Solutions and Electrochemistry Crib Sheet
The Gibbs-Duhem Equation
The total Gibbs free energy of a binary mixture is
G = nA µA + nB µB
which implies that at constant p, T :
dG = nA dµA + µA dnA + nB dµB + µB dnB
But from the definition of chemical potential, at constant p, T :
dG = µA dnA + µB dnB
For both equations to be true, we must have:
nA dµA + nB dµB = 0
which generalises to the Gibbs-Duhem equation:
X
nJ dµJ = 0
J
i.e. the chemical potential of one component of a mixture cannot change independently of the chemical potentials of the others. The same principal applies
to partial molar volumes:
X
X
∂V
nJ dVJ = 0 where V =
nJ VJ , VJ =
∂nJ p,T,n′
J
J
The Thermodynamics of Mixing
Consider nA and nB moles of two perfect gases, A and B, both at the same p, T .
p
Before mixing, they have chemical potentials given by µJ = µ⊖
J + RT ln p⊖ . So
initially,
p
p
⊖
⊖
Gi = nA µA + RT ln ⊖ + nB µB + RT ln ⊖
p
p
After mixing, the partial pressures of the gases are pA and pB , where p = pA +pB ,
so
pA
pB
⊖
⊖
Gf = nA µA + RT ln ⊖ + nB µB + RT ln ⊖
p
p
Now, ∆mix G = Gf − Gi , so
∆mix G = nA RT ln
pA
pB
+ nB RT ln
p
p
But nJ = xJ n and from Dalton’s Law, pJ /p = xJ , so
∆mix G = nRT (xA ln xA + xB ln xB )
NB: because xJ < 1, ∆mix G < 0 - the mixing is spontaneous.
Entropy of mixing: from S = −(∂G/∂T )p,T ,
∆mix S = −nR(xA ln xA + xB ln xB )
Enthalpy of mixing: for two ideal gases, there are no interactions between the
molecules and (as can be seen from ∆G = ∆H − T ∆S),
∆mix H = 0
Ideal Solutions
Ideal solutions obey Raoult’s Law :
pA = xA p∗A
where p∗A is the vapour pressure of pure liquid A, and pA is its vapour pressure
in a mixture in which it has mole fraction xA . Solutions are ideal if the average
interaction between molecules in the mixture (A–B) is the same as the interaction between pure liquid molecules (A–A and B–B). This is often the case when
A and B are similar molecules (e.g. benzene and toluene).
The above expressions for ∆mix G, ∆mix S, and ∆mix H all apply to ideal solutions
as well. The composition of the vapour is given by:
yJ = pJ /p
p∗
A
The chemical potential of pure liquid A can be written as µ∗A = µ⊖
A + RT ln p⊖
because it is in equilibrium with its vapour. Similarly, in the mixture, the
pA
chemical potential of liquid A is µA = µ⊖
A +RT ln p⊖ . Combining these equations
pA
∗
gives µA = µA + RT ln p∗ , whence
A
µA = µ∗A + RT ln xA
for an ideal solution.
Ideal-dilute solutions obey Henry’s Law :
pB = xB KB
for
xB ≪ 1
where KB is an empirically-determined constant with the dimensions of pressure.
Real Solutions
The chemical potential of a solute in a real solution is defined in terms of activity:
µ = µ⊖ + RT ln a
Where the standard state is a hypothetical solution with molality b⊖ = 1 mol kg−1
in which the solute is behaving ideally. Defined this way, the activity can be
thought of as a kind of “effective mole fraction”, related to the molality by
a=γ
b
b⊖
where γ → 1 and a →
b
as b → 0
b⊖
γ is called the activity coefficient and depends on the composition, molality,
and temperature of the solution. The above equation for chemical potential
then becomes
µ = µideal + RT ln γ
because ideal solutions obey µideal = µ⊖ + RT ln(b/b⊖ ). Since γ is usually less
than 1, it can be seen as a measure of the reduction of the chemical potential
of a solute caused by its attractive interactions in the solution.
Debye-Hückel Theory
Debye-Hückel Theory gives a formula for calculating γ for ions in solution, in
the limit of low molality. It assumes that each charged species is surrounded
by an oppositely charged ionic atmosphere. The solute is assumed to be fullydissociated (no ion-pairs) and ion-solvent interactions are ignored. The result
is
√
1 X 2 bi
log10 γ± = −|z+ z− |A I where I =
z
2 i i b⊖
I is the ionic strength, and A = 0.509 for an aqueous solution at 298 K.
Colligative Properties
Colligative properties of solutions depend only on the amount of solute, not on
its identity.
Elevation of boiling point: the presence of a solute at mole fraction xB causes
an increase in normal boiling point from Tb∗ to Tb∗ + ∆Tb , where:
∆Tb =
RTb∗2
xB
∆vap H
In practice, xB can be taken to be proportional to the molality of B in the
solution, b, and ∆Tb = Kb b, where Kb is the ebullioscopic constant of the
solvent.
Depression of freezing point: the presence of B decreases the freezing point from
Tf∗ to Tf∗ + ∆Tf m where:
∆Tf =
RTf∗2
xB
∆fus H
and, as before, this is often written ∆Tf = Kf b, where Kf is the cryoscopic
constant of the solvent.
Solubility: the mole fraction of B in the solution at equilibrium with its solid
(i.e. a saturated solution), is given by
∆fus H
ln xB =
R
1
1
−
∗
T
T
where T ∗ is the melting point of the solute. NB: don’t get too excited about
this equation - solubility is clearly not a colligative property!
Electrode Potentials
The standard half-cell potential of a couple is the emf of a cell constructed
with the electrode of interest on the right-hand side, and a standard hydrogen
electrode (SHE): Pt(m) |H2(g) (1 bar)|H+
(aq) (a = 1) on the left-hand side.
Standard conditions correspond to solutions of unit activity (a = 1) and gases
at 1 bar pressure (p = p⊖ ). Pure solids and liquids always have unit activity.
Conventions for electrochemical cells:
1. Write the cell diagram, clearly indicating the left- and right- hand half
cells.
2. Write each half-cell reaction as a reduction (i.e. Xm+ + ne− → X(m−n)+ )
3. Balance the half-cell reactions so that they involve the same number of
electrons, n.
4. The conventional (formal) cell reaction is defined as
(RHS half-cell reaction) – (LHS half-cell reaction)
5. The cell potential is that of the RHS measured relative to the LHS:
⊖
⊖
⊖
Ecell
= ERHS
− ELHS
The Nernst equation:
⊖
−
Ecell = Ecell
RT
ln Q,
nF
Q=
Y
aJ νJ
J
where νJ is the (signed) stoichiometric coefficient of constituent J, and aJ is
understood to be the activity of species in solution, unity for pure liquids and
solids, and equal to pJ /p⊖ for gases.
For a cell operated at constant p, T and reversibly, dG = dwe , the non-expansion
(i.e. electrical) work done. When the cell reaction proceeds by an extent dξ,
the total charge transported is −nF dξ. The work done in moving a charge q
through a potential difference V is qV , so dwe = −nF Edξ. But dG = ∆r Gdξ,
so
∆r G = −nF E
The standard change of Gibbs free energy is therefore ∆r G⊖ = −nF E ⊖ , and
the equilibrium constant can be found from:
∆r G⊖ = −RT ln K
⇒
ln K =
nF E ⊖
RT
If E ⊖ is known as a function of temperature, the standard reaction entropy and
enthalpy changes may be obtained from:
∂∆r G⊖
∂E ⊖
∆r S ⊖ = −
= nF
∂T
∂T p
p
∆r H
⊖
⊖
= ∆r G + T ∆r S
⊖
#
" ∂E ⊖
⊖
−E
= nF T
∂T p