Chemical Thermodynamics IIPhase Equilibria

Valentim M. B. Nunes

ESTT - IPT

April 2015

We live in a world of mixtures: the air we breathe, the petrol in the deposits of our automobiles, etc. Many of the chemical technology activities are related with the transfer of substances from a mixture to another.

Phase equilibrium: transfer of substances between phases. Many of the operations in the industry, such as extraction, distillation, etc., involve transfer between phases. To scale these units it is necessary to characterize the equilibrium properties of the several phases.

Determines the number of degrees of freedom, g, or number of intensive properties necessary to specify a given state of equilibrium, with F phases and C components.

2 CgF

Gibbs Phase Rule

,..., 21 xx

,..., 21 xx

T

p

T)(or p and ,....,, p)(or T and ,....,, 2121 xxxx

The basic problem:

Solution of the problem: Gibbs!

For any component i:

ii To relate with T and p and x1, x2, etc., it’s necessary to introduce the concepts of fugacity and activity. For instance if = vapor and = liquid, then, as we will see later:

satiiiii pxpy

Classic thermodynamics of phase equilibria

For a homogeneous closed system, the combination of the first and second law of thermodynamics gives, for a reversible process in which the State of equilibrium is maintained:

pdVTdSdU “Work done by the system”

“Heat absorbed by the system”

For a finite change:

2

1

2

1

12

V

V

S

S

pdVTdSUUU

Applies to any process, reversible or irreversible, since the initial and final State are equilibrium States.

Prausnitz et al., Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, New Jersey, 1986

For an open system, there may be an exchange of matter (and also energy), with the exterior.

ii

i

ii

i

dnVdpSdTdG

dnpdVTdSdU

jnTpii n

G

,,

For a heterogeneous closed system, we can consider each phase as a homogenous system open. Resuming:

ii

idnpdVTdSdU

Integrating from a state of zero mass (U=S=V=…=0), until a given finite state at constant temperature, pressure and composition we obtain:

ii

inpVTSU

Differentiating again:

ii

iii

i dndnVdppdVSdTTdSdU

Then:

0 ii

idnVdpSdT

Gibbs-Duhem equation: fundamental equation of the thermodynamics of solutions, restricts the simultaneous variation of T, p and i in a given phase.

For a pure substance, dpVdTSd iii

i

T

i Vp

For a ideal gas, Vi = RT/p

00 ln

p

pRTii

To generalize, Lewis introduced the concept of fugacity.

00 ln

i

iii f

fRT

Valid for an isothermal process, for any component, whether solid, liquid or gas, pure or in mixtures, ideal or not!

For a ideal gas, f = p.

For a component i in an ideal gaseous mixture, the fugacity of i is the partial pressure, fi = pi = yi p.

For all systems:1lim

0

py

f

i

i

p

Lewis called the relation f/f0 as activity.

0i

ii f

fa

An important consequence:

,0,0

,0,0

ln

ln

i

iii

i

iii

f

fRT

f

fRT

If we consider: ,0,0,0,0

iiii ff

,0

,0,0,0 ln

i

iii f

fRT

Assuming the standard states of the two phases at the same temperature but not to the same pressure and composition

ii ff

Equilibrium accordingly with Lewis

Consider the equilibrium between a liquid phase and a vapor phase. For the component 1 the condition of equilibrium implies that:

Lpuro

L

Vpuro

V

LV

fxf

fyf

ff

,111

,111

11

A simple case, the Raoult's Law

satLpuro

Vpuro

pf

pf

1,1

,1

satpxpy 111

The Raoult's law in practical calculations its of reduced utility. It is necessary to use molecular thermodynamics to relate the fugacity of compounds with its molecular characteristics.