School of Chemical & Biological Engineering, Konkuk University

Lecture 16

• The chemical potentials of liquids

• Liquid mixtures

Ch. 5 Simple Mixtures

• To discuss the equilibrium properties of liquid mixtures, we need to know how the Gibbs energy of a liquid varies with

composition.  the chemical potential of liquid

• We can start with the fact that, at equilibrium, the chemical potential in a vapor phase of a substance must be equal to its chemical potential in the liquid phase.

o o

p RT p

p) ln

(   



where is the standard chemical potential (the chemical potential of the pure gas at 1 bar).

o

For a convenience, let’s denote the relative pressure with p. _o

p p p

ln )

(p  ^o  RT



• For a perfect gas, Lecture 12 & 15

• We shall denote quantities relating to pure substance by a

superscript *, so the chemical potential of pure A is written . ex) : chemical potential of pure liquid A

*

A

 

A

*

• At equilibrium, the chemical potential of the gaseous form of a substance A is equal to the chemical potential of its condensed phase. ^_A^*

 

 

• Because the vapor pressure of the pure liquid is , the chemical potential of A in the vapor is: p^*_A

 

 

* _{A A}^o ln _A

A g   l    RT p

 A(g)

A(l)

o A

A p

p^*

*  where p

• The equality in the above equation is preserved if a solute (B) is also present.  The vapor and solvent are still in equilibrium.

• However, the notation should be changed as below:

 

* ^o_A ln _A

A l    RT p



 

A l    RT lnp



o A

A p

 p where p

• These two equations can be combined to eliminate as below: A^o

   

A

o A

A A

A

A p p

p RT p

RT RT

l

l  ^*  lnp  lnp^*  ln _*



 

 

A A A

A p

RT p l

l   



 

 

A A A

A p

RT p l

l   



• The French chemist F. Raoult found that the ratio of is proportional to the mole fraction of A ( ) in the liquid.

* A A

p p

xA

Raoult’s Law: pA  xAp^*A

However, this law is only applicable to mixtures of closely related liquids. Ex ) benzene and methylbenzene.

• Mixtures that obey the Raoult’s law throughout the

composition range from the pure A to pure B are called ideal solution.

• For an ideal solution, ^_A

 

 

Another definition of the Raoult’s law

* B B

B x p

p 

* A A

A x p

p 

B

A P

P p  

(solvent) (solute)

• In ideal solutions, the solute (B) as well as the solvent (A) obeys Raoult’s law.

• Raoult’s law reveals that the vapor pressure of a component in an ideal solution is proportional to its mole fraction.

• Note that and are the slopes (or constants of proportionality) of the p-x_A and p-x_B plots, respectively.

*

pA p^*_B

• Some solutions depart significantly from Raoult’s law.

• Nevertheless, even in such solutions, the Raoult’s law is

obeyed increasingly for the component in excess (solvent) as it approaches purity (dilute solution).

• Therefore, the Raoults’ law is a good approximation for the properties of the solvent if the solution is dilute.

• Consider a solution consisting of solvent A and a solute.

The rate of vaporization is proportional to the number of A molecules at the surface, which in turn is proportional to the mole fraction:

kxA

 ion vaporizat of

rate

The rate of condensation is proportional to their concentration in the gas phase, which in turn is proportional to their partial pressure:

pA

k' on

condensati of

rate 

• Therefore, the rate of vaporization of the

solvent molecules is reduced by the presence of the solute molecules but the rate of

condensation of the solvent is not hindered.

where k and k’ are proportional constants.

• At equilibrium, the rates of vaporization and condensation are equal:

A

A k p

kx  '

A

A x

k p k

 '

For a pure liquid, , so in this special case ^x_A ^1

'

*

k p_A  k

Therefore, _{pA  p}^*_AxA

• W. Henry found experimentally that, for real solutions at low concentrations, although the vapor pressure of the solute (B) is proportional to , the constant of proportionality is not . x_B p^*_B

solutions real

dilute for

but

_B ^*_{B B}

B

B x p p x

p  

• Henry’s law is: _{pB  xBKB}

where x_B is the mole fraction of the

solute and K_B is an empirical constant with the dimensions of pressure.

• K_B is chosen so that the p-x_B plot is

tangent to the experimental curve at x_B=0.

• Mixtures for which the solute and the solvent obey Henry’s law and Raoult’s law, respectively, are called ideal-dilute solutions.

• The origin of the difference in behavior of the solute and solvent at low concentration:

In dilute solution, the solvent molecules are in an environment very much like the one they have in the pure liquid.

The solute molecules are surrounded by solvent molecules.

• For practical applications, Henry’s law is expressed in terms of the molality (b) of the solute,

B B

B b K

p 

Note that the unit of K_B in the equation expressed with molality is .

  



Molality: the amount of solute divided by the mass of solvent.

• Now we consider the thermodynamics of mixing of liquids.

• First we consider the simple case of mixtures of liquid which is an ideal solution.

• Similar to the same way for mixing ideal gases, the total Gibbs energy of two liquid (A and B) before mixing them is:

) ( )

( ^*

* l n l

n

G_i  _A_A  _B_B

• The total Gibbs energy of two liquid after mixing is:

 





A

B B

A A

mix

x x

x x

nRT

n x n

x nRT

x x

nRT x

x RT

n x

RT n

G

ln ln

ln

ln

ln ln



















 

 

A l  ^* l  RT ln x



• Consequently, the Gibbs energy of mixing is:



 



A

B B A

A f

x RT

l n

x RT

l n

l n

l n

G

ln )

( ln

) (

) ( )

(

*

*   



















mixG  nRT x ln x  x ln x



• Because , S T

G

n p



 



 









,

• The entropy of mixing is: ^_mix^{S  nR}





• Because , the enthalpy of mixing is zero: G  H TS

 0

_mixH

• The (molar) volume change of mixing is zero:

 0

_mixV

• The _mixG is the same as that for two perfect gases.

• Therefore, the entropy change in its surroundings is: ^_mix^S_sur ^{ 0}











  

 







 V

p G



• For the mixing of two liquids which results in an ideal solution,  The driving force for mixing is the increasing entropy of the system, and the enthalpy of mixing is zero.

• However, note that in a perfect gas there are no forces

between molecules, but in ideal solutions there are interactions.

 The average energy of A-B interactions in the ideal solution is the same as the average energy of A-A and B-B interactions in the pure liquids.





mixG  nRT x ln x  x ln x







mixS  nR x ln x  x ln x

 _mixH  0

• For both mixture of perfect gases and ideal solutions consisting of A and B,

• The Gibbs energy of mixing is

negative for all compositions and T, so the mixing is spontaneous in all proportions.

• The entropy of the system (and also total entropy) increases for all compositions and T, so the mixing is spontaneous in all proportions.

• In real solutions, the A-A, B-B, and A-B interactions are all different.

• Not only may there be enthalpy and volume changes when liquids mix, but there may also be an additional entropy

contribution, originated from clustering of molecules such as solvation.

• If _mixH > 0 and _mixS < 0 (due to an orderly mixture), then the

_mixG > 0.

 Separation is spontaneous and the liquids are immiscible.

• Sometimes, two liquids are partially miscible, which means

that they are miscible only over a certain range of compositions.

S T H

G    



• The thermodynamic properties of real solutions are expressed in terms of the excess functions (X^E).

• The X^E is defined as the difference between the observed

thermodynamic function of mixing and the function for an ideal

solution. _ideal

mix mix

E X X

X   

• For example, the excess entropy (S^E) is defined as:

ideal mix

mix

E S S

S   

where . _mixS^ideal  nR





• For the excess enthalpy and excess volume,





 



 _{mix mix} ^ideal

E V V

V 





 



 _{mix mix} ^ideal

E H H

H 

• Deviation of the excess energies from zero indicate the extent to which the real solutions are non-ideal.

• Among the real solutions, the regular solution is a solution for which H^E  0 but S^E = 0.

• In a regular solution, the two kinds of molecules are randomly distributed (without clustering) as in an ideal solution but have different energies of interactions with each other.

Benzene/cyclohexane

Tetrachloroethene/

cyclopentane

Endothermic mixing Contradiction Expansion

• The excess enthalpy depends on composition as below:

B A mix

E H n RTx x

H    

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.

• The plot of the above equation resembles the experimental curve.

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

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

• For a regular solution (H^E  0 but S^E = 0),

)

( G^E  H^E TS^E  H^E

H H

G G

G^E  _mix _mix ^ideal  ^E  _mix

• Therefore, the Gibbs energy of mixing can be derived as below:





ideal mix

mix

x RTx n

x x

x x

nRT

H G

G



















ln ln





mixG  nRT x x  x x x x

 ln ln

For a regular solution,

E

E H

G 

• If a mixture This figure shows the variation of Gibbs energy of mixing for different values of the parameter .

• For  > 2, two minima separated by a maximum.  The system will separate spontaneously into two phases with

compositions corresponding to the minima.

• Reading: page 143 ~ 150

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