Chem 340 Fall 2013 – Lecture Notes 11- Solution equilibria (Chap. 6)
Recall for gases:
where
at equilibrium: Grxn = 0, then 0 = Gorxn + RT ln QP
or Gorxn = - RT ln KP and KP = exp (-Gorxn / RT)
For liquid reactions same form but different standard state, c0 in molar or b0 in molal
For Solution recall: At equilibrium know: soln = 0 = soln + RT  ln(aieq)
Gorxn = - RT  ln(aieq) = -RT ln Keq
Keq =  (aieq) =  (ieq)(cieq/ co)
This equilibrium formulation in terms of activities is now general, it has the same form as
KP but applies to all solutions
To use these need to get Gorxn =Gof as before, but now depend on solvent for K
In concentration units Kc =  (ieq)(cieq/ co)~  (cieq/ co) as ci0
book has molarity: cieq/ co = [J]eq/co , but molality as: bieq/ bo, others use: mieq/ mo
Must know standard states to get K or Gorxn, dilute neutrals I ~1, but charged deviate
Recall the exact relation between free energy and equilibrium is Gorxn = - RT  ln(aieq)

so use of Kc or Km or Kb or Kx is approximate, assumes j~1, which often works
for neutrals, dilute, but will have problems at high concentrations and for ions
Example: dissociation reaction: N2O4  2 NO2 where KP = [(PNO2)2/(PN2O4)](Po)-1
Extent of dissociation can be represented as , so start with n moles N2O4
At equilibrium have: (1-)n of N2O4 and 2n of NO2
Mole fractions: (1-)/(1+) for N2O4 and 2/(1+) for NO2
Partial pressure: xPtot  KP = [(2)2/(1+)(1-)] Ptot/Po= [42/(1-)] Ptot/Po
Can solve this if know KP or Gorxn (note will be quadratic eqn in ) and Ptot (or n)
If a <<1 then  KP ~ [42] Ptot/Po recall Po term due to standard state, KP unitless
Biological standard states. Consider reaction: R + nH+  P
Grxn = Gorxn + RT ln(aP)/(aR)(aH+)= Gorxn + RT ln(aP)/(aR) - RTln(aH+)
Combine to get standard state at pH7: G+rxn = Gorxn + 7RT ln10
Text example (p.221): NADH + H+  NAD- + H2 at T = 310 K, Gorxn = -21.8 kJ/mol
Here n = 1, so standard state correction: 7ln10 = 16.1
(lnx = log10x ln10)
+
Thus G rxn = -21.8 kJ/mole +16.1(8.314 J/molK)(310 K) = 19.7 kJ/mol
See it changed sign, but this means that reaction is not spontaneous as written at pH 7,
but would have been at pH 0 ([H+] = 1 molar) since using up H+ makes sense
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Reaction equilibrium under different T,P conditions
K determined as : K = exp (-Gorxn / RT)  this referenced to standard state, P=1 bar
So change of pressure not affect K, however it can affect extent of reaction
Consider our N2O4  2 NO2 where KP = [(PNO2)2/(PN2O4)](Po)-1
Partial pressures unaffected if increase pressure by adding say He (inert, ideal)
(mole fraction decrease but Pi = xiPtot so increasing Ptot not change Pi if inert)
However if compress volume and raise pressure in that way, reaction must adjust
LeChatlier principle, system react to minimize disturbance from equilibrium
So would decrease fraction of NO2, since that decreases pressure by reduce particles
For our N2O4 example rearrange K = [42/(1-)] Ptot/Po to give = [1/(1+4Ptot/KPo)]½
Clearly an increase in Ptot will decrease , will favor reactants over products
Just as for gases, the van’t Hoff equation works to describe temperature effect on K:
Change KPKP & standard states for liquid solutions, Ho or H+
Determine from slope (-Ho/R) of lnK vs. 1/T plot
For exothermic reactions, ln K and hence K will decrease with
increasing T, d lnK/dT and dK/dT < 1
endothermic , K increase, again fits LeChatlier: exothermic
give off heat, so higher temperature favors less reaction,
less product form, more reactant stay
Think about how G = H –TS vary, may be easier (text) as: G/T = -H/T +S determine Ho from temperature
variation K, plot lnK vs 1/T slope is Ho/R
Ion formation in solution -- Imagine reaction HCl(g)  H+(g) + Cl-(g)
Expect G(+) for this since the formation of gas phase ions is not favored
For HCl(g)  H+(aq) + Cl-(aq) expect G(-) (from tables: -131-0+95 = -36kJ/mol)
Ions in solution more stabilized, charged species interact with solvent, esp. H 2O
Can look at formation reaction: H2(g) + Cl2(g)  H+(aq) + Cl-(aq)
For this Horxn = Hfo(H+(aq)) +Hfo(Cl-(aq)) = -167kJ/mol (elem. H2 and Cl2:Hfo=0)
Problem, determining either Hfo for ion, need the other, not independent,
so define a reference state where Gfo (H+aq) = 0 for all temperatures, T
Then So(H+aq) = 0 = - Gfo/T and Hfo(H+aq) = 0 = Gfo +TSo
Once define reference for H+ then can compute values for Cl- (all ions aqueous)
using Horxn from calorimetry, Gorxn = – RTln K, and Sorxn = (HorxnGorxn)/T
then Hof(Cl-) = Horxn (-167kJ/mol), Gof(Cl-) = Gorxn & Hof(H+) = Gof(H+) = 0
but Sorxn(Cl-) = So(Cl-) – ½ So(H2) -1/2 So(Cl2) to account for absolute entropy
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Once you have established H+ ref values and use to calculate e.g. Cl- formation values
Then those can be used for other ions: NaCl (s)  Na+ + ClHorxn = Hfo(Na+(aq)) +Hfo(Cl-(aq)) - Hfo(NaCl (s)) = 3.9 kJ/mol
Tables give NaCl value (- 411kJ/mol), have Cl- value, so Hfo(Na+(aq)) = -240 kJ/mol
This process can continue build up reaction by reaction, each having one ion the same
Same idea for Gfo and Sfo for ion wanted (assume complete dissociation)
Note patterns, Hfo for positive cations normally negative since forming complex with
water, but must be stronger interaction than water, so some are positive. Smaller size
ions (e.g. F- compared to Br-) and multiply charged species generally more negative,
due to higher charge-to-radius values. Gfo(ion) and Hfo(ion) generally track together
for various ions. Negative ions a bit variable, but same idea. S o a bit different, since
reflects organizing the water molecules around the ion, Normally S is always positive,
but So(ion) is defined relative to H+ so could be more organized and thus negative, e.g.
in cases of more charge, such as Ca+2, Mg+2, PO4-3. (see table ↓)
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While the above is a practical way of developing thermodynamic variables for ions,
also can view it as a thermodynamic cycle as before:
So the top part is what we are discussing, but Go a state function, so can view as sum
of the bottom (tan) parts. The relative values indicate solvation is a big part of Go
The Gosolv can be estimated as the difference in reversible work to charge the ion in
solution, compared to that in vacuum, using Born equation. Represent solvent as a
dielectric: r; ion as charge: q; potential is then:  = q/4r, where r is ion radius
wvac = ∫ q’dq’ = ∫ q’dq’/4r = q2/8r where 0 = vacuum permittivity, q = ze

Gosolv = (z2e2NA/8r)(r–1-1) because (dielectric const>)  r > 1, Gosolv < 0
reaction spontaneous. This implies Gosolv should be linear with z/r (charge to radius)
As shown lower left, linear but not on line, if use effective radius, account for distance to
the solvent molecules, add 0.085Å to positive, 0.1Å to negative ion radii, now fits. Gives
idea of the source of the contribution of solvation, but not quantitative, especially for S o
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Activity coefficients
When we discussed solutions we recognized that solvent and solute could interact
differently, i.e. A-A and B-B and A-B interactions could be inequivalent, and this gave
rise to Raoult and Henry law behaviors that worked only for dilute solutions, we had
general form of chemical potential for solute: B = Bo + RTln aB where reference state
is to Henry law constant, KB, of solute B, and activity aB accounts for deviation from
ideal behavior with aB = B,s bB/bo for concentration in molal, for example
Can write Gibbs energy as: G = nAA + nBB where A = solvent, B = solute and
if B is electrolyte that dissociates: G = nAA + n++ + n-- = nAA + nB(++ + --)

+ and - are stoichiometric coefficients of ions, can differ e.g. CaCl2 (+ ; -)
Mean ionic chemical potential ± = B/ = (++ + --)/where = (+ + -) can be
determined experimentally, but individual values(+ & -) cannot
+ = +o +RTln a+ and - = -o +RTln a-  ± = ±o +RTln a± where a± = (a+ a- )
For activity coefficients same: a+ = +,s b+/bo and a- = -,s b-/bo here b+ = +b and b- = -b
Substitution gives: a± = [(b+/bo)(b-/bo)](+ - )
Simplify form with mean molality: b± = (b+ b- ) where b± = (+ - )b
and ± = (+-)so mean values all related: a± = (b±/bo)±  or a± = (b±/bo)±
chemical potential: B = ±o +RTln a± = [±o + RTln + ] +RTln(b/bo) +RTln ±
Effectively a new standard state,±oo, Henry plus stoichiometry correction  brackets [ ]
next is concentration made up and last is the activity coefficient correction ±
B = ±oo +RTln(b/bo) +RTln ±
Note: can do in c rather than b, differ. ±oo
First two terms, ideal ionic solution, last term is the important issue for deviation from
ideal behavior. Reason is long range interaction of charged (ionic) species. Goes
as 1/R where R is distance between ions, while other interactions in solution like
van der Waals go as 1/R6 or so, i.e. much shorter range
Debye – Huckel limiting law was derived to try to explain the ± values, its derivation is
beyond the needs of this course, but the idea is that the electron ion cloud and
other ions effectively shield a given ion from the rest of the species in solution, or
damp the potential at large r (distance) with an exp(-r) term – falls off fast with r
soln = (±ze/4rr) exp(-r)
1/ - Debye Huckel screening length, larger  - more screening, faster  fall off, set as:
2 = e2NAsolv (1000 L/m3) (+z+2 + -z-2)/rkT
varies with ionic strength, I, and conc.: I = (b/2)i (i+zi+2 + i-zi-2) = ½ I (bi+zi+2+bi-zi-2)
Resulting in: ln ± = -|z+z-|e2/8r0kT, which for T = 298 K and aqueous solution is:
log10 ± = -0.5092|z+z-|I1/2 or ln ± = -1.173|z+z-|I1/2
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Limiting law - since works only at low ionic strengths, model says ± decreases with
ionic strength, and it does, but the deviation is less than predicted (two figures
below AgNO3 and CaCl2), and at high ionic strengths actually turns around (figure
at bottom, ZnBr2 even gets >1). Empirical corrections (Davies equation) can be
made to simulate observed data (figure below right for 1-1, 1-2 and 1-3
electrolytes) but the point is that charges interact in solution over long ranges, and
that as concentration grows they shield each other so simple electrostatics needs
modification to model complex behavior, shows ions not ideal-dilute solutions
3:1
ZnBr2
2:1
1:1
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Solubility Products
Dissociation of salts upon dissolving can be partial, represent as an equilibrium, activity
of pure solid salt = 1, so equilibrium constant just product of ions – solubility
e.g.: MgF2  Mg+2 + 2 F- for which Ksp = aMgaF2 = (cMg/co)(cF/co)2±3 = 6.4x10-9
2 cMg = cF are related, but do not know ± or cF so need to solve iteratively:
guess ± = 1, solve for cMg (=1.2x10-3 mol/L) then compute ionic strength
I = ½(z+2c+ + z-2c-) = 3.5x10-3 mol/L and use Debye-Huckel to get± = 0.87,
substitute into Ksp and resolve for cMg and reiterate method until value constant
Eventually ± = 0.87, and cMg =1.36x10-3 mol/L
Salting in/out. As change concentration at fixed T, the Ksp is constant, but ± is changing.
This will make the concentrations change in order to keep Ksp constant, which can
affect solubilities of macromolecules by change of
ionic strength of solution (via added salts)
If macromolecule is 1:1 electrolyte Ksp = ±2s2
where s is its solubility (molar)
Rearrange: log (s/Ksp½) = - log± = 0.5092|z+z-|I1/2
Debye Huckel predicts increased solubility, s, as ionic
strength increased (salting in)
but at high concentration, water becomes tied up
hydrating ions and s decreases (salting out)
Often use (NH4)2SO4 for high solubility, and can use to
purify proteins due to variation in s
Empirical relation (plot): log (s/Ksp½) = B – K’I
Tinoco concentration determination examples
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