chemical engineering tongji university · partial molar property •partial molar property (q )...

Solution thermodynamics

Min HuangChemical Engineering

Tongji University

Mixture/solution

• Chemical Potential• Partial Molar Property• Partial Pressure• Ideal-Gas Mixtures– enthalpy of an ideal gas– entropy of an ideal gas

• Gibbs energy of an ideal-gas mixture Gig = Hig – TSig,

Mixture/solution

• Because the properties of systems in chemical engineering depend strongly on composition as well as on temperature and pressure,

• Need to develop the theoretical foundation for applications of thermodynamics to gas mixtures and liquid solutions

Mixture/solution

• definition of a fundamental new property called the chemical potential, upon which the principles of phase and chemical-reaction equilibrium depend.

• This leads to the introduction of a new class of thermodynamic properties known as partial properties.

Chemical potential

• Gibbs free energy of a multicomponent mixture is a function of T, P and each species mole number, therefore, the total differential of the Gibbs free energy,

, , ,

i i i

i

iP N P N i P N

i i

i

G G GdG dT dP dN

T P N

SdT VdP G dN

æ ö¶ ¶ ¶æ ö æ ö= + + ç ÷ç ÷ ç ÷

¶ ¶ ¶è ø è ø è ø

= - + +

å

å

C

C

(CP1)

Chemical potential

• Recall

• in extensive function, therefore we have

i

r

i

i

i

r

i

i

dn

pVTSdnpVTSdG

å

å

=

=

=

--÷ø

öçè

æ+-=

1

1

µ

µ

( )

jinTpi

in

nG

¹

úû

ùêë

é

¶

¶º

,,

µ

Chemical Potential and Phase Equilibrium

• Consider a closed system consisting of two phases in equilibrium.

• Within this closed system, each individual phase is an open system, free to transfer mass to the other, that is

( ) ( ) ( )

( ) ( ) ( ) bbbbb

aaaaa

µ

µ

i

i

i

i

i

i

dndpnVdTnSnGd

dndpnVdTnSnGd

å

å

=

=

++-=

++-=

1

1


• The change in the total Gibbs energy of the two-phase system is the sum of these equations.

( ) ( ) ( )

011

11

=+

+++-=

åå

åå

==

==

bbaa

bbaa

µµ

µµ

i

i

ii

i

i

i

i

ii

i

i

dndn

dndndpnVdTnSnGd


• And• Therefore,

• Thus, multiple phases at the same T and P are in equilibrium when the chemical potential of each species is the same in all phases.

ba

ii dndn -=

( )

),,2,1(

01

Ni

dn

ii

i

i

ii

!==

=-å=

ba

aba

µµ

µµ

Partial Molar Property

• Partial Molar Property (q )

• It is a response function, representing the change of total property q due to addition at constant T and P of a differential amount of species i to a finite amount of mixture/solution.

( )

jinTpi

in

nG

¹

úû

ùêë

é

¶

¶º

,,

//

qqµ


• Let q be any molar property (molar volume, molar enthalpy, etc.) of a mixture consisting of Nimoles of species i,

• And partial molar thermodynamic property

• Therefore,

( )( )

, ,

, ,

j i

i i

i T P N

NT P x

N

qq q

¹

¶= =

¶

iiN N=å

C

( )1

, ,i ii

x T P xq q=

=åC


• Or

So that, by the product rule of differentiation,

• substitute G with Nq into Eq. CP 1

1

i i

i

N Nq q=

=åC

( ) i i id N N d dNq q q= +å å PM 1


• substitute G with Nq into Eq. CP 1

, , ,

, ,

i i i

i i

i

iP N T N i P N

i i

iP N T N

N N NdN dT dP dN

T P N

N dT N dP dNT P

q q qq

q qq

æ ö¶ ¶ ¶æ ö æ ö= + + ç ÷ç ÷ ç ÷

¶ ¶ ¶è ø è ø è ø

¶ ¶æ ö æ ö= + +ç ÷ ç ÷

¶ ¶è ø è ø

å

å

C

C

PM 2


• Subtracting PM 2 from PM 1,

• Or,, ,

0

i i

i i

iP N T N

N dT N dP N dT P

q qq

¶ ¶æ ö æ ö- - + =ç ÷ ç ÷

¶ ¶è ø è øåC

, ,

0

i i

i i

iP N T N

dT dP x dT P

q qq

¶ ¶æ ö æ ö- - + =ç ÷ ç ÷

¶ ¶è ø è øåC

This is the generalized Gibbs-Duhem Equation


• For constant temperature and pressure,

• Substitute q with G

,1

,1

0

0

i i T Pi

i i T Pi

N d

x d

q

q

=

=

=

=

å

å

C

C

1

1

0

0

i i

i

i i

i

SdT VdP N dG

SdT VdP x dG

=

=

- + =

- + =

å

å

C

C


• At constant temperature and pressure,

• Recall Gibbs-Duhem Equation

1

1

0

0

i i

i

i i

i

N dG

x dG

=

=

=

=

å

å

C

C

, ,

0

i i

i i

iP N T N

dT dP x dT P

q qq

¶ ¶æ ö æ ö- - + =ç ÷ ç ÷

¶ ¶è ø è øåC


• Rearrange and times n on both sides,

• Derivative with respect to Ni

å-

=++÷÷ø

öççè

æ¶¶

-÷÷ø

öççè

æ¶¶

-1c

jjjii

NT,Np,

0θdNθdNdppθndT

Tθn

ii

å-

=++÷÷ø

öççè

æ¶¶

-÷÷ø

öççè

æ¶¶

-1c

j i

jji

NT,

i

Np,

i 0dNθd

dNθddppθdT

Tθ

ii

å-

=¶¶

++÷÷ø

öççè

æ¶¶

-÷÷ø

öççè

æ¶¶

-1c

j j

iji

NT,

i

Np,

i 0NθdNθddp

pθdT

Tθ

ii


• Rearrange,

• Substitute back to the Gibbs-Duhem Equation

å-

=÷÷

ø

ö

çç

è

æ

¶

¶+÷÷

ø

öççè

æ

¶

¶+÷÷

ø

öççè

æ

¶

¶=

1

1,,,

C

j

j

pTj

i

xT

i

xp

ii xd

xdp

pdT

Td

qqqq

å å=

-

= úú

û

ù

êê

ë

é

÷÷

ø

ö

çç

è

æ

¶

¶+÷÷

ø

öççè

æ

¶

¶+÷÷

ø

öççè

æ

¶

¶+

÷÷ø

öççè

æ

¶

¶-÷÷

ø

öççè

æ

¶

¶-=

C

j

C

j

j

pTj

i

xT

i

xp

ii

NT

i

Np

i

xdx

dpp

dTT

x

dpp

dTT

ii

1

1

1,,,

,,

0

qqq

qq


• Since

• Finally

• For Binary

1

1 1,

0ii ji j j T P

x dxx

q-

= =

æ ö¶=ç ÷

ç ÷¶è øå åC C

å å= =

÷ø

öçè

æ

¶

¶=÷

ø

öçè

æ

¶

¶=÷÷

ø

öççè

æ

¶

¶C

i xp

C

i

ii

xpxp

ii dT

TdTx

TdT

Tx

1 ,1,,

qq

q

00

,1

22

,1

11

2

1

1

1

1

,1

=÷÷

ø

ö

çç

è

æ

¶

¶+÷

÷

ø

ö

çç

è

æ

¶

¶=÷

÷

ø

ö

çç

è

æ

¶

¶å å=

-

= pTpTi

C

j pT

ii

xx

xx dx

xx

qqq

Ideal-Gas Mixtures

• Ideal gas

• pi is known as the partial pressure of species i

t

ii

V

RTnp =

tV

nRTp =

),,2,1( Ni pyp or yn

n

p

piii

ii!====

( ) ( )

P

RT

n

n

P

RT

n

pnRT

n

nVV

j

ii

ni

npTinpTi

igig

i

=÷÷

ø

ö

çç

è

æ

¶

¶

úû

ùêë

é

¶

¶=ú

û

ùêë

é

¶

¶=

,,,,

/

Ideal-Gas Mixtures

• A partial molar property (other than volume) of a constituent species in an ideal-gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.

ig

i

ig

ii

ig

i

ig

i VM whenpTMPTM ¹= ),(),(

Ideal-Gas Mixtures

• Since the enthalpy of an ideal gas is independent of pressure,

• For ideal gases, this enthalpy change of mixing is zero.

ig

i

i

i

ig

ig

i

ig

i

ig

i

ig

i

ig

ii

ig

i

HyHp T, mixture at value pureHH

pTHpTH pTHpTH

å==

=

=

)(),(),(),(),(

Ideal-Gas Mixtures

• The entropy of an ideal gas does depend on pressure, is given as

• Therefore, integration from pi to p

T const at pRddS igi ln=

i

i

i

ig

i

ig

i yRp

pRpTSpTS lnln),(),( =-=-

1

2

1

2p p

pnR lnTTlnncS -=D

Ideal-Gas Mixtures

• Recall• Therefore,

• Where Siig is the pure-species value at the mixture T and P.

ig

i

ig

ii

ig

i

ig

i VM whenpTMPTM ¹= ),(),(

i

ig

i

ig

i

i

ig

i

ig

i

yRSS

yRpTSpTS

ln

ln),(),(

-=

-=

Ideal-Gas Mixtures

• By the summability relation

• the left side of the second equation is the entropy change of mixing for ideal gases. Since l/yi > 1, this quantity is always positive, in agreement with the second law.

ii

i

ig

i

i

i

ig

i

i

i

i

ig

i

i

i

ig

i

yyRSyS

yyRSyS

1ln

ln

åå

åå

=-

-=

Ideal-Gas Mixtures

• For the Gibbs energy of an ideal-gas mixture Gig = Hig – TSig,

• for partial properties is

i

ig

i

ig

i

ig

i

i

ig

i

ig

i

ig

i

ig

i

ig

i

ig

i

yRTGG

yRTTSHG

STHG

ln

ln

+=º

+-=

-=

µ

Recap Mixture/solution• Gibbs free energy of a multicomponent mixture• Chemical Potential• Partial Molar Property• Partial Pressure of Ideal-Gas• Ideal-Gas Mixtures– enthalpy of an ideal gas– entropy of an ideal gas

• Gibbs energy of an ideal-gas mixture Gig = Hig –TSig,

chemical engineering tongji university · partial molar property •partial molar property (q )...

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