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Thermodynamics of Ions in Solutions
Electrolyte solutions – deviations from ideal behaviour occur at molalities as low as 0.01 mole/kg.
How do we obtain thermodynamic properties of ionic species in solution?
mRT ln
Thermodynamics (II)
For the H+(aq) ion, we define fH = 0 kJ/mole at all temperatures S = 0 J/(K mole) at all temperatures fG = 0 kJ/mole at all temperatures
Activities in Electrolyte Solutions
For the following discussion Solvent “s” Cation “+” Anion “-“
Consider 1 mole of an electrolyte dissociating into + cations and - anions
G = ns s + n = ns s + n+ + + n- -
Note – since = + + - = + + + - -
The Mean Ionic Chemical Potential
We define
= / We now proceed to define the
activities
= + RT ln a
+ = + + RT ln a+
- = - + RT ln a-
= + RT ln a
The Relationship Between a and a
Since = /
= + RT ln a = ( + RT ln a)
Since = / This gives us the relationship
between the electrolyte activity and the mean activity
(a)= a
The Relationship Between a , a- and a+
We note that = + + + - -
and = / This gives us the following relationship
( + RT ln a) = + (+ + RT ln a+) +
- ( - + RT ln a-)
Since = + + + - - (a) = (a+)+ (a-)-
Activities in Electrolyte Solutions
The activities of various components in an electrolyte solution are defined as follows
a+ = + m+
a- = - m-
a+ = + m+
As with the activities () = (+)+ (-)-
(m) = (m+)+ (m-)-
The Chemical Potential Expression
This can be factored into two parts
lnlnln
lnlnln
lnln
RTRTmRT
RTmRTmRT
mRTaRT
lnlnln RTRTmRT
The ideal partDeviations from ideal behaviour
KCl
CaCl2
H2SO4
HClLaCl3
Activity Coefficients As a Function of Molality
Data obtained from Glasstone et al., Introduction to Electrochemistry, Van Nostrand (1942). CRC Handbook of Chemistry and Physics, 63rd ed.; R.C. Weast Ed.; CRC
Press, Boca Raton, Fl (1982).
Determination of Activity Coefficients in Solution
Two ways Use the Gibbs-Duhem equation and for
the solvent to estimate for the solute. Determination of osmotic coefficients from
colligative properties vapour pressure measurements
Js
ss
Js
s
mRTMmMa
*ln
Estimates of Activity Coefficients in Electrolyte Solutions
A few have been proposed to allow the theoretical estimation of the mean activity coefficients of an electrolyte.
Each has a limited range of applicability.
21
cIzz 5100 .log
This is valid in the up to a concentration of 0.010 molal!
2 ½ m j j
j
I Ionic strength z m
The Debye Hűckel Limiting Law
Z+ = charge of cation; z- = charge of anion
Debye Hűckel Extended Law
This equation can reliably estimate the activity coefficients up to a concentration of 0.10 mole/kg.
1
2
1
2
0 510
1
. log m
m
z z I
B I
B = 1.00 (kg/mole)1/2
The Davies Equation
This equation can reliably estimate the activity coefficients up to a concentration of 1.00 mole/kg.
1
2
1
2
0 5101
log . mm
m
Iz z kI
I
k = 0.30 (kg/mole)
The Equilibrium Constant
For a nonideal system, the nonstandard Gibbs energy of reaction is written
J
eqJrrJaRTGG ,
The Equilibrium Condition
If we apply the equilibrium conditions to the above equation
J
eqJrJaRTG0 ,
J
oJJ
JeqJr
JaRTG ,
J
eqJo JaK
,
The Autoionization of Water
Water autoionizes (self-dissociates) to a small extent
2H2O(l) ⇌ H3O+(aq) + OH-(aq)
H2O(l) ⇌ H+(aq) + OH-(aq) These are both equivalent
definitions of the autoionization reaction.
Water is amphoteric.
The Autoionization Equilibrium
From the equilibrium chapter
22
3
2
OHaOHa OHa
or
OHaOHa( Ha
K =
)()()(
)())(
But we know a(H2O) is 1.00!
The Defination of Kw
Kw = a(H+) a(OH-)
Ion product constant for water, Kw, is the product of the activities of the H+ and OH- ions in pure water at a
temperature of 298.15 K
Kw = a(H+) a(OH-) = 1.0x10-14 at 298.2 K
The pH scale
Attributed to Sørenson in 1909
We should define the pH of the solution in terms of the hydrogen ion activity in solution
pH -log a(H+)
Single ion activities and activity coefficients can’t be measured
Determination of pH
What are we really measuring when we measure the pH?
pH -log a(H+)
a (H+) is the best approximation to the hydrogen ion activity in solution.
How do we measure a(H+)?
For the dissociation of HCl in water
HCl (aq) Cl-(aq) + H+(aq) We measure the mean activity of
the acid a(HCl) = a(H+) a(Cl-)
a(H+) a(Cl-) = (a(HCl))2
Equilibria in Aqueous Solutions of Weak Acids/ Weak Bases
By definition, a weak acid or a weak base does not ionize completely in water ( <<100%).
How would we calculate the pH of a solution of a weak acid or a weak base in water?
Equilibria of Weak Acids in Water: The Ka Value
Define the acid dissociation constant Ka For a general weak acid reaction
HA (aq) ⇌ H+ (aq) + A- (aq)
HAa
Aa HaKa
Equilibria of Weak Acids in Water
For the dissolution of HF(aq) in water.HF (aq) H+ (aq) + F- (aq)
4a 10x17
HFaFa Ha
K
.
The Nonelectrolyte Activity
HF (aq) ⇌ H+ (aq) + F- (aq) The undissociated HF is a nonelectrolyte
a(HF) = (HF) m[HF] m[HF] (HF) 1
4a 10x17
HFmFa Ha
K
.
Equilibria of Weak Bases in Water
Calculate the percentage dissociation of a weak base in water (and the pH of the solutions)
CH3NH2 (aq) + H2O ⇌ CH3NH3+(aq) +
OH- (aq)
The Kb Value
Define the base dissociation constant Kb
For a general weak base reaction with water
B (aq) + H2O (aq) ⇌ B+ (aq) + OH- (aq)
Ba
OHa BaKb
BmOHaBa
Kb
Calculating the pH of Solutions of Strong Acids
For the dissolution of HCl, HI, or any of the other seven strong acids in water
HCl (aq) H+ (aq) + Cl- (aq) The pH of these solutions can be
estimated from the molality and the mean activity coefficient of the dissolved acid
pH = -log ( (acid) m[H+])
Calculating the pH of Solution of Strong Bases
For the dissolution of NaOH, Ba(OH)2, or any of the other strong bases in water
NaOH (aq) Na+ (aq) + OH- (aq)
pOH = -log ( (base) m[OH-])
Calculating the pH of a Weak Acid Solution
The pH of a weak acid solution is obtained via an iterative procedure.
We begin by making the assumption that the mean activity coefficient of the dissociated acid is 1.00.
We ‘correct’ the value of (H+) by calculating the mean activity coefficient of the dissociated acid.
Repeat the procedure until (H+) converges.
The Definition of a Buffer
Buffer a reasonably concentrated solution of a weak acid and its conjugate base
Buffers resist pH changes when an additional amount of strong acid or strong base is added to the solutions.
The Generalized Buffer Equation
The pH of the solution determined by the ratio of the weak acid to the conjugate base.
Henderson-Hasselbalch equation often used for buffer calculations!
acid weakm
base conjapKpH a
).(log
Buffer CH3COONa (aq) and CH3COOH (aq))
CH3COOH (aq) ⇄ CH3COO- (aq) + H+ (aq)
The Equilibrium Data Table
n(CH3COOH) n(H+) n(CH3COO-)
Start A 0 B
Change -eq + eq +eq
m (A-eq) (eq) (B+ eq)
The pH of the solution will be almost entirely due to the original molalities of acid and base!!
][log
AmBm
pKpH a
Solubility Equilibria
Examine the following systemsAgCl (s) ⇌ Ag+ (aq) + Cl- (aq)
BaF2 (s) ⇌ Ba2+ (aq) + 2 F- (aq) Using the principles of chemical
equilibrium, we write the equilibrium constant expressions as follows
The Common Ion Effect
What about the solubility of AgCl in solution containing NaCl (aq)?
AgCl (s) ⇌ Ag+ (aq) + Cl- (aq)NaCl (aq) Na+ (aq) + Cl- (aq)AgCl (s) ⇌ Ag+ (aq) + Cl- (aq)
Equilibrium is displaced to the left by LeChatelier’s principle (an example of the common ion effect).
Solubility in the Presence of an Inert Electrolyte
What happens when we try to dissolve a solid like AgCl in solutions of an inert electrolyte (e.g., KNO3
(aq))? We must now take into account of
the effect of the ionic strength on the mean activity coefficient!
The Salting-In Effect
AgCl (s) ⇌ Ag+ (aq) + Cl- (aq). Designate the solubility of the salt in
the absence of the inert electrolyte as so = m(Ag+) = m(Cl-) at equilibrium.
2o2
102
sp
s
10x81ClmAgm
Cla AgaK
.
For a dilute solution
2osp sK1 Designate s as the solubility of the salt in the
presence of varying concentrations of inert electrolyte.
o2sp
sssK
Reaction Equilibria in Nonideal Gaseous Systems
For a nonideal system gaseous, the nonstandard Gibbs energy of reaction is written
J
eqJrrJfRTGG ,
JoeqjJ
JoeqJo
JJ
P
Px
P
fK
,,
The Equilibrium Condition
Calculate the equilibrium composition from the fugacity coefficients from compression factor data
Jo
eqJ
JJ
o J
J P
PxK
,