School of Chemical & Biological Engineering, Konkuk University

Lecture 19

• The activities of regular solutions

• The activities of ions in solution

Ch. 5 Simple Mixtures

• Now consider the activities of regular solutions (H^E  0 but S^E = 0).

• For a regular solution, the Gibbs energy of mixing is:





mixG  nRT x x  x x  x x

 ln ln

where  is a dimensionless parameter which is a measure of the energy of A-B interactions relative to those of the A-A and B- B interactions.

B A mix

E H n RTx x

H    

• If  < 0, mixing is exothermic (A-B interactions more favorable).

• If  > 0, mixing is endothermic (A-B interactions less favorable).

 0 SE

 

 

   





1

ln ln

ln ln

2 2

B A B

A B

B A

A

B A

B A

B A B

B A

A

B A B

B A

A mix

x x x

x x

x x

x nRT

x x

x x

x x x

x x

x nRT

x x x

x x

x nRT G







































Set and , which are called Margules equations.

ln_B  x2_A

ln _A  x_B2

 





mix

a x

a x

nRT

x x

x x

x x

nRT G

ln ln

ln ln

ln ln













  

• For a regular solution, the Gibbs energy of mixing is expressed with activities in place of mole fractions in the Gibbs energy of mixing for ideal solutions.





mixG  nRT x ln a  x ln a





mixG  nRT x ln x  x ln x



Ideal solutions Regular solutions

• Note that, According to the Margules equations, for dilute solutions, _A  1 as x_B  0 and _B  1 as x_A  0.

ln_B  x2_A

ln_A  x_B2

• Using the Margules equations, the activity can be expressed as:

 ²

2 1 _A

B x

A x

A A

A

A x x e x e

a   ^  ^{ }

Because the activity of solvent A is: ^*

A A

A p

a  p

 





A x e p

p  ^{  A}

• When =0, the straight line corresponds to the Raoult’s law. ^*

A A

A x p

p 

• When  > 0 (endothermic mixing,

unfavorable A-B interactions), higher vapor pressure than ideal.

• When  < 0 (exothermic mixing, favorable A-B interactions), lower vapor pressure

than ideal.

• As x_A  1, all the curves approach linearity and coincide with the Raoult’s law.

• When x_A << 1,

Endothermic

Exothermic

 





A x e p

p  ^{  A} pA  xAe^ p^*A

The form of Henry’s law ( ) pA  xAKA

 





A x e p

p  ^{  A}

• Interactions between ions are so strong that the

approximation of replacing activities by molalities is valid only in very diluted solutions (total ion concentration < 10^-3 mol/kg).

• If the chemical potential of a univalent cation (M⁺) and a

univalent anion (X^-) are denoted and , respectively, the molar Gibbs energy of the ions in the electrically neutral solution is the sum of these partial molar quantity ( ).

• For a ideal ionic solution of MX,

 _

ideal ideal

ideal

Gm  _  _

• For a real ionic solution of MX,



 

   Gm











     

     RT ln RT ln

G_m ^{ideal ideal}



 

G RT ln  G_{m m}^ideal

• All the deviations from ideality are contained in the last term.



 

G RT ln  G_{m m}^ideal

• There is no experimental way to separate the product ₊_- into contributions from the cations and the anions.

• The best way is to assign responsibility for the non-ideality equally to both kinds of ions.

• For a 1,1-electrolyte (MX), we introduce the mean activity coefficient ( ) as the geometric mean of the individual

coefficients:







   



Therefore, the individual chemical potentials of the ions is:





    

 ^ideal RT ln _  _^ideal  RT ln_

 



 G_m G_m^ideal 2RT ln

• For a compound M_pX_q which dissolves to give a solution of p cations and q anions, the molar Gibbs energy of the ions is:

   





























































ln ln

ln ln

qRT pRT

q p

RT q

RT p

q p

G

ideal ideal

ideal ideal

m

Introducing the mean activity coefficient (geometric mean),

q

p p q





   



and writing the chemical potential of each ion as:

 

  

_{i i}^ideal RT ln

 





























ln

ln

ln RT q

p G

q p

G qRT

pRT G

G

ideal m

ideal ideal

ideal m ideal

m

m 

Then,







G p q RT ln G_{m m}^ideal

Here, both types of ion share equal responsibility for the non- ideality.

• For a compound M_pX_q,

q

p p q





   



ideal ideal

ideal

m p q

G  _  _

• Consider an ionic solution which shows the nonideal behavior.

• Oppositely charged ions attract one another.

• As a result, anions are more likely to be found near cations in solution, and vice versa.

• Overall, the solution is electrically neutral, but near any given ions there is an excess of counter ions.

• A time-averaged, spherical haze around the central ion, in which counter ions outnumber ions of the same charge as the central ion.  called ionic atmosphere.

• The chemical potential of any given central ion is lowered as a result of its electrostatic interaction with its ionic atmosphere.

• The lowering of Gibbs energy appears as the difference between and the ideal value of the solute.

• In 1923, Debye and Hückel proposed a theory (called Debye- Hü ckel theory) to explain the nonideal behavior of ionic

solutions.

• The theory assumes that electrolytes in solution are fully

dissociated, and that the nonideal behavior arises because of the electrostatic interactions (Coulombic interaction) between ions.

ideal

Gm

Gm







G p q RT ln G_{m m}^ideal

• According to the theory, the activity coefficient can be calculated from the Debye-Hückel limiting law at very low concentrations ( < 1 mmol/kg):

I A z z_{ }

  

 ln

where A = 0.509 for an aqueous solution at 25 ^oC, z_i is the

charge number of an ion i, I is the dimensionless ionic strength of the solution, and b_i is its molality.





i

o i i b z b

I ²

2 1

• Ionic strength is a function expressing the effect of the charge of the ions in a solution.

 

 

o o

o

b b b

b

b I b

2 15 12 18

) 2 3

3 2 2

1 _{2 2}

 











) 

( 3

) (

2 ^{3 2}

3

2X M aq X aq

M  ^  ^

ex) The ionic strength of an M₂X₃ solution of molality b (not b_i).

(See Table 5.4)

• Although there are marked deviations for moderate ionic strengths, the limiting slopes (as I  0) are in good agreement with the Debye-Hückel theory.

• Therefore, the limiting law can be used for extrapolating data to very low molalities.

• Nevertheless, the approximations are valid only at very low concentrations (< 1 mmol/kg).

• When ionic strength of the solution is too high for the limiting law to be valid, the activity

coefficient may be estimated from the extended Debye-Hückel law:

I CI B

I A z

z 

 

 ^{ }

 1

ln ^



i

o i

i b

z b

I ²

2 1

where B and C are dimensionless constants (adjustable empirical parameters).

• The B can be interpreted as a measure of the closest approach of the ions.

• The extended law accounts for some activity coefficients over a moderate range of dilute solutions (up to ~ 0.1 mol/kg)

• Nevertheless it remains very poor near 1 mol/kg.

• Current theories of activity coefficients for ionic solutes take an indirect route.

1. Set up a theory of the dependence of the activity coefficient of the solvent on the concentration of the solute.

2. Use the Gibbs-Duhem equation to estimate the activity coefficient of the solute.

• This indirect method is reasonably reliable for solutions with molalities greater than ~ 0.1 mol/kg.

• Reading: page 174 ~ 184

• Problem set (Ch. 5): Discussion 5.4

5.5a, 5.7a, 5.15a, 5.19a Due dates: 3215 (May 20)

3996 (May 21)

• The 2^nd exam: May 23 (Fri), 19:00, B566, Ch. 3~5