Thermodynamic properties: phase equilibria Ireneo Kikic DICAMP - SFLAB - University of Trieste [email protected]

Theo de Loos PCMT-DelftChemTech – Delft University of Technology [email protected] Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

DICAMP

Lisbon, July 2006 Socrates Intensive Course

Delft University of Technology

The Netherlands

Lisbon, July 2006 Socrates Intensive Course

Head office

Lisbon, July 2006 Socrates Intensive Course

TU Delft campus

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

Phase Equilibrium relationships For a closed system at constant T and P at equilibrium the total Gibbs energy G is minimum

System of

phases and N component

i

.........

i

(

for i 1, 2,...., N

i

ni g ) i

i

ni P ,T ,n j

i

Lisbon, July 2006 Socrates Intensive Course

i

Phase Equilibrium relationships equations

F

2

N 1

N

1

2

N

variables

Lisbon, July 2006 Socrates Intensive Course

From the Chemical Potential … to the Fugacity

d

i

RT ln fi

Fugacity coefficient

fi lim P 0 P i

at constant T

i

fi Pi

Fugacity of component i in a mixture

f

i

f xP i

Lisbon, July 2006 Socrates Intensive Course

1

i

Phase Equilibrium in terms of fugacity

fi

fi

.........

RT ln

RT ln

V i

P i

P ni

0

V ,T ,n j

fi

for i 1, 2,..., N

Vi

RT dP P RT V

ni RT RT ln

i

PV

i

Lisbon, July 2006 Socrates Intensive Course

fi 0

Phase Equilibrium and activity f i ( P, T , x ) f i 0 ( P, T , x 0 )

ai

id i

a

xi

i

ni g E RT ln

fi

i i

ni P ,T , n j

i

fi 0

ai id ai

xi i f i

0

standard state fugacity Lisbon, July 2006

Socrates Intensive Course

Standard state fugacity of solids and liquids fi S ( Pi sub , T )

f i S ( P, T )

f iV ( Pi sub , T )

d iS Pi sub RT P

f i S ( Pi sub , T ) exp

f i S ( P, T )

f i L ( P, T )

V i

V i

( Pi sub , T ) Pi sub

fi S ( Pi sub , T ) exp

( Pi sub , T ) Pi sub exp

V i

( Pi sat , T ) Pi sat exp

P Pi sub

P Pi sat

P Pi sub

viS dP RT viL dP RT Lisbon, July 2006

Socrates Intensive Course

viS dP RT

Critical phenomena Two phases in equilibrium can became identical

CRITICAL POINT Gibbs conditions for pure components

P/ v

T

0 and

2

P/ v

2 T

0

Lisbon, July 2006 Socrates Intensive Course

Critical phenomena For N components

molar Helmholtz energy 2

2

a v2 2

2

Dsp

Dsp

Dsp

Dsp

v

x1

x2

xN

2

2

2

a v x1 ( Dc )T

a x12

2

2

a v xN

1

a x1 xN

a x2 xN

2

1

a x2 xN

2

1

2

1

a

xN2

a/ v

1

2

a x22

2

a x2 xN

a x2 xN

2

0 1

T ,x

2

a v xi

1

a

2 N 1

P

P xi

1

Lisbon, July 2006 Socrates Intensive Course

x

1

0

1

a x1 xN

2

a x1 x2

1

a x1 xN

2

a x22

2

1

2

a x1 x2

2

a v x2

a v xN

a x1 xN

2

a x1 x2

1

2

a x1 x2

2

a v x2

a v xN

2

a x12

2

2

a v x2

2

a v x1 ( Dsp )T

2

a v x1

v ,T

Critical phenomena KT

KT

1 v

v P

T

goes to infinity approaching the critical point

the way KT and other thermodynamic properties diverge when approaching the critical point is described in a fundamentally wrong way by all classical, analytical equations of state like the cubic equations of state and is path dependent. The reason for this is that these equations of state are based on mean field theory, which neglects the strong density fluctuations (and concentration fluctuations) close to the critical point

Lisbon, July 2006 Socrates Intensive Course

Critical phenomena Distance from critical point:

P Pc , Pc

P

Property

Power law

T

T Tc , Tc

Path

Isothermal compressibility

KT

T

critical isochore

Isochoric heat capacity

CV

T

critical isochore

Coexistind densities

(

Pressure Exponent

L

V

)

T

P

Classical fluid

two phase critical isotherm Real fluid

0

0.110

½

0.326

1

1.239

3

4.80 Lisbon, July 2006

Socrates Intensive Course

c c

Phase rule for binary systems (N = 2) Number of phases

Degree of freedom

Representation in P, T, x

F= 4 Examples

1

3

Region

L/V/L1/L2/S

2

2

Two surfaces: x (P,T); x (P,T)

LV/SL/LL

3

1

Three curves: x (P,T); x (P,T); x (P,T)

Critical curve

1

xc(P,T)

Azeotropic curve

1

xaz(P,T)

4

0

Four point at one P and T

SL2L1V

Critical end point

0

Two points at one P and T

L2=L1V/L2=L1V/SL =V

Critical Azeotrope

0

One point

SLV / L2L1V

L=V/L2=L1

Lisbon, July 2006 Socrates Intensive Course

Classification of phase behaviour Often the essentials of phase diagrams in P,T,x-space are represented in a P,T-projection. In this type of diagrams only non-variant monovariant (F=1) equilibria can be represented.

(F=0)

and

Since pressure and temperature of phases in equilibrium are equal, a four-phase equilibrium is represented by one point and a three-phase equilibrium with one curve. The critical curve and the azeotropic curve are projected as a curve on the P,T-plane. A four-phase point is the point of intersection of four threephase curves. The point of intersection of a three-phase curve and a critical curve is a so-called critical endpoint. In this intersection point both the three-phase curve and the critical curve terminate. Lisbon, July 2006 Socrates Intensive Course

Classification of phase behaviour Phase equilibria can be represented in isothermal P,x-sections, isobaric T,x-sections and isoplethic P,T-sections.

The equilibrium composition of the liquid phase and vapour phase are represented by curves, the so-called binodal curves.

At a given pressure a mixture with an overall composition in between the two binodal curves will split in two phases with compositions given by the binodal curves. Mixtures outside this region will be in a one-phase state (either liquid or vapour).

Lisbon, July 2006 Socrates Intensive Course

Classification of phase behaviour P

a

P

b

l

l

P

c l=g l

x=y l+g l+g

g x

l+g

g

g

x

x

Two binodals can intersect, which occurs in a at the pure component vapour pressures. They also can be tangent as is the case in b in the azeotropic point or they can merge as is the case ic in the critical point l=g. Critical points and azeotropic points are always extrema in pressure or temperature in P,x- or T,x-sections, respectively. Lisbon, July 2006 Socrates Intensive Course

Classification of phase behaviour P

P

a

b

+ +

+

+

+

+

x

x

In a P,x- or T,x-section a three-phase equilibrium is represented by three points. All mixtures with composition on the line through these points will split into three phases. Three two-phase regions, , and are found.

,

, and

and three one-phase regions,

The one- and two-phase regions have to be arranged around the three-phase equilibrium as is shown in a or as is shown in b. Lisbon, July 2006 Socrates Intensive Course

Type I

Type I I

P

P l 2 =l 1

Classification of

l=g

l=g

van Konynenburg - Scott

lg

lg l 2l1 g

lg

lg T

T

Type III

Type IV

l 2 =l 1

P

P

l 2 =l 1

l 2 =l 1

l 2=g

l 2 =l 1

l 1=g

lg

l 1=g

lg

l 2l1 g

l 2l1 g

l 2l1 g

lg

l 2=g

lg

T

T

Type V

Type VI P

P

l 2 =l 1 l 2 =l 1 lg

l 1=g

l 2=g

l 2l1 g

l=g lg l 2l1 g

lg T

lg T

Lisbon, July 2006 Socrates Intensive Course

Classification of van Konynenburg - Scott The curves lg are the vapour pressure curves of the pure components which end in a critical point l=g. The curves l=g, l1=g and l2=g are vapour-liquid critical curves and the curves l1=l2 are curves on which two liquid phases become critical. The points of intersection of a critical curve with a three-phase curve l2l1g is a critical endpoint. Distinction can be made between upper critical endpoints (UCEP) and lower critical endpoints (LCEP). The UCEP is highest temperature of a three-phase curve, the LCEP is the lowest temperature of a three-phase curve. The point of intersection of the l2l1g curve with a l1=g curve is a critical endpoint in which the l1 liquid phase and the vapour phase are critical in the presence of a noncritical l2 phase (l2+(l1=g)) and the point of intersection of the l2l1g curve with a l2=l1 curve is a critical endpoint in which the two liquid phases l2 and l1 are critical in the presence of a non-critical vapour phase ((l2=l1)+g). Lisbon, July 2006 Socrates Intensive Course

TYPE I

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

Retrograde condensation In the critical point the bubble point curve (l+g l) and the dew-point curve (l+g g) P merge at temperatures between Tc and Tmax an isotherm will intersectl=g the dewpoint curve twice. If we lower the pressure l=g on this isotherm we will pass the first dewpoint and with decreasing the l +pressure g amount of liquid will increase. Then the amount of liquid will reach a maximum l + gof the and upon a further decrease pressure the amount of liquid will decrease until is becomes zero at the x second dew-point.

P

l

l=g

g

l +g g

Tc

Tmax T

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

TYPE II

UCEP

a

bc d Lisbon, July 2006

Socrates Intensive Course

a

b

c

d

Lisbon, July 2006 Socrates Intensive Course

TYPE V

UCEP LCEP

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

TYPE IV

Lisbon, July 2006 Socrates Intensive Course

TYPE IV

Combination of type II and V Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

TYPE III

Lisbon, July 2006 Socrates Intensive Course

TYPE III

The l2=l1/l2=g branch of the critical curve can have the shape as is shown Lisbon, July 2006 Socrates Intensive Course

but it is also possible that this curve goes from the critical point of component B to high pressure via a temperature minimum or that dP/dT is always positive

TYPE III

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

TYPE VI

Lisbon, July 2006 Socrates Intensive Course

Lisbon, July 2006 Socrates Intensive Course

Phase behaviour In binary systems of a light gas and members of the n-alkane homologuous series in almost all cases type II phase behaviour is found for the lower members of the n-alkane series. With increasing carbon number type IV phase behaviour is found followed by type III phase behaviour at high carbon numbers. Systems of CO2 + n-alkanes show type II phase behaviour for nalkane carbon numbers n= 12, type IV phase behaviour for n=13 and type III phase behaviour for n= 14. This transition from type II to type III phase behaviour via type IV phase behaviour seems to be the rule, although in most binary families type IV phase behaviour is only found at broken values of n in so-called quasibinary systems.

Lisbon, July 2006 Socrates Intensive Course

Critical endpoint for CO2-nalkane systems If in a particular binary system the three-phase curve l2l1g is followed to low temperature then at a certain temperature a solid phase is formed (solid n-alkane or solid CO2 at low carbon numbers). This occurs at one unique temperature because we now have four phases in equilibrium in a binary system, so according to the phase rule F=0. Below this so-called quadruple point temperature the l2l1g curve is metastable. The quadrupole points curve intersects the curve through the critical endpoints at a carbon number 23
Lisbon, July 2006 Socrates Intensive Course

320

300

T/K

280

260

240

220

200 6

8

10

12

14

16

18

20

22

n

The family of carbon dioxide with n-alkanes. Critical endpoint temperatures and quadruple point temperatures as a function of carbon number n. : quadruple point l2l1gsCO2; ¦ : quadruple point sn-alkanel2l1g; : UCEP l2=l1g; : LCEP l2=l1g; : UCEP l2l1=g. Lisbon, July 2006 Socrates Intensive Course

P

sl

l 2 =l 1

P

sl 2l 1

sl 2l 1 l 2=g

l 1=g

lg

sl 2g sg

a

T

P

sl l 2 =l 1

l 1=g

l 2=g

lg sl 1g

sl 2g

lg sg

c

T

l 2=g

l 2l1 g

sl 1g

lg

sl 2l 1

l 1=g

lg

l 2l1 g

sl 1g

sl

l 2 =l 1

sl 2g sg

lg

T

b

As discussed for CO2 + n-alkane systems at nc<24 the threeends at low temperature in a quadruple q Pphase curve l2l1g sl point s2l2l1g. In the quadruple point three other threephase curves slterminate. The s2l2l1 curve runs steeply to l 1=g p 2g l 2=g lg high pressure and ends in a critical endpoint where this sl 1g intersects the critical curve. The s2l2g curve runs to curve lg the triple point of pure component B and the s1l1g curve sg T d runs to lower temperature and ends at low temperature in a second quadruple point s2s1l1g (not shown).

q P sln-alkane the With increasing nC of the l 2=g to higher melting curve of the n-alkane shifts sl 2g temperature. Also thel1quadruple point s2l1l2g =g p lg shifts to higher temperature (b) and sl 1g lg eventually coincides with the critical endpoint sg l2+(l1=g) of the l2l1g curve (c)

Socrates Intensive Course

e

Solid phases

T

Lisbon, July 2006

sl P P sl disappeared and l 2 =l 1 At higher carbon numbers the l2l1g curve l 2 =l 1 has 2l 1 sl 2l 1 the sls2l2l1 and the s2l2g curve form now one curve (d). This l 2=g l 1=g curve endsl1=gat high pressure in a critical endpoint q where the lg lg l 2=g l 2l1 g l 2l1 g curve intersects the critical three-phase curve. The s2l1g curve sl 1g sl 1g lg sl 2g a critical sl 2g critical lg also shows endpoint p. In both endpoints the sg sg solid is in equilibrium with a critical fluid phase. In d the three a T T b phase curve s2l2g shows a temperature minimum which disppears at even higher carbon numbers (e). P

The p, q systems sl(b, sl l 1 l =l 2 1 d) are of 2importance l 2=g for thel1=g extraction lg of sl1g solids sl 2g lg with supercritical sg gases. c T

q

P

l 1=g p sl 2g

lg

lg sg T

d

q

sl l 2=g

sl 2g l 1=g p

lg sl 1g

lg sg

e

l 2=g

sl 1g

P

Socrates Intensive Course

sl

Lisbon, July 2006 T

P-x sections of a p,q-system at temperatures around the critical temperature of component A a

P

b

P g=l

l

l+s 2

l+s 2

c

P fl fl+l

l g+l

g+l g + s2

g+s2 g x

g

x

g

x

a: T
P,T-sections for a p,q-system at temperature close to Tq P

P fl fl

s2 +fl

s2 +fl sl 2g

l+g

g

g a

T

P

T

b

q

s2 +l l l=g

s2 +g

sl 2g

l+g

g c

T

At very low concentration of the heavy component no vapour-liquid equilibria are found (a). At intermediate concentration of the heavy component a part of the three-phase curve s2l2g is found and a vapour-liquid region with only dew points. At higher concentration of the heavy component the three-phase curve ends at high pressure in the point q and the vapour liquid region shows a critical point.

Lisbon, July 2006 Socrates Intensive Course

Thermodynamic properties: phase equilibria

The point of intersection of a three-phase curve and a critical curve is a so-called critical endpoint. In this intersection point both the three-phase curve and the ...

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