1

Gibbs free energy and

Equilibrium

Gibbs Free Energy

Gibbs free energy is a measure of chemical

energy

Gibbs free energy for a phase:

G = H - TS

Where:

G = Gibbs Free Energy

H = Enthalpy (heat content)

T = Temperature in Kelvins

S = Entropy (can think of as randomness)

Entropy Change in the Universe

The universe is composed of the system and the surroundings.

Therefore,

Suniverse = Ssystem + Ssurroundings

For spontaneous processes

Suniverse > 0

Entropy Change in the Universe

This becomes:

Suniverse = Ssystem +

Multiplying both sides by T,

TSuniverse = Hsystem TSsystem

Hsystem

T

2

Gibbs Free Energy

TSuniverse is defined as the Gibbs free

energy, G.

When Suniverse is positive, G is

negative.

Therefore, when G is negative, a

process is spontaneous.

Under standard conditions

∆Gosys = ∆Ho

sys - T∆Sosys

An isolated system is said to be in chemical equilibrium if no changes occur in the chemical composition of the system. The criterion for chemical equilibrium is based on the second law of thermodynamics, and for a system at a specified temperature and pressure it can be expressed as For the reaction where the v's are the stoichiometric coefficients.

© The McGraw-Hill Companies, Inc.,1998

Equilibrium Criterion Equilibrium Criterion Criterion for Chemical Equilibrium Criterion for Chemical Equilibrium

3

The four Gibbs fundamental equations

Relations from the fundamental equations

PdVSdTda

VdPTdSdh

VdPSdTdg

PdVTdSdu TS

u

V

P

V

u

S

TS

h

P

V

P

h

S

PV

a

T

S

T

a

V

VP

g

T

S

T

g

P

Inspection of above equations gives the eight

Coefficient Relations below

Pv s

h

s

uT

Ts v

a

v

uP

Pv T

g

T

as

Ts P

g

P

hv

Maxwell Relations

4

Deriving Maxwell Relations

First, start with a known equation of state such as that of internal energy

Next, take the total derivative of with respect to the natural variables. For example, the

natural of internal energy are entropy and volume.

dVV

UdS

S

UdU

SV

Deriving Maxwell Relations

TS

U

V

P

V

U

S

Now that we have the total derivative with respect to its natural variables, we can refer

back to the original equation of state and define, in this example, T and P.

dVV

UdS

S

UdU

SV

Maxwell Relations

f(x,y)

x

y

df

1

2

dx

dy

yxxy y

f

xx

f

y

yxx

M

y

L

Now, let’s assume that the partial derivatives themselves are state

functions of x and y, as well, where:

yx

fL

xy

fM

MdyLdxdf

yxff ,

dyy

fdx

x

fdf

xy

Let’s consider an arbitrary state function of two variables,

let’s write the total differential of that function.

Deriving Maxwell Relations

We must now take into account a rule in partial derivatives

When taking the partial derivative again, we can set both sides equal

and thus, we have derived a Maxwell Relation

5

Derivation of Maxwell's Equations from Mixed Second

Derivatives

sv

u

s

u

vv

T

svs

2

vs

u

v

u

ss

P 2

vv

vs s

P

v

T

By comparison we get

1st Maxwell equation

Maxwell Relations

VSU , dVV

UdS

S

UdU

SV

PdVTdSdU

VS S

P

V

T

PSH ,

VTA ,

PTG ,

dPP

HdS

S

HdH

SP

dVV

AdT

T

AdA

TV

dPP

GdT

T

GdG

TP

VdPTdSdH

PdVSdTdA

VdPSdTdG

PS S

V

P

T

VT T

P

V

S

PT T

V

P

S

yxx

M

y

L

dy

y

fdx

x

fdf

xy

MdyLdxdf yxff ,

We now apply this analysis to our thermodynamic state functions …

and obtain the well known Maxwell relations.

• The Maxwell relations are

• The equations that relate

the partial derivatives of

properties P, v, T, and s of

a simple compressible

substance to each other

are called the Maxwell

relations. They are

obtained from the four

Gibbs equations,

Variation of G with P

G

pV

V V V

T

gas liquid solid

gas

liquid

solid

p

G

6

Pressure dependence of G

)()( liquids, and solidsfor

)/ln()()( :gasesfor

)()(

12

1212

12

2

1

2

1

2

1

PGPG

PPnRTPGPG

VdPpGpG

VdPdGVP

G

P

P

P

P

P

PT

Variation of G with T

G

TS

S S S

p

gas liquid solid

gas

liquid

solid

T

G

Gibbs-Helmholz Equation

2

22

)/(

1

T

H

T

TG

T

H

T

G

T

G

T

T

HGS

T

G

p

p

p

T Dependence of G

121

1

2

2

2

2

11)()(

)/(

)/(

2

1

2

1

TTH

T

TG

T

TG

T

dTHTGd

T

H

T

TG

T

T

T

T

p

7

Chemical Potential

• The chemical potential of substance A in a

mixture of nA moles A, nB moles B, nC

moles C, etc. is the ratio of dG to dnA

when the composition, temperature, and

pressure remain constant.

,...,,, CB nnPTA

An

G

Ideal Gas Mixture

A

pure

AA

pure

A

pure

mm

xRTPPRT

PPRTP

PPRTPGPG

PPnRTPGPG

ln/ln

:mixture gas idealan in A substancefor

)/ln()(

)/ln()()(

)/ln()(=)(

:substance pure afor

0

A

0

00

00

Chemical Potential

Chemical potential is defined as

, , j i

i i

i T P n

GG

n

• The last quantity in the above expression is the partial molar

Gibbs energy.

• This expression shows how the free energy changes as one

changes the number of moles of component i holding T, P and all

other components fixed.

• This is the potential that drives the flow of mass during a

chemical process or a phase change.

Activities (solvent)

Start with the definition of the chemical potential of a real or ideal

solvent

0

0ln A

A A

A

pRT

p

• If the solvent obeys Raoult’s law, , we can write

0 lnA A ART x

• If the solvent does not obey Raoult’s law (i.e. not ideal) we can

preserve the above equation by writing

0 lnA A ART a

*

A A AP x P

8

Activities (solvent) The activity of the solvent approaches the mole fraction as xA

1

1A A Aa x as x

• A convenient way of expressing this convergence is to introduce the

activity coefficient

1 1A A A A Aa x as x

• The chemical potential of the solvent becomes

0 ln lnA A A ART x RT

Gibbs Free Energy

Chemical Reactions

If G is negative, the reaction is spontaneous

in the forward direction.

If G is zero, the reaction is at equilibrium.

There is no driving force tending to make the

reaction go in either direction.

If G is positive, the reaction is

nonspontaneous in the forward direction.

G allows us to predict whether a process is

spontaneous or not (under constant

temperature and pressure conditions):

G < 0 spontaneous in forward direction

G > 0 non-spontaneous in forward

direction/spontaneous in reverse

direction

G = 0 at equilibrium

© The McGraw-Hill Companies, Inc.,1998

Equilibrium Criterion Equilibrium Criterion

9

Criterion for Chemical Equilibrium Criterion for Chemical Equilibrium

Gibbs free energy and

Phase Equilibrium

Clapeyron Equation

Liquid/Vapour Phase Equilibrium Liquid/Vapour Phase Equilibrium

The multicomponent multiphase system is in phase equilibrium when the specific Gibbs function of each

component is the same in all phases

10

Equilibrium Diagram for a Two-Phase Mixture of Oxygen and Nitrogen at 0.1 MPa

Equilibrium Diagram for a Two-Phase Mixture of Oxygen and Nitrogen at 0.1 MPa

Equilibrium between 2 phases (pure components)

Liq or solid

p

I

gas, liq, or sol

II

nI moles of phase I; nII moles of phase II

G = nIgI + nIIgII = a minimum at equilibrium

gI, gII = molar free energies of phases I and II

dGT,p = dnIgI + dnIIgII = dnI(gI – gII)

Because dnI = - dnII

gI = gII

chemical equilibrium: analogous to thermal equilibrium

(TI = TII) and mechanical equilibrium (pI = pII)

dG caused by transferring dnI moles to II

No inert gas

Phase equilibria Fixed P, T processes and the Gibbs function

'' '' ''' ''' '' '' ''' '''

1 2 1 1 2 2

'' ''' '' '''

1 1 2 2also (conservation of particles)

G G n g n g n g n g

n n n n

Combined, these constraints lead to: g'' = g'''

P, T P, T

V1 V2

Gas

Liquid

Two phases are said to be in phase equilibrium

when there is no transfor-mation from one phase

to the other. Two phases of a pure substance are

in equilibrium when each phase has the same

value of specific Gibbs function. That is,

gf = gg

11

Clapeyron equation for phase change

Since on a co-existence curve,

Therefore, for a first order

phase transformation,

Where is the latent heat from phase 1 to 2.

• The Clapeyron equation enables us to determine

the enthalpy change associated with a phase

change from a knowledge of P, v, and T data

alone. It is expressed a

12

12

ldP

dT T v'' v'

23

23

ldP

dT T v''' v''

13

13

ldP

dT T v''' v'

solid-liquid

(melting)

liquid-vapor

(vaporization)

solid-vapor

(sublimation)

The Clapeyron equation

Temperature

Pre

ssu

re

Liquid

Gas

Solid

-.007oC atm-1

Vapor

Water

Water &

Vapor

Volume (V)

Pre

ssu

re (

e)

Ice & Vapor 0oC

T

Ice

Ice & Water

Triple Line

(courtesy F. Remer)

12

• For liquid-vapor and solid-vapor phase-change

processes at low pressures, the Clapeyron

equation can be approximated as

Integrated Clausius-Clapyron equation

For modest temp. range, CP term can be neglected, so hvap ≈ constant

• Integrate the Clausius-Clapyron equation:

T

1

T

1

R

h

p

pln

ref

v ap

ref,sat

sat

• alternate form:

T/BApln sat

R

hBpln

RT

hA

v ap

ref,sat

ref

v ap

lnpsat

slope = -hvap/R

1/T

Vapor-Liquid Equilibrium

(VLE) of Binary Systems

13

Overall all Objectives

Qualitative VLE analysis.

The thermodynamics of solutions: The ideal

solution reference states.

Raoult’s law

Deviation from Raoult’s law and the introduction

of the activity coefficient

Azeotropes

Thermodynamics and distillation

The description for binary

mixtures

The thermodynamics of solutions

Our goal is to calculate pressure, temperature

and compositions of a phase in equilibrium.

We need to define a model solution system.

Ideal gas

Ideal solution

(macroscopic definition)

0

ln

ln

ideal solution ideal

mixing mixing

ideal solution

mixing i i

ideal solution

mixing i i

V H

S R x x

G RT x x

14

Phase Equlibrium in Multicomponent Systems

Two-component liquid at equilibrium with its vapor.

At constant P and T,

2 1 21 2 2

1 1 21 2 1

1 1

1 1, , , , , , , ,

2 2

2 2, , , , , , , ,

g g gl l l

g g gl l l

l g

l g

P T n n n P T n n n

l g

l g

P T n n n P T n n n

G GdG dn dn

n n

G Gdn dn

n n

1 1 2 2Since 0, 0, andl g l gdn dn dn dn

1 2,g gn n

1 2,l ln n

2 1 2

1

1 , , , ,

, etc.g gl

l

l

P T n n n

G

n

1 1 2 2,l g l g

Assuming the vapor behaves as an ideal gas, the chemical potential of substance

j in the solution is

0

0( ) ln

jl g

j j j

PT RT

P

For the pure substance j,

*

* * 0

0( ) ln

jl g

j j j

PT RT

P

0because ( ) doesn't depend on mole fractions. It follows thatj T

*

*ln

jl l

j j

j

PRT

P

Ideal Solutions and Raoult’s Law

If the partial vapor pressure of each component in a solution obeys the relation

*

j j jP x P

where is the mole fraction of component in the liquid phase,j

j

i

i

nx j

n

sol * lnj j jRT x

the solution is called ideal. Ideal solutions follow Raoult’s law,

sol

*

Here is the chemical potential of (liquid) component in the solution

and is the chemical potential of the pure substance.

j

j

j

15

Ideal Solutions/Raoult’s Law

Ap*

p*B

Mole Fraction of A, x A

Pressure

Total Pressure

Partial

Pressure of A

Partial

Pressure of B

Mixtures which obey Raoult’s

Law throughout the

composition range are Ideal

Solutions

Phenomenology of Raoult’s

Law: 2nd component inhibits

the rate of molecules leaving

a solution, but not returning

rate of vaporization XA

rate of condensation pA

at equilibrium rates equal

implies pA = XA p*A

Vapor Pressure of Ideal Two-Component Solutions

* * * * * * *

1 2 1 1 2 2 1 1 1 2 1 1 2 2 1(1 ) ( ) (linear in )P P P x P x P x P x P x P P P x

The mole fraction of

component 1 in the liquid

phase is

*

21 * *

1 2

P Px

P P

P *

1P

*

2P

P

1P

2P

1x 10

11

* * * * *

1 1 2 1 1 2

* * * * * *

1 1 2 2 1 2 1 2

(Dalton's law)

1

Py

P

x P P P P P P

x P x P P P P P P P

Calculate the mole fraction y1 of component 1 in the vapor phase at a given

value P of the vapor pressure using Dalton’s law of partial pressures:

1

1

1

depends linearly on .

depends nonlinearly

(hyperbolically) on (and on )!

x P

y

P x

*

1P

*

2P

1 vs. P x

1 1 or x y10

1 vs. P y

liquid-vapor coexistence

Applying Henry’s Law & Raoult’s Law

Henry’s law applies to the

solute in ideal dilute

solutions

Raoult’s law applies to

solvent in ideal dilute

solutions and solute &

solvent in ideal solutions

Real systems can (and

do ) deviate from both

Bp*

BK

Mole Fraction of B, x B

Pressure

B is solvent

B is solute

Henry's Law

Raoult's Law

16

Deviations from Raoult’s Law Raoult’s Law works well when components of a mixture are structurally similar

Wide deviations possible for dissimilar mixtures

Ideal-Dilute Solutions

Henry’s Law (William Henry)

For dilute solutions, v.p. of solute is proportional to the mole fraction (Raoult’s

Law) but v.p. of the pure substance is not

the constant of proportionality

Empirical constant, K, has dimensions

of pressure

pB = xBKB (Raoult’s Law says pB =

xBpB)

Mixtures in which the solute obeys Henry’s Law

and solvent obeys Raoult’s Law are called Ideal

Dilute Solutions

Differences arise because, in dilute soln,

solute is in a very different molecular

environment than when it is pure

The thermodynamics of ideal solutions

On the microscopic scale Interactions between species is the same

Non-Ideal Solutions

P *

j j jP x P

*

1P

*

2P

P

1P

2P

1x 10

attractive interactions between

different molecules dominate

* as 1 onlyj j j jP x P x

*

1P

*

2P

P

1P

2P

1x 10

repulsive interactions between

different molecules dominate

Azeotropes

Sometimes interactions between the molecules distort the phase diagram. If the liquid’s free energy is less concave than that of the gas, the curves can intersect in two places.

Therefore, at this T, there are two composition ranges at which a combination of gas and

liquid is more stable. At lower T, G of gas moves up faster than G of liquid due to the entropy

difference, so the intersections move closer together until finally the two curves touch each

other at a single point. The composition at this point is the so-called azeotrope; at this concentration, the mixture boils at a well-defined boiling temperature, just as a pure

substance would.

17

Water-Ethanol Mixture

For the water-ethanol mixture, the azeotrope concentration corresponds to ~95% of ethanol

in the mixture. This is the limit that can be

reached by distillation of a less-alcohol-rich

mixture.

Activities (solvent) The activity of the solvent approaches the mole fraction as xA

1

1A A Aa x as x

• A convenient way of expressing this convergence is to introduce the

activity coefficient

1 1A A A A Aa x as x

• The chemical potential of the solvent becomes

0 ln lnA A A ART x RT

The Phase Rule

• The Gibbs phase rule is a fundamental relation between the number of components in a chemical system, the number of phases present, and the number of variables that can be independently varied while maintaining equilibrium (the variance, D).

• Consider a system of C components with f coexisting phases. How many free parameters are there?

• Total number of parameters: P, T, and C–1 compositional parameters for each phase = (C+1)f

• Total number of constraints: • P must be equal in all phases: f–1 constraints • T must be equal in all phases: f–1 constraints • for each component must be equal in all phases: C(f–1) • in special cases (critical, singular points, etc.), other constraints

• Remaining degrees of freedom: (C+1)f – (C+2)(f–1) = C – f + 2

D C f 2 other

Spontaneity involves

S

H

G = H – TS

Spontaneity is favoured by increasing S and

H is large and negative.

18

Tabulated data of Gfo can be used to calculate

standard free energy change for a reaction as

follows:

)(reactantsGm(products)GnG of

of

o

Stoichiometric coefficients

Free Energy and Temperature

How is change in free energy affected by

change in temperature?

G = H – TS

H S -TS G = H - TS

+

+ +

+

+

+ - - -

-

-

-

-

+ at all temp

- at all temp

- at high temp

+ at low temp

+ at high temp

- at low temp

Class test

A steam power plant operates on the ideal reheat

Rankine cycle. Steam enters the high pressure

turbine at 8 Mpa and 500 C and leaves at 3 Mpa.

Steam is then reheated at constant pressure to

500 C before it expands to 20 kPa in the low

pressure turbine. Determine the turbine work

output, in kJ/kg and the thermal efficiency of the

cycle. Also show the cycle on a T-s diagram with

respect to the saturation lines.