calculate ± (i.e. the non idealities) for extremely dilute electrolyte solutions. In the plot below, the solid line is a pure Coulomb potential, the dashed and dotted lines are shielded Coulomb potentials. Lets see if we can understand how the simple 1/r Coulomb potential is altered when the central charge, and the test charge sensing it, are placed in a two-component charged fluid 3. 3 This logic is also relevant for discussing plasmas and Quantum field theory. The point of mentioning this is only to point out how connected science is. The same solid ideas come up again and again in seemingly unrelated fields. 78

4 5. The Poisson equation relates Coulomb (heavy solid) and shielded Coulomb potentials for =33 and.1 nm 1 (3. and 9.6 nm peak of counter-ion shields) are shown in dashed and dotted, respectively. These shielded potentials are appropriate for.1m and.1m 1:1 electrolytes in water. 1. Consider a collection of fixedcharges,eachproducing a potential at r (r) ganguptoproducean overall electric potential (r) = P (r). The relative probability of an ion of charge being at this point r is given by a Boltzmann factor, a general result from statistical mechanics. =exp{ } =exp{ }. The net charge density (charge/volume) is, [ 1 ]=( ) =[ ] 3. When the numerator = ± we can expand the about in the exponent, = (1 + )+ (1 ) =( ) [+ + + ] The () = by overall charge neutrality, thus, [ ] (1.) This gives us one relationship between and If we can get another, we will have two equations and two unknowns, and thus life will be good. To get the second relation we have to use a result from EM physics. I will review something covered in Physics 118. I may cover it differently than you did in 118, but the end result should make sense. (a) Overview: variation in produces (b) Outward flux from a surface = P (net losses from all pts inside). That is, for a vector field E, which in this case is the Electric (force) field, H E S = R E (c) However, it is also true, from Gauss Law 4, Outward electric flux from a surface = H E S = R 1 where (= ) is the (total) permittivity. (d) As the volume integrals are equal, the arguments must be the same, thus E = (e) The final step is appreciating that the electrostatics is conservative, not right wing rather that electric (force) field is the gradient of a scalar (potential) field. (See the intro. chapter on the structure of physical science.) Calling this field the electric potential φ, E = φ therefore, E = ( φ) = φ = φ = (1.3) A result often called the Poisson equation. I remind you that the Laplacian is a differential operator used in the kinetic part of the S.E. and that in Cartesian and spherical polar coordinates is, φ = P 1 = =3 1 1 φ φ (φ) (f) Now we employ the charge density deduced previously. φ = e o r = [ ] () {[ ] } φ = {K } (1.4) 4 This is sometimes called the 1st law of E&M and is the content of the first of Maxwell s equations. (Not to be confused with Maxwell s relations.) 79

5 with K ={[+ + + ] } A result called the Poisson-Boltzmann (PB) equation. (g) Evaluating the constant, K =( )() q K[ 1 ] =98 = 91 [] = The solution of the PB DEQ gives what is sometimes called a Shielded Coulomb potential. () = ( K ) (1.5) with = () ( ) 144 ( ) 7. The shielded Coulomb potential can be approximated by the (unshielded) Coulomb potential a correction term important at larger distances. () = central ion atmosphere terms. The ionic solution forms an atmosphere around each ion. The effect is such that very close to any ion you have the standard Coulomb 1 potential. However at larger distance, the atmosphere shields, and to a large extent cancels, the charge of the central ion. As a result the potential weakens with distance MUCH faster than the standard (unshielded) 1/r Coulomb potential. One can think of the shielding as creating a shell of opposite charge ( if the central charge is +)atamostprobable distance of 1K 8. Now one can use Max Born s logic to calculate + ln + = work to form atmosphere. Let = work to add an infinitesimal charge to the central ion in the presence of a counter ion charge at a most probable distance 1K We can identify this work with the non-ideal portion of the chemical potential, i.e. ln on a per ion basis. = = R = R ( K) R 4 = K 4 = ( + 8 )K Making the association to the non-ideal part of the CP, ln + = and using the defined quantity K, = 1 ( + 8 )[+ + + ] 1 ( )1 = 1 8 ( 1 )3 3 +[ ] 1 asallthetermsare+ 9. Using the ionic strength, (+ + + )[ # ]=[ ] [ ] [ multiplying by ( ) 3,andusingF = ln + = 1 8 ( 1 ) ( ) 1 ln + 8 ( 1 )3 F +( ) 1 = C+ 1. # ] 1. C is a collection of constants times a weak function of T. As a reminder: F = [C/mol], = [ ]and = and for water, =7854 = 99778[ ] (at 5 ) Evaluation of C ( ) yields, [ ] C Finally, the quantity of physical relevance, ln ± =( + ln + + ln )( + + ) = C 1 ( ) [ + + ] = C 1 + ( ) Using the fact that + + = = C 1 + ( As ( + )= 1, ) ln ± =+C 1 + = C 1 + (1.6) 13. As ± 1 i.e. there is a contribution to the CP and thus the electrolytic nonidealities are attractive, in the limit of very small This enhances the solubility ( salting-in ) of sparingly soluble molecules (e.g. proteins.) 14. A better treatment leads to the the Davies equation, with another coefficient ( at 98 ) 1 ln ± = C + [ 1+ ] 1 (1.7) This equation is often suitable up to the saltingout ionic stengths. 8