E 01 Mean Activity Final
E 01 Mean Activity Final
E 01 Mean Activity Final
mi
a
m0 1.
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).
We also know that we can write Equation 2. again with standard chemical
potential and activity
0
μ NaCl,aq μNaCl, aq RT ln a NaCl,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
mi
ai
m0
mi
i i0 RT ln
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
mi
ai i
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
i i0 RT ln mi i
We will also refer to the ionic compound simply as the "salt." With this notation
we can rewrite former Equations as,
salt 4.
0
or 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.
RT ln asalt RT ln m m
asalt m m
Geometric mean.
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,
1
2 or 2 6.
asalt m 2 2 7.
The chemical potential of salt given in terms of activity coefficient and molality
salt salt
0
RT ln m 2 2 8.
Keep in mind that this is for NaCl, but it is correct for any one-to-one ionic
compound.
Try MgCl2,
0
salt RT ln asalt 0Mg 2 2 0Cl RT ln m m2 2
so
asalt m m2 2
3
1
2
asalt m 2m 2 3 4m 3 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
aAx a By A mA 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
HA → H+ + A-
a a m m 1
Ka
aHA mHA HA mo
Knowing
2
and m+ = m- = m
a a m2 2
Ka o
aHA m mHA HA 9.
K a K c K
where
m2 2
Kc K
mo mHA HA
Theoretical calculation of γ ± .
The Debye Hückel limiting law gives the γ ± in terms of the ionic strength, I,
defined as,.
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+ or 2- ion, like Ca2+) 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:
2 2
mH+ = mCl- = 0.01 mol/kg, and zH z 1
Cl
I
1
2
0.01 1 2 0.01 1 2 0.01 mol kg -1
Notice that for a simple salt of two monovalent ions, the ionic strength is just the
concentration of the salt.
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.
lg A z z I 1 / 2
0.511 z z I 1 / 2
lg
1 b I1/ 2
Let’s try to make sense of our equation for the mean activity coefficient by
taking it apart
I 1/ 2
10 0.511 AB , where A z z ; B
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.
100.000511 0.9988
I
1
2
mK 12 mCl 1 2 mCr2 O 7 2 2
mK mKCl 2 mK 2 Cr2 O 7 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)
0.511 2 1.81 / 2
lg 0.5855
1 1.81 / 2
0.26