Chemistry 232

Electrolyte Solutions

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

Under the assumption a(H+) = a(Cl-)

We obtaina´(H+) = (a(HCl))1/2 = a(HCl)

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.

How would we calculate the pH of a buffer solution?

HCOOHm

HCOOa HaKa

HCOOHmHCOOa Ha

KpK aa loglog

HCOOHmHCOOaHapKa logloglog

note pH = -log a(H+)

Define pKa = -log (Ka )

The Buffer Equation

Substituting and rearranging

HCOOHmHCOOa

pHpKa

)(log

HCOOHmHCOOa

pKpH a

)(log

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

622sp 10x01Fa BaaK .

10

sp 10x81Cla AgaK

1AgCla note

AgClaCla Aga

K

.

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

,

Temperature and Pressure Dependence of Ko

As a function of temperature

2r

o

RTH

dTKd

ln

As a function of pressure

RTV

dPKd r

T

o

ln