School of Chemical & Biological Engineering, Konkuk University

Lecture 18

• The solvent activity

• The solute activity

Ch. 5 Simple Mixtures

where the fugacity (f) is an effective pressure of the real gas and  is the dimensionless fugacity coefficient .

• Now we consider how to take into account deviations of real solutions from ideal behavior.

• In chapter 3, the imperfections of real gases are taken into account by introducing a quantity called fugacity.

• Here we introduce the concept of activity to express the real solutions.

• For a perfect gas, _{m m}^o _o

p RT p

G p

G ( )   ln

• For real gases,

o o

m

m p

RT f G

p

G ( )   ln f p

Chapter 3

• The general form of the chemical potential of a real or ideal solvent is given by:

 

 

A A A

A p

RT p l

l   



where p_A* is the vapor pressure of pure A and p_A is the vapor pressure of A when it is a component of a solution.

• For an ideal solution (of which the solvent and solute obey the Raoult’s law),

 

 

A l  ^* l  RT ln x

 For the ideal solvent:

* A A

A p

x  p

 Raoult’s law

• For a real solution (of which the solvent and solute does not obey the Raoult’s law),

 

 

A l  ^* l  RT ln a For the real solvent: 

where the quantity a_A is the activity, a kind of effective mole fraction of the real solvent A.

By comparing two equations,

  ^*  ln _*

A A A

A p

RT p l

l  



* A A

A p

a  p

The activity can be determined by measuring p_A and p_A*.

• Because all solvents obey Raoult’s law (p_A/p_A* = x_A), when x_B0 (that is, x_A1),

1 as



 _{A A}

A x x

a

where  is the activity coefficient.

• By definition, at all T and p.

1 as



 _{A A}

A x x

a

A A

A x

a  

1 as

1 

 _A

A x



• The chemical potential of the solvent is:

 

 

A l  l RT x RT 

  ^*  ln  ln

 

 

 

A l  l RT a  l RT x RT 

  ^*  ln  ^*  ln  ln Ideal solvent

Correction for real solvent and

• The solute in an ideal-dilute solution (the solute and solvent obeying the Henry’s law and Raoult’s law, respectively),

• The chemical potential of B in the ideal-dilute solution,

B B

B K x

p 

     

 

B B B

B B B B

B B B

B

x p RT

RT K l

p x RT K

p l RT p

l l

ln ln

ln ln

*

*

*

*

*

*























The 2^nd term may be combined with the 1^st to give a new standard chemical potential ( ) of B:

 

 

B B B

o

B p

RT K l

l   



 

o

B

  ^*  ln _*

B B B

B p

RT p l

l   



 

 

B l   l  RT ln x



 

 

B B B

o

B p

RT K l

l   



• For ideal-dilute solutes,

If the solution is ideal, (the solute also obeys the Raoult’s law, K_B = p_B* and . _B^o

 

 

• The standard state ( ) can be regarded as one in which the pure solute is giving

rise to a vapor pressure of magnitude of K_B.

• The standard state is a hypothetical state in which the solute is pure, but behaving as though it still obeyed Henry’s law (the

extrapolation of behavior at lower concentrations).

o

 

 

B B B

o

B p

RT K l

l   



• Now we permit the solute to deviate from ideal-dilute behavior (Henry’s law .

• For real solutes, we introduce a_B in place of x_B,

 

 

B l   l  RT ln a



 

 

B B B

o

B p

RT K l

l   



The standard state remains unchanged in this equation, and all the deviations from ideality are captured in the activity a_B. The activity of solute B is: ^{cf) *}

A A

A p

a  p

B B

B K

a  p

As for the solvent, the activity coefficient is introduced as:





B B

B K x

p 

and at all T and p.

• Now all the deviations from ideality (Henry’s law) are captured in the activity coefficient (_B).

• Because the solute obeys Henry’s law as its concentration goes to zero,

 

 

B B B

o

B p

RT K l

l   



   

 

 

 

B B B

B B

B B B

B B

B B B

B B

o B B

p RT RT p

l

p RT x RT K

l

RT x

p RT RT K

l

RT x

RT l

l



















ln ln

ln ln

ln ln

ln

ln ln

*

*

*

*

*

*



























0 as



 x x 0 a

as

1 

 x



 

 

B l   l  RT lna



• The selection of a standard state is entirely arbitrary.

• In chemistry, compositions are often expressed as molalities (b) in place of mole fractions.

 

 

B l   l  RT lnb b



Note that here has a different value from the standard values introduced earlier.

 

o

B

• According to this definition, the chemical potential of the solute (B) has its standard value when the molality (b) is equal to 1 mol/kg ( )

• Note that as b_B  0, _B  ; that is, as the solution becomes diluted, so the solute becomes increasingly stabilized.  difficult to remove the last trace of a solute from a solution.

 

o

B

bo

• Now, as before, we introduce deviations from ideality by

introducing a dimensionless activity (a_B) and its coefficient (_B).

0 as

1

where  

 _B ^B_{o B B}

B b

b

a  b  at all T and p.

 

 

B l   l  RT ln a



• The standard state remains unchanged in this stage, and all deviations from ideality are captured in the activity coefficient (_B).

 

 

B l  l RT b b RT 

   ln  ln

• In dilute solutions, . Therefore ⁿ_B ^n_A

A B B

B n

n n

x  n 

A m A

B

m A

m A

B B

B n

M n

M M n

b n ^,

, ,

kg kg 1

1  

 ^{B A m}

A B B

B b M

n n n

x  n   _,



m A B

B b M

x  _,

• Because the Henry’s law is obeyed in dilute solutions,

 

 

B l   l  RT ln x



 

 

B B B

o

B p

RT K l

l   



   

 

  ^{o B}

o

m A o o o B

B

m A B o

B B

RT b M

b RT l

M b b

RT b l

M b RT l

l

ln ln

ln

ln

, ,























 

 



New standard state

• The conventional standard state of hydrogen ions (unit activity, corresponding to pH = 0, ) is not appropriate to normal biological conditions.

• In biochemistry, it is common to adopt the biological standard state, in which pH = 7 (an activity of 10^-7, neutral solution).

• The corresponding standard thermodynamic functions are denoted as , , , .

• The relation between the thermodynamic and biological standard values of the chemical potential of hydrogen ions:

 

 a_{H O}

3

log pH

G H^ ^ _S^





ln   



   

 _H^o RT a_{H H}^o RT

H  



when pH = 7, _H^{ } _H^{o  7RT ln10}

• At 298 K, 7RT ln10 ~ 40 kJ/mol

• Reading: page 162 ~ 169

• Problem set (Ch. 4): Discussion 4.4 4.2a, 4.8a, 4.10a Due dates: 3215 (May 6)

3996 (May 7)

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