Electrochemistry
Ionic activity, mean activity coefficient
Activity
Activity of a molecule or ion in a solution is referred to as effective
concentration. Recalling the definition of activity from equilibrium
thermodinamics
a
mi
m0
1.
m0 is the standard concentration, (e.g.: 1 mol kg-1),
mi the molality of ith component of a solution,
γ is the concentration dependent activity coefficient. Both a and γ have no unit.
On diluting the solution, i.e.
c→0
γ→1
then a = c
activity becomes equal to the concentration. In very dilute solution interactions
are insignificant and concentration and effective concentration are the same.
Though, activity and activity coefficient have no unit their magnitude depend on
the concentration units used. E.g., concentration of the same solution can be
given in units mol·kg-1, mol·dm-3, molar fraction and so on.
Ion activity
Activities of ionic solutions serve us for calculating
accurate chemical potentials
accurate equilibrium constants
Taking into account that the activity has a greater importance in ionic solution
than that of non-electrolytes. Interaction forces are greater among ions
(Coulombic interaction).
Ions in aqueous solution.
Strong electrolytes are the ionic compounds that completely dissociate into ions
when they are dissolved.
NaCl(aq) → Na+(aq) + Cl−(aq).
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Weak electrolytes are ionic compounds that partially dissociate
The chemical potentials for the formation of ions in aqueous solution
μNaCl, aq  μ
Na  , aq
μ
Cl  , aq
2.
which also implies that standard chemical potentials
0
0
μNaCl,
aq  μ
Na  , aq
 μ0
Cl  , aq
We also know that we can write Equation 2. again with standard chemical
potential and activity
0
μNaCl, aq  μNaCl,
aq  RT ln aNaCl, aq
When working with ionic solutions we would like to be a little more specific
about the activities of the species in solution. It is customary to use units of
molality, m, for ions and compounds in aqueous solution. If the solutions were
ideal we could write for the activity of ith component
ai 
mi
m0
and the chemical potential would be written
 i   i0  RT ln
mi
mi0
3.
Two things must be said about Equation 3. First, it must be understood that there
is an implied mio dividing the mi inside the logarithm, and second, the standard
state is the solution at concentration mi = mio. Usually we set mio = 1 mol kg-1.
However, ionic solutions are far from ideal so we must correct this expression
for chemical potential for the nonidealities. As usual, we will use an activity
coefficient, γ , and write the activity as
ai   i 
mi
m0
and consistent with what we have been doing, we will set mio to 1 molal and not
write it in the equation. The form of equation indicates that activity and activity
coefficient are dimensionless, i.e. they have no unit. Thus the chemical potential
will be written
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i  i0  RT ln mi  i
The standard state for this equation is a hypothetical standard state. The
standard state is not actually realizable. We are using the so-called Henry's law
standard state in which the solution obey's Henry's law in the limit of infinite
dilution. That is,
γ i → 1 in the limit when mi → 0.
We will simplify matters by writing m+ , γ + and m− ,γ − for the molalities and
activity coefficients of the positive and negative ionic species, respectively.
We will also refer to the ionic compound simply as the "salt." With this notation
we can rewrite former Equations as,
 salt      
or
4.
0
salt
 RT ln asalt   0  RT ln m    0  RT ln m  .
In a simplified form
0
salt
 RT ln asalt   0   0  RT ln m   m 
5.
The sum of standard chemical potentials is equal to that of the salt
0
 salt
  0   0
The logarithmic term in Equation 5. on the left is identical to the logarithmic
term on the right,
RT ln asalt  RT ln m   m 
from which we conclude that
asalt  m    m 
The activity coefficients, γ + and γ − can't be measured independently because
solutions must be electrically neutral. In other words, you can't make a solution
which has just positive or just negative ions. You can't calculate the individual
activity coefficients from theory, either. However, you can measure a
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"geometric mean" activity coefficient and, within limits, you can calculate it
from theory.
Geometric mean.
Geometric mean. Numbers like x1, x2,…xn form a geometric mean which can be
given as, m  n x1  x2  ...xn
The actual form of the geometric mean depends on the number of ions produced
by the salt. Right now we will define it for NaCl and then give more examples
later. For NaCl we define,
   
2

or
     
1
2
 
6.
     which says that
It turns from the form of Equation 6. that
equal amount of “activity coefficient property” is divided to positive and negativ
ions. So,
asalt  m  m   2
But for a NaCl solution of molality, m, we have m+ = m− = m so that
asalt  m2   2
7.
The chemical potential of salt given in terms of activity coefficient and molality
0
salt  salt
 RT ln m2 2
8.
Keep in mind that this is for NaCl, but it is correct for any one-to-one ionic
compound.
Try MgCl2,
MgCl2 → Mg2+ + 2 Cl−
0
salt
 RT ln asalt  0Mg2  RT ln mMg2    20Cl   2RT ln mCl   
0
salt
 RT ln asalt  0Mg2  20Cl   RT ln m    m2  2
so
asalt  m m2   2  
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But for MgCl2 at molality, m, we know that m+ = m and m− = 2m. Further, we
define the geometric mean activity coefficient by,
 


1
2
 3
Then
asalt  m  2m2   3  4m3 3
Other ionic compounds are done in a similar manner. After some practice you
can probably figure out the expression for asalt just by looking at the compound.
Until then, or when in doubt, go back to the expressions for chemical potentials
as we have done here. (For practice you might want to try Al2(SO4)3.).
General formula
Absolute activities of cations and anions can not be determined experimentally.
The definition of the mean activity coefficient depends on the number of ions
into which a molecule dissociates when it is dissolved.
In the laboratory it is impossible to study solutions which only contain one kind
of ion. Instead, solutions will have at least one positive and one negative type of
ion. For the generic electrolytic compound dissolving in water:
A x By  xA z   yBz 
we will find terms such as a Ax a Bx in equilibrium constant, Ka and in reaction
quotient, Q values. Expressing such products in terms of molalities and activity
coefficients, we first note that for a solution containing A x By component of
concentration m, the ion concentrations are [A] = xm and [B] = ym. The activity
product then becomes:



aAx aBy   AmA x  BmB y   A xm x  B ym y  x x y y m x y   xA By

We can define a mean activity coefficient as:     xA  yB


1/x y 
and this allows us to rewrite equation as: aAx a By  x x y y m  x  y  x  y 
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
Relationship between Ka and Km
The equilibrium constant given in terms of activities is the thermodynamic
equilibrium constant.
Use an example of dissociation equilibrium of a weak acid to show this
relationship;
HA → H+ + Athe equilibrium constant of the reaction, (introducing the standard concentration,
mo again)
Ka 
a a m  m  1


aHA mHA   HA mo
Knowing
       2
and m+ = m- = m
a a
m 2   2
Ka 
 o
aHA m  mHA   HA
Ka  Kc  K
where
m2
Kc  o
m  mHA
 2
K 
 HA
The equilibrium constants given in terms of molality and activity coefficient.
Ka = f(T,p) but independent of molalities.
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9.
Debye-Hückel Limiting Law.
Theoretical calculation of γ ± .
The Debye Hückel limiting law gives the γ ± in terms of the ionic strength, I,
defined as,.
I
1
mi zi2

2 i
1.
where zi is the charge on ion i, and mi is the molality of ion i. The ionic strength
of a solution is a measure of the amount of ions present. The ionic strength is a
measure of the total concentration of charge in the solution. A divalent ion (a
2+
2+ or 2- ion, like Ca ) does more to make the solution ionic than a monovalent
+
ion (e.g., Na ). The ionic strength, emphasizes the charges of ions because the
charge numbers occur as their squares.
Examples
a. What is the value of ionic strength of HCl solution with molality 0.010:
mH+ = mCl- = 0.01 mol/kg, and z 2   z 2
I

H
Cl
1

1
0.01  12  0.01   12  0.01 mol  kg -1
2
Notice that for a simple salt of two monovalent ions, the ionic strength is just the
concentration of the salt.
b. the ionic strength of 0.10 molal Na2SO4
c.What molality of CuSO4 has the same ionic strength as a 1 mol/kg molality
solution of KCl?
We must include all ions in the solution. Notice that it includes contributions
from both the number of ions in the solution and the charges on the individual
ions.
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Mean activity coefficient determination via calculation: Debye-Hückel
theory
The chemical potential or activity of ions cannot be determined on a purely
thermodynamic basis.
This is due to the fact that the effects of an ion cannot be separated from the
effects of the accompanying counter-ion, or in other terms, the electrochemical
potential of the ion cannot be separated into the chemical and the electrical
component. Such a separation must necessarily be based on a nonthermodynamic convention.
The mean activity coefficient can be calculated at very low concentrations by
the Debye-Hückel Limiting Law
lg     A  z  z  I 1 / 2
where A = 0.509 / (mol kg-1)1/2 for an aqueous solution at 25 oC, in general, A
depends on the relative permittivity of solvent and the temperature.
The log10 of mean activity coefficient also depends on the product of
z  z
the absolute value of cation and anion charge number.
When the ionic strength of the solution is too high for the limiting law to be
valid, it is found that the activity coefficient may be estimated from the
extended Debye-Hückel law.
lg   
 0.511 z  z I 1 / 2
1  b  I1/ 2
Where b is a measure of distance between ions.
1/2
In the limit of small concentration I << 1, and in the denominator of Eq. the
1/2
term b·I
again.
can be neglected, therefore we receive Debye-Hückel Limiting Law
Let’s try to make sense of our equation for the mean activity coefficient by
taking it apart
   100.511 AB , where A  z  z ; B 
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I 1/ 2
1  I 1/ 2
A
Greater charges produce more negative the argument for the exponential,
  is farther away from 1, its ideal value.
B
The term B varies between 0 and 1. When I is small B ~ 0, and   ~ 1.
Example: for A = 1, B = 0.001, the argument for the exponential is -0.000511,
and
   100.000511  0.9988
d.Calculating ionic strength
Calculate I for a solution that is 0.3 molal in KCl and 0.5 molal in K2Cr2O7.

1
 mK   12  mCl   12  mCr2O 7   2 2
2
mK   mKCl  2  mK 2Cr2 O 7
I

mol/kg
I  0.5  1.3  0.3  2   1.8
  (KCl)
 0.511 1 1.81 / 2
lg   
 0.2928
1  1.81 / 2
   0.5096,
  (K2Cr2O7)
lg   
 0.511 2 1.81 / 2
   0.26
1  1.81 / 2
 0.5855
Therefore ionic activities are clearly dependent on the overall composition
of the solution.
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