1. Introduction

2.Postulates of thermodynamics

3. Thermodynamic equilibrium in isolatedand isentropic systems

4. Thermodynamic equilibrium in systemswith other constraints

5. Thermodynamic processes and engines pp. 1–92

6.Thermodynamics of mixtures(multicomponent systems)

7. Phase equilibria

8. Equilibria of chemical reactions

9. Extension of thermodynamics foradditional interactions (non-simple systems)

10. Elements ofequilibrium statistical thermodynamics

11.Towards equilibrium – elements of transport phenomena

pp. 92–303

Appendix pp. 305-328

Table of contentsTable of contents

1. Introduction2. Postulates of thermodynamics3. Thermodynamic equilibrium in isolated and isentropic systems

4. Thermodynamic equilibrium in systems with other constraints

5. Thermodynamic processes and engines pp. 1–926. Thermodynamics of mixtures (multicomponent systems)7. Phase equilibria

8. Equilibria of chemical reactions9. Extension of thermodynamics for additional interactions

(non-simple systems)

10.Elements of equilibrium statistical thermodynamics11.Towards equilibrium – elements of transport phenomena pp. 92–303 Appendix pp. 305–328

Table of contentsTable of contents

Appendix

F1. Useful relations of multivariate calculus

F2. Changing extensive variables to intensive ones:

Legendre transformation

F3. Classical thermodynamics: the laws

Fundamentals of Fundamentals of postulatorypostulatory thermodynamicsthermodynamics

An important definition: the thermodynamic system

The objects described by thermodynamics are called thermodynamic systems. These are not simply “the partof the physical universe that is under consideration” (or in whichwe have special interest), rather material bodies having aspecial property; they are in equilibrium.The condition of equilibrium can also be formulated so that thermodynamics is valid for those bodies at rest for which the predictions based on thermodynamic relations coincide with reality (i. e. with experimental results). This is ana posteriori definition; the validity of thermodynamic description can be verified after its actual application.However, thermodynamics offers a valid description for an astonishingly wide variety of matter and phenomena.

PostulatoryPostulatory thermodynamics thermodynamicsA practical simplification: the simple system

Simple systems are pieces of matter that are macroscopically homogeneous and isotropic, electrically uncharged, chemically inert, large enough so that surface effects can be neglected, and they are not acted on by electric, magnetic or gravitational fields.

Postulates will thus be more compact, and these restrictions largely facilitate thermodynamic description without limitations to apply it later to more complicated systems where these limitations are not obeyed. Postulates will be formulated for physical bodies that are homogeneous and isotropic, and their only possibility to interact with the surroundings is mechanical work exerted by volume change, plus thermal and chemical interactions.

Postulates Postulates of thermodynamicsof thermodynamics1. There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy U, the volume V, and the amounts of the K chemical components n1, n2,…, nK . 2. There exists a function (called the entropy, denoted by S ) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states.3. The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a strictly increasing function of the internal energy.4. The entropy of any system is non-negative and vanishes in the state for which the derivative (∂U /∂S )V,n= 0. (I. e., at T = 0.)

Summary of the postulatesSummary of the postulates

(Simple) thermodynamic systems can be described by K + 2 extensive variables.

Extensive quantities are their homogeneous linear functions.Derivatives of these functions are homogeneous zero order.

Solving thermodynamic problems can be done using differential- and integral calculus of multivariate functions.

Equilibrium calculations – knowing the fundamental equations – can be reduced to extremum calculations.Postulates together with fundamental equations can be used directly to solve any thermodynamical problems.

Relations of the functions Relations of the functions SS and and UU

S (U, V, n1, n2,… nK) is concave, and a strictly monotonous function of U

S

U

X

S = S 0 p lan e

U = U 0 p lan e

i

Identifying (first order) derivativesIdentifying (first order) derivatives

We know:at constantS and n (in closed, adiabatic systems):

(This is the volume work.)

K

ii

nVSiSV

dnn

UdV

V

UdS

S

UdU

ij1 ,,,, nn

PdVdU

PV

U

S

n,Similarly:

at constantV and n (in closed, rigid wall systems):(This is the absorbed heat.)

Properties of the derivative confirm:

TdSdU

TS

U

V

n,

at constant S andV (in rigid, adiabatic systems):(This is energy change due to material transport)The relevant derivative is called chemical potential:

iidndU

i

nVSiij

n

U

,,

Identifying (first order) derivativesIdentifying (first order) derivatives

K

ii

nVSiSV

dnn

UdV

V

UdS

S

UdU

ij1 ,,,, nn

PV

U

S

n,

TS

U

V

n,

i

nVSiij

n

U

,,

is negative pressure, is temperature,

is chemical potential.

The total differential

can thus be written (in a simpler notation) as:

K

iiidnPdVTdSdU

1

Equilibrium calculationsEquilibrium calculationsisentropic, rigid, closed system

impermeable, initially fixed,thermally isolated piston,

then freely moving, diathermal

S α, V α, n α S β, V β, n β

S α + S β = constant; – dSα = dS β

Consequences of impermeability (piston):

n α = constant; n β = constant → dn α = 0; dn β = 0

V α + V β = constant; – dV α = dV β

Equilibrium condition:

dU= dUα + dU β = 0

U α U 

β

0,,,,

dVV

UdS

S

UdV

V

UdS

S

UdU

nSnVnSnV

0 dVPdVPdSTdSTdU

0 dVPPdSTTdU

Equilibrium: Tα = T β and Pα = P 

β

Equilibrium calculationsEquilibrium calculationsisentropic, rigid, closed system

S α, V α, n α S β, V β, n β

Condition of thermal andmechanical equilibriumin the composit system:

U α U 

β

Tα = T β and Pα = P 

β

4 variables Sα , Vα , S β and V β are to be known at equilibrium.

They can be calculated by solving the 4 equations:

T α (S α, V α, n 

α) = T β (S β, V β, n 

β )

P α (S α, V α, n 

α) = P β (S β, V β, n 

β )

S α + S 

β = S (constant)

V α + V 

β = V (constant)

Equilibrium at constant temperature and Equilibrium at constant temperature and pressurepressure

isentropic, rigid, closed system

T = T r and P = P

r (constants)

S r, V r, n r

T r, P 

r S, V, n

T, P

equilibrium condition:the „internal system” is

closedn r = constant and n  = constant

d (U+U r ) = d U + T r

dS r – P

r dV

r = 0S 

r + S = constant; – dS r = dS

V r + V = constant; – dV r = dV

d (U+U r ) = d U + T r

dS r – P

r dV

r = d U + T r

dS – P r

dV = 0

T = T r and P = P

r d (U+U r ) = d U – TdS + PdV = d (U – TS + PV ) = 0

minimizing U + U r is equivalent to minimizing U – TS + PV

Equilibrium condition at constant temperature and pressure:

minimum of the Gibbs potential G = U – TS + PV

Rankine vapor cycle and enginesRankine vapor cycle and engines

refrigerator

B

A A’ E

CD ’

D

B ’

T

S

P u m p

B o ile r

Tu rb in e

C o n d en se r

Wp u m p W o u t

Q in ,1 Q in , 2

Q o u t

A E

D

B C heat engine

B

AE

CD

T

S

W in

AE

BD

Q F ro m co o led ro o m

in

To h o t ro o m Q o u t

E v ap o ra to rL o w p re ssu rev ap o r

C o m p resso rT h ro ttlin g v a lv e

H ig h p ressu re v ap o r

H ig h p ressu reliq u id

L o w p re ssu re liq u id

C o n d en se r

Fugacty Fugacty and interrelation of activitiesand interrelation of activities

Illustration of thethermodynamic definitionof fugacity

Fugacity and Fugacity and interrelation of interrelation of activitiesactivities

Relation of the activities fi (referenced to infinite dilution)

and γi (referenced to pure substance

for the same system

Overview of different activitiesOverview of different activities activity ai name meaning of the standard condition

fi xi

relative activity (activity referenced to Raoult’s law)

(chemical potential of the pure substance)

at anyconcentration 0 ≤ xi ≤ 1

rational activity (activity referenced to Henry’s law)

(chemical potential of the hypothetical pure substance in the state identical to that at infinite dilution)

at any concentration in existingmixtures

molality basisactivity

(chemical potential of the hypothetical ideal mixture at concentration = 1 mol/kg in the state identical to that at infinite dilution)

in solutions

concentrationbasis activity

(chemical potential of the hypothetical ideal mixture at concentration = 1 mol/dm3 in the state identical to that at infinite dilution)

in solutions

fugacity (chemical potential of the hypothetical ideal mixture at a reference pressure φi pi = P)

in everygaseous mixture

),(* PTi

iix x,

iiixxi xRTxPTPT ln),,(lim),(0,

i

i

iim m

m,

i

iiimim m

mRTmPTPT ln),,(lim),(

0, i

i

iic c

c,

i

iiicic c

cRTcPTPT ln),,(lim),(

0, i

P

pii

P

pRTpTT ii

iiii

ln),,,(lim)(0

xip

Phase diagram of a van der Waals fluidPhase diagram of a van der Waals fluid

)13(ln3

8

)13(3

86

)13(

246),( r

r

r

r

r2

r30,rr0

r

0r,

VT

V

T

VdV

V

T

VVVT

V

V

0 1 2 3 4

-7.5

-7.0

-6.5

-6.0

-5.5

D

Tr = 1.1

Tr = 0.75

F

E

C

B

– 0

reduced volume

A

0 1 2 3 40 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

B

Tr = 1 .1

T r = 0 .8 FE

R educed volum e

A

C r

Red

uce

d pr

essu

re

Equilibrium condition: ),(),( ErrBrr VTPVTP ),(),( ErBr VTVT and

2rr

rr

3

13

8

VV

TP

P (V, T ) phase diagram of a pure phase diagram of a pure substancesubstance

V

P

V

P

T

12

4

T

P

12

4

VT

12

4

124

C r

contractingwhen freezing

P (V, T ) phase diagram of a pure phase diagram of a pure substancesubstance

T

P

V

T

P

P

T

V

expanding when freezing

Thermodynamics of phase separationThermodynamics of phase separation

molar Gibbs potential ( g) of(heterogeneous)mechanical dispersionand(homogeneous) mixture

Common tangents

2 components, liquid-liquid

0 .2 0 .4 0 .6 0 .8 10

2 .0

1 .0

x 1

0 .0

0 .5

g R T/

*2

*1

m ech an ica l d isp e rs io n

m ix tu re

10 x 1

mol

ar G

ibbs

pot

enti

al

10 x 1

mol

ar G

ibbs

pot

enti

al

Tc r

T T< c r

T T> c r

Tc r

Thermodynamics of phase separationThermodynamics of phase separation

1 1 10 0 0x B

T 1 T 2 T 3

x B x B

1 1 10 0 0x B

T 4 T 5 T 6

x B x B

g g g

g g g

L1

L

2 b 2 a

2

L3 d3 c

3 b

3 a

L

4 c4 b

4 a

L

5 b6 b

5 a

L

6 a

2 components, solid-liquid

Thermodynamics of phase separationThermodynamics of phase separation2 components, solid-liquid

10 x B

p h ase sep a ra tio n in to

p h ase sep a ra tio n in to + liq u id

liq u id +

T 1

T 2

T 3

T 4

T 5

T 6

1

2 b 2 a 2

3 d 3 c 3 b 3 a

4 c 4 b 4 a

5 b 5 a

6 b 6 a

so li

dso lu tio n

so li

dso lu tio nTe

mpe

ratu

re

liqu id

Other binary solid-liquid phase Other binary solid-liquid phase diagramsdiagrams

ba

a ba b

peritectic reaction compound formation

monotectic reaction syntectic reaction

Three-component Three-component phase diagramsphase diagrams

T A

d ew su rface

p o in t

p ro je c tio n o f th e b o ilin g p o in tcu rv e a t tem p era tu re T

liq u idA

a )

B

CT B

T C

b o ilin g p o in t su rface

va p o r

b )

A B

C

liq u id

va p o r

p ro jec tio n o f th e d ew p o in t cu rv ea t tem p era tu re T

A B

C

T A T B

T Cfreez in g p o in tsu rface

A B

C

liq u id

p h a se

a ) b )

m eltin g p o in tsu rface o f p h a se

p h a se

p h a se

3D diagram

3D diagram

2D projection

2D projection

Factors influencing chemical Factors influencing chemical equilibriaequilibria

Example: 1 ½ H2 + ½ N2 NH3

0 ,2 0 ,4 0 ,6 0 ,8 10

2G /

kJ m

ol –

1

/ m o l eq u ilib riu m

0

– 2

– 4

– 6

4

mixing

reaction

Extension for additional interactionsExtension for additional interactions

surface effects(elements of surface chemistry)

Zn2+

p m em b ran e to av o id m ix in g

o ro u s

s ilv e r p la te

co p p e r w ire

Ag+

co p p e r w ire

z in c p la te

electrically charged phases(elements of electrochemistry)

Energy distribution in canonical Energy distribution in canonical ensemblesensembles

density function ofmultiparticle

energy distribution

density function ofsingle particle

energy distribution

EM E( )

P E( )

( )E

e– E

E N T 1p ( )

T 2

T 1 < T 2 < T 3

T 3

i

i

General interpretation of entropyGeneral interpretation of entropy

disordered ordered

smaller entropy greater entropy

Misunderstandings due to the interpretation as “order–disorder”

Viscuous flow as momentum transferViscuous flow as momentum transfer

Lagrange-transformation (Appendix)Lagrange-transformation (Appendix)

the envelope of the tangent linesdetermines the curve

y

x

Special terms and notation explainedSpecial terms and notation explained

The name comes from the German freie Energie (free energy). It also has another name, Helmholtz potential, to honor Hermann Ludwig Ferdinand von Helmholtz (1821-1894) German physician and physicist. Apart from F, it is denoted sometimes by A, the first letter of the German word Arbeit = work, referring to the available useful work of a system.

The words diabatic, adiabatic and diathermal have Greek origin. The Greek noun διαßασις [diabasis] designates a pass through,e. g., a river, and its derivative διαßατικος [diabatikos] means the possibility that something can be passed through. Adding the prefix α- expressing negation, we get the adjective αδιαßατικος [adiabatikos] meaning non-passability. In thermodynamic context, diabatic means the possibility for heat to cross the wall of the container, while adiabatic has the opposite meaning, i. e. the impossibility for heat to cross. ….

Összefoglalás

• easy-to-follow basis of thermodynamics

• postulates ready-to-use in equilibrium calculations

• detailed discussion of multicomponent systems

• sound thermodynamic foundations ofphase transitions & related equilibriachemical reactions (homogeneous &

heterogeneous)surface chemistryelectrochemistry

• exact explanation of statistical thermodynamics

• elements of nonequilibrium thermodynamics (transport)

• Appendix: calculus + laws of classical thermodynamics

Summing upSumming up

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