Chapter 3.9
Chapter 3.9
Chapter 3.9
3:9:3
where zi is the charge number (signed) of the ion and F is the Faraday constant. For reactions in a single conducting phase, fi is constant everywhere in the phase and thus shows
no effect on a chemical equilibrium. The chemical potentials of ions defined by
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Tetsuji Hirato
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Since the composition of the solution is described commonly in terms of the molality
scale, usually the hypothetical one-molal solution of the ion is chosen as the standard
state. The standard state is an imaginary solution with molality m01 mol/kg in which
the ions behave ideally. The activity is related to the molality, mi, by
ai gi
mi
m0
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where gi is the molal activity coefficient. The activity and activity coefficient have no
dimensions. As the molality approaches zero, obeying Henrys law, gi tends to 1:
gi ! 1 and ai ! mi ! 0
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mideal
i
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yi ! 1 and ai ! ci as ci ! 0:
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The molarity can be related to the molality by measuring the density of the solution.
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the anion X have charges z1e and z2e, respectively, the following equation is obtained
for electrical neutrality:
v1 z1 v2 z2 0
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The molar Gibbs energy for a real solution of Mn1 Xn2 is as follows:
mMn1 Xn2 n1 mMz1 n2 mXz2
n1 m0Mz1 n1 RT ln mMz1 n1 RT ln gMz1 n2 m0Xz2
n2 RT ln mXz2 n2 RT ln gXz2
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More simply, taking A for the formula of the electrolyte and denoting the cation by 1
and the anion by 2:
mA m0A RT ln mn11 mn22 RT ln gn11 gn22
m0A
n1 m0Mz1
n2 m0Xz2
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The mean molarity and mean activity coefficient are defined as follows:
m mn11 mn22 1=n
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Thermodynamic data of electrolytes are often reported and tabulated as the mean
activity coefficient.
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change appears as the difference between the chemical potential and the ideal value of the
solute, and hence can be identified as RT ln g.
In very dilute solutions, the activity coefficient can be calculated from the Debye
Huckel limiting law:
logg jz z jAI 1=2
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where A 0.509 for an aqueous solution at 298 K and I is the ionic strength of the
solution:
I
1X 2
mi zi
2 i
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The ionic strength is used widely where ionic solutions are discussed.
For more concentrated solutions, where the ionic strength of the solution is too high
(I > 0.2) for the limiting law to be valid, the activity coefficient may be estimated from the
extended DebyeHuckel law:
logg
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where B and C are other dimensionless constants. Although B can be related to effective
ion radius, it is better considered as an adjustable empirical parameter.
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The reaction Gibbs energy, DGr, is expressed by using chemical potentials of the
species, A, B, C, and D:
DGr cmC dmD amA bmB
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acC adD
ac ad
DG0 RT ln Ca D
a b
aA aB
aA abB
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The value of DG0 can be calculated by using Gibbs energy of formation, DG0f , instead
of chemical potentials, as in the following equation:
DG0 cDGf0 C dDGf0 D aDGf0 A bDGf0 B
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Here, DGr 0 35:9kJ=mol and DG0f (HCl,g) 95.3 kJ/mol; thus, the standard
free energy of formation of Cl(aq) can be calculated as follows:
DGf0 Cl ,aq 131:2kJ=mol
In the same way, by taking the standard enthalpy of formation and entropy of H(aq)
as zero:
DHf0 H ,aq 0
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DS H ,aq 0
0
the standard enthalpy of formation and entropy of individual ionic species can
be obtained.
The data of the standard free energy and enthalpy of formation and entropy of individual ionic species were tabulated by Latimer [3].
The standard free energy of formation at temperature T, DG0T, is given as follows:
DGT0 DH298
T
298
T
298
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DCp0
T
dT
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where DH0, DS0, and DC0p are changes of enthalpy, entropy and specific heat at 1 atm.,
respectively. Knowledge is limited about the specific heat of ionic species in aqueous
solution, so we have to calculate the value of DC0p approximately. First, DC0p is approximated to be zero in the considered temperature range.
0
0
0
0
TDS298
DG298
DS298
T 298
DGT0 DH298
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If DH0T is measured at a temperature other than 298 K, the value of DC0p can
be calculated.
As the third approximation, the CrissCobble correspondence principle [4] is commonly applied for simple cations and anions, oxyanions (XOz
n ), and acid oxyanions
z
(HXOn ). The relationship between ionic entropies at 298 K and those at elevated
temperatures can be expressed as follows:
:
ST: aT bT S298
S.T
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S.298
where
and
are absolute entropies at the temperature and at 298 K, respectively,
and aT and bT are constants dependent on the temperature and type of ion. The absolute
entropy is related to the conventional entropy (based on S0298 0 for H(aq)) as in the
following equation:
0
S298
i S298
i zS298
H
S0298(i)
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Here, HA(aq) is an acid and H2O(l) is a base, and A(aq) is a conjugate base and
H3O(aq) is a conjugate acid:
Baq H2 O HB aq OH aq
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Here, B(aq) is a base and H2O(l) is an acid, HB(aq) is a conjugate acid, and OH(aq)
is a conjugate base.
Although free protons do not exist in aqueous solutions, H(aq) or H is used as a
representation of the state of the proton in this section for simplicity. Accordingly, the
equilibrium constants for reactions (3.9.34) and (3.9.35) are expressed as follows:
aH aA
aHA
aHB aOH
Kb
aB
Ka
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3:9:37
3:9:38
KW aH aOH
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Since acidbase reaction is a proton transfer reaction, pH is one of the most important
factors to describe reactions in aqueous solutions, where
pH logaH
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aA
aHA
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aA
aHA
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where L represents a ligand. Here the metal ion, Mn(aq), is a Lewis acid and the ligand
is a Lewis base. The equilibrium constants of complex formation reactions have been
tabulated by Martell and Smith [5]. The standard free energy of formation for a complex
species, MLn
m (aq) can be calculated by using its equilibrium constant.
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where F is the Faraday constant, and E is potential, the potential, E, is the potential based
on the hydrogen standard electrode as described later.
The potential can be obtained on the basis of Equations (3.9.21) and (3.9.22):
E
DG0 RT aRed
ln
zF
zF
aOx
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DG0
zF
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RT aRed
ln
zF
aOx
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where
E0
DGf0 H2 2DGf0 H
0
2F
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where
DGf0 Zn DGf0 Zn2
0:76V
2F
DGf0 Cu DGf0 Cu2
0:34V
E0 Cu2 =Cu
2F
E0 Zn2 =Zn
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and the electromotive force (emf) of the total reaction can be obtained as follows:
emf ECu2 =Cu EZn2 =Zn 1:10
RT aZn2
ln
zF aCu2
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and when z 6 0, one can obtain the potential as a function of pH and activities of species
in aqueous solutions, on the basis of the above discussion.
DG0 2:303RT
aRs
npH
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E
log
zF
aOm
zF
When n 0, the reaction is a simple redox reaction.
At 298 K,
DG0 0:0591
aRs
npH
E
log
zF
aOm
z
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When z 0, the reaction is an acidbase reaction and we can obtain the following
relationship:
aRs
0
npH
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DG 2:303RT log
aOm
As DG0 2.303RT log K, the following equation is obtained:
1
aRs 1
logK
pH log
aOm n
n
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3:9:67
O2 g 4H aq 4e 2H2 O
E 1:23 0:059pH 0:015logpO2
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To obtain Equation (3.9.69), the standard free energy changes of formation of H2O
shown in Table 3.9.1 are used. At potentials higher than Equation (3.9.67) and lower
than Equation (3.9.69), water is stable.
The potentialpH diagram for the ZnH2O system is constructed by using the standard free energy changes of formation shown in Table 3.9.1, as shown in Figure 3.9.1.
The numbered lines in Figure 3.9.1 represent the following reactions:
1,2: Zn2 2e Zn E 0.761 0.0295 log a(Zn2)
3,4: Zn2 2H2O Zn(OH)2 2H, pH 5.86 0.5 log a(Zn2)
5: ZnOH2 HZnO2 H , pH 15:7logaHZnO2
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DG0f (kJ/mol)
Zn
Zn
aq
147.03
Zn(OH)2
553.58
HZnO
2
aq
464.0
ZnO2
2
aq
389.2
aq
237.2
H2 O
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REFERENCES
[1] E.A. Guggenheim, J. Phys. Chem. 33 (1929) 842, 34, 1540(1930).
[2] P. Atkins, J. de Paula, Atkins Physical Chemistry, eighth ed., Oxford University Press, Oxford, 2006,
pp. 163165.
[3] W.M. Latimer, The Oxidation States of the Elements and Their Potentials in Aqueous Solutions, second
ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1952.
[4] J.W. Cobble, et al., J. Am. Chem. Soc. 86 (1964) 5385, 5390, 5396.
[5] A.E. Martell, R.M. Smith, Critical Stability Constants, Plenum Press, New York, NY, 1974.
[6] M. Pourbaix, Atlas of Electrochemical Equilibria in Aqueous Solutions, Pergamon Press, Oxford, 1966.