Introduction to thermodynamics - PowerPoint Presentation

Slide1

Introduction to thermodynamics

1. Basics of chemical thermodynamics2. Methods of thermodynamic value measurements

Slide2

Materials Chemistry: Thermodynamic Modeling

Phase diagram Microstructure Properties

Mechanical properties;

Oxidation resistance;Thermal conductivity;Durability;Ionic Conductivity;

Kinetics

Micro-

structure

design

Graphical presentation of phase/assemblage stability Z(

T

,

p

,x

)

Phase diagram can be constructed:

1) based on phase equilibrium studies2) calculated based on thermodynamic information

Equilibrium corresponds to the minimum o

f

the Gibbs energy of the system including all possible phases from 1 to k;

 

ni is amount of phase i and is the Gibbs energy of phase i, which depends on temperature pressure and composition xj

Slide3

Materials Chemistry: Thermodynamic Modeling

The aim of this course is to consider different models describing G=f(T,p,x

j) for various solid, liquid and gas phases. Some thermodynamic functions which are model parameters can be determined experimentally. The experimental methods of thermodynamic parameters determinations will be considered.

Phase diagrams

Experimental thermodynamic values

CALPHAD method

Thermodynamic description of all phases possible in the system stored in the databases

Slide4

Basic

definitionsSystem is a part of Universe chosen for investigation.

Properties are defined and measurable features of a system.Intensive variables are independent on amount of materials present. They are not additive.Extensive variables are proportional to the amount of materials. They are additive.

System boundaries: System is separated from surroundings by boundaries.Isolated systems no exchange of energy or matter with the surroundings.Closed systems can exchange energy with surrounding, but transfer of matter is prohibited.Open system can exchange both energy and matter with the surroundings.

Phase

is homogeneous part of the system having the same properties and composition and separated by interface from other part of the system.

Components

are minimal numbers of constituents, which can describe composition of any phase of the system and can be separated from the system and exist independently. Elements or compounds can be selected as components

.

Constituent

(Species) are particles which can exist in the system. They can be elements, molecules or ions.

Slide5

A phase possesses internal energy due to kinetic and potential energy of it molecules or atoms. Change in internal energy is manifested as change of temperature.

The first law is the principle of conservation of energy. In the closed system the heat transferred to a system is spent to change of internal energy and work done by the system: Q=DU+A,

where Q is heat transferred to the system, DU is the change of internal energy and A isthe work done by the systemWork A is positive when the work is done by the system and

negative when work is done on the system (convention), but it can be accepted vise versa.Internal energy U is function of state of the system, since it does not depends on the way passed by the system. Work and heat are not functions of state, since they depend on the way passed by the system.dQ=dU+d

A

Law of Hess: Heat effect of the reaction does not depend on intermediate stages and it is determined by nature and state of initial substances and final products. Reactions can be exothermic (releasing heat ) and endothermic (consuming heat).

Law of Joule: When gas expands without doing work into a chamber that has been evacuated and without taking in or giving out heat, the temperature of gas does not change. Therefore internal energy is not changing though the volume of gas changes. The law is valid for ideal gas.

First law of thermodynamics

Slide6

The

isochoric process V= const, dV=0 , work A=0

Q=DU=nCVDT;CV

is heat capacity at constant volume; for ideal gas it does not depend on temperature and volume; The isobaric process P=constThe

isothermal

process T=

const

A=P

D

V

Q=

D

U+PDV

The adiabatic process d

Q=0 dU+PdV=0 dA=-nCvdT A=nC

v(T1-T2)=CV(P1V

1-P2V2)/RDT=0 D

U=0 Q=A=

 For ideal gas PV=nRTQ=dV=nRTln

 

PV

=const at T=constQ=nRTln

 

Kinds of thermodynamic processes

Ideal gas

is a theoretical gas composed of many randomly moving point particles that do not interact (no repulsion or attraction between atoms) except they collide elastically. Internal energy U depends only on T for ideal gas.

PV=

nRT

(

gas law for ideal gas) equation of state of ideal gas; heat capacity at constant P and constant V are related as C

P

=C

V

+R ; C

P

/C

V

=

g

; R=8.314 J

.

K

-1

mol

-1

is universal gas constant

For

n=1

C

V

dT+PdV

=0

can be rearranged to

=0;

after integration

;

;

 

When the system undergoes change from one thermodynamic state to final state due change in properties like temperature, pressure, volume etc., the system is said to have undergone thermodynamic process.

Slide7

Second law of thermodynamics introduces

entropy S as function of state dQ/T=

dSThe heat can not spontaneously transfer from cold to the hot body. It is not possible to reach temperature of 0 K.Classical thermodynamics approach:Carnot cycle: considering reversible processes consisting from two isothermal processes at T1 and T2 (T1

>T2) and two adiabatic processes (in isolated systems).

Carnot cycle on P-V diagram

Second law of thermodynamics

At the first stage gas gets heat Q

1

from warm reservoir and expands up to V

2

at constant temperature T

1

. The work done by the gas is A

1

. Then gas is isolated from warm reservoir and expands adiabatically up to volume V3. The work done by the gas is A2 and temperature decreases to T2

. At the third stage gas is in contact with cold reservoir with temperature T2. Gas gives heat Q2 to the cold reservoir and is compressed to volume V4 isothermally. The volume V4 is selected in a way that during adiabatic compression gas returns in it initial state with volume V1 and temperature T1.

;

(

);

;

(

);

;

 

At the first and second stages work is done by the system and at the third and fourth stages work is done on the system to return it in initial state. Efficiency of Carnot cycle is equal to:

Carnot cycle can be reversed , in which it operates as refrigerator.

Slide8

Statistical thermodynamics approach

Physical meaning : S=kBlnWkB – Boltzmann

constant equal to R/NA, where NA is Avogadro number, W – number of different microscopic states available in the system.

Example:A and B atoms can randomly occupy N positions W=

Using

Stirling’s

formula ln(N!)



Nln

(N)-N and neglecting –N for large N, taking into account

x

A

=N

A/N and xB=NB

/N and kN=R (N is Avogadro number)S=k  ln(W)  k[N  ln(N)-N

A  ln(NA)-NB  ln(N

B)]=-R(xAlnxA+xBlnxB)

 Second law of thermodynamics

The entropy is a measure of the amount of energy which is unavailable to do work. It is impossible to construct a heat engine that operating in the cycle produce no effect other than adsorption of energy from one reservoir and performance of an equal amount of work.Entropy is a measure of disorder and a measure of multiplicity of the system.

Slide9

Third law of thermodynamics

Theorem of Nernst: When T  0 K, heat capacity CP0, thermal expansion coefficient

aV 0 and entropy S0.The entropy of pure element or substance in a perfect crystalline form is zero at absolute zero (Planck).At absolute zero, the system must be in a state with the minimum energy state and the perfect crystal has only one minimum energy state.

It is impossible to reduce the temperature of any system to zero temperature in a finite number of finite operations.

Assuming that the temperature can be reduced in adiabatic process by changing the parameter B from B

2

to B

1

. If there is entropy difference at absolute zero then it can be reached in a finite number of steps. However there is no entropy difference so infinite number of steps would be needed

.

Cp

,

Cv

, J/(

mol

K)

C

p is heat capacity at constant P Cv is heat capacity at constant V

Thermal expansion coefficient

a

V 0when

T  0 K

Slide10

Functions of state

and interrelations between themInternal energy

U=TS-PV dU=TdS-PdVEnthalpy

H=U+PV dH=TdS+VdPHelmholtz energy F=U-TS dF=-SdT-PdV

Gibbs

energy

G=U-TS+PV=H-TS

dG

=-

SdT+VdP

Maxwell

equations (second derivative):

Slide11

Thermodynamic equilibrium and reversible processes

Thermodynamic equilibrium is internal state of single thermodynamic system, or relation between several systems connected by permeable or impermeable walls. In thermodynamic equilibrium there is no net macroscopic flows of matter or of energy, either within a system or between systems. In a system in the state of internal thermodynamic equilibrium, no macroscopic change occurs.

A reversible process is a process whose direction can be "reversed" by inducing

infinitesimal changes to some property of the system via its surroundings, while not increasing entropy. Throughout the entire reversible process, the system is in thermodynamic equilibrium with its surroundings. Since it would take an infinite amount of time for the reversible process to finish, perfectly reversible processes are impossible. However, if the system undergoing the changes responds much faster than the applied change, the deviation from reversibility may be negligible. A quasi-static process is a

thermodynamic process

that happens slowly enough for the system to remain in internal

equilibrium

. Any

reversible process

is necessarily a quasi-static one. However, some quasi-static processes are irreversible, if there is heat flowing (in to or out of the system) or if entropy is being created in some way.

In a

reversible cycle, the system and its surroundings will be returned to their original states.

Thermodynamic processes can be carried out in one of two ways: reversibly or irreversibly. Reversibility refers to performing a reaction continuously at equilibrium. In an ideal thermodynamically reversible process, the energy from work performed by or on the system would be maximized, and that from heat would be minimized.

Slide12

Conditions for equilibrium and spontaneous change

dG=0 (minimum G) Equilibrium at constant T and P (T and P are characteristic state variable)

dG<0 Spontaneous change at constant T and PdS=0 (maximum S) Equilibrium

at constant U and V – isolated system(U and V are characteristic state variables)dS>0 Spontaneous change at constant U and VdF=0 (minimum F) Equilibrium at constant T and V(T and V are characteristic state variables) dF<0 Spontaneous change at constant T and V

Functions of state have an

extremum

at equilibrium if the characteristic state variables are kept constant

Slide13

Heat capacity

Compressibility and expansion

Heat capacity is amount of heat needed to transfer to a body to raise its the temperature by 1 K

b

is compressibility,

a

is thermal expansion

Kirchhoff‘s law

Standard

entropy at

298 K

-

=

T

 

Slide14

The Gibbs-Helmholtz

equation The Clausius-Clapeyron

equation

G

=

H

-

TS

Maxwell equation

For transformation at constant P and T

For transformation

liq

=gas

V

gas

>>

V

liq

using equation of state for ideal gas RT=PV

after

integration

Ideal

gas approximation at temperatures much below critical point

1/T

lnP

D

H is latent heat of evaporation

Slide15

Partial

molar properties and chemical potentialsFundamental equations for open systems

m

i

–

chemical

potential

- partial molar

volume

dU

=

TdS-PdV+

Sm

i

dn

i

dH

=

TdS+VdP+SmidnidF=-SdT-PdV+Smidni

dG=-SdT+VdP+Smidni

Partial molar properties are intensive parametersThese equations are valid for open or close systems where composition changes due to mass transfer or reaction between species

Composition as a variable

n

i

– number of moles of component or species

Thermodynamic functions can depend on composition e.g.

G(P, T, n

1

, n

2

, …

n

k

)

Slide16

Gibbs-

Duhem equationdG=

Smidni+Sni

dmi

in

equilibrium at constant

T

and

P

-

SdT+VdP

=

S

n

i

dmi

The Gibbs energy change due to adding or removing material at constant P and T

other forms of the Gibbs-

Duhem

equation

Slide17

Raoult‘s

law, ideal solutionsPa

rtial pressure of solvent (component 1) is proportional to its mole fraction in the solution. The law is valid for dilute solution of not volatile component in volatile solvent.

P

1

is pressure of solvent over solution, x

1

– mole fraction of solvent,

P

0

1

is pressure of saturated vapour of solvent over pure solvent

Ideal solution is a solution for which each component

i

obeys

Raoult‘s

law and

chemical

potential for each component i is expressed:

In real

solutions

a

i

=

g

i

x

i

a

i

is activity,

g

i

is coefficient of activity

Slide18

Activity, coefficient of activity, standard state

Definition of activity of component i is equal

:

m

i

–

chemical potential

-

chemical potential in standard state

Standard state can be selected as pure component at given P and T or as state of infinite dilute solution

Coefficient of activity

g

i

relates activity to measured concentration

a

i

=gixi

Activity is “effective” concentration at which phase would behave as ideal solutionSubstituting mi in equation of Gibbs-Duhem by

different form of Gibbs-

Duhem

equation can be derived

Slide19

System

Cu-Ni

When x

i

1

g

i

1

and solution is close to ideal

Slide20

Ideal solutions and real solutions: positive and negative deviation from ideal

behaviour

Ideal

solution

G=x

i

G

0

i

+x

j

G

0

j

+RT(xilnxi+x

jlnxj)

Real

solutions

G

m

=xiG0i

+xjG0j+RT(xilnxi+xjlnxj)+0L

ijxixj

x

1

x

2

0

L>0

0

L<0

x

1

x

a

a

x

G

x

G

G

x

x

Model of regular solution

Slide21

Henry‘s law for diluted solutions

C=K∙P, where C is concentration of component in liquid (mol

/l), K – temperature dependent parameter and P is pressureConcentration C=ngas/Vliq; For ideal gas nRT

=PVgas and law of Henry for ideal gas is expressed as PVgas/(RTVliq)=KP and Vgas/Vliq=KRTCoefficient of solubility

V

gas

/

V

liq

of ideal gas does not depend on gas pressure.

activity

concentration

Roult‘s

law

x

i

1 and solutions are ideal;

Henry‘s law

x

i

0 and solutions

are not ideal, but activity linearly depends on concentration

a

i

=

g

0

i

x

i

,

where

g

0

i

is

Henrian

parameter or activity coefficient in infinite dilute solution

Solubility of gas in liquid is proportional to pressure of this gas over liquid

Slide22

Models for very dilute solutions

ai

=g0ixi

,where g0i is Henrian parameter or activity coefficient in infinite dilute solution

Wagner‘s formalism

(

i

2)

g

0

i

is activity coefficient at infinite dilution x

1

1,

e

ij

are first-order interaction parameters

Comparison with regular solution model:

Wagner‘s dilute solution modelRegular solution model

For very dilute solutions

Darken‘s

quadratic formalism

1

is solvent,

a

12

and C

2

are constants

If C

2

=0 the

Darken‘s

model is equivalent to regular solution model

Slide23

Chemical equilibrium, constant of reaction

Reaction can occur in multicomponent homogeneous systemat constant P and T

Smidni=0mi=m

0i+RTlnaiaA + bB = cC + dD

D

r

G

0

+RTlnK

a

=0

i-

reactant, j- products,

n

-stoichiometric coefficients

for gas phase

for condensed phase

Slide24

Examples: heterogeneous and homogeneous reactions

Fe + CO2 =

FeO + CO3 phases participate in the reaction Fe, FeO and gas (CO, CO2)

2 CO + O

2

= 2 CO

2

Reaction occurs in single phase gas

Slide25

The elevation of boiling point and the depression of freezing point

using Gibbs-

Helmholz

equation

for dilute solutions

T

0

f

∙T

f

=T

0

f

2

A –

is

solvent

lnx

A

=

ln

(1-x

B

)=-x

B

-1/2(

x

B

)

2

-1/3(

x

B

)

3

….

D

T

f

=T

f

0

-T

f

D

T

f

=

K

f

∙x

B

D

H

f

- latent

heat of fusion

Similar way

D

T

b

=

K

b

∙xB, where DTb=Tb-Tb0, DHv - heat of vaporization

lnxA can be expanded as a Taylor series

Kf is cryoscopic constantKb is ebullioscopic constant

Slide26

D

Tb=Kb∙xB, where

DTb=Tb-Tb0,

D

H

v

- heat of vaporization

K

b

is

ebullioscopic

constant

pure

P

eq

T

b

0

T

b

The elevation of boiling point

Clausius-Clapeyron

equation

Raoult‘s

law

lnx

A

=

ln

(1-x

B

)

ln

(1-x)=-x-x

2

/2-x

3

/3-x

4

/4…. (Taylor

series

)

T

b



T

b

0

Simplifications

P

1

0

P

1

pure

solution

Slide27

Gibbs-

Konovalov rule

p

1

/p

2

=y

1

/y

2

,

where y

1

and y

2

are mole fractions of components 1 and

2 in gas

Vapour

is enriched by component, which increases pressure

at constant T

or decreases temperature of boiling at constant pressure

In extremum points composition of liquid and solid are equal

A

B

A

B

P

Liq

Gas

Gas

Liq

Using

d

ln

p

=

dp

/

p

Total

pressure P=p

1

+p

2

(p

1

and p

2

are partial pressures

)

For equilibrium liquid + gas in two-component system, the Gibbs –

Duhem

equation can be presented as

x

1

d

ln

p

1

+

x

2

d

ln

p

2

=0

, where x

1

and x

2

are mole fractions in liquid phase, p

1

and p

2

are partial pressures of component 1 and 2 in gas phase.

Slide28

Gibbs-

Konovalov rule for ternary system

x

l

x

s

B

A

C

T

B

>T

A

>T

C

s

l

BAC

A

BC

T

A

<T

1

<T

B

T

C

<T

2

<T

A

T

3

=T

A

l

s

x

s

x

l

A has little influence to T, C decrease T, B increase T, tie lines are parallel to side B-C or slightly deviate

l

s

x

l

x

s

A and C decrease T, B increases T

s is enriched by B,

l is enriched by C and A, but ratio A/C and B/A are larger in s.

C decrease T, B and A increases T

s is enriched by B and A,

l is be enriched by C, ratio A/C and B/A are larger in s

Slide29

Heterogeneous

equilibria,Gibbs phase rule

f=C+2-P, where f is degrees of freedom, C- number of component, P- number of phases

m

i

(

P,T,x

i

)

S

x

i

=1

Number of equations C(P-1)

Number of unknown parameters P(C-1)+2, 2 parameters are P and T

f= Npar-Neq=P(C-1)+2-C(P-1)=C-P+2at constant pressuref=C-P+1

a

b

g

x

a

B

A

x

b

B

x

g

B

B

Slide30

Methods of measurements of thermodynamic quantities

Adiabatic calorimetry

CP from very low temperatures to temperatures around 400 K Q is heat measured in calorimeter with sample of mass m Q0 is heat in empty crucible, DT is increase in temperature

Standard entropy can be calculated by CP/T integration

at

T< 20 K C

P

=aT

3

Low temperature heat capacity measurements and entropy determination

Slide31

Solution

calorimetry:measure enthalpy of

solution (in HF, HF-HCl, 2PbO∙B2O3, 3Na2O ∙ 4MoO3, (Na0,48

,Li0,52)BO2Direct reaction calorimetry: measure heat of reaction initiated in closed insulated containerEnthalpy

of

formation

N

Reaction

D

r

H (kJ)

(1)

(2)

(3)

(4)(5)La2O

3(s, 298.15 K)→La2O3(s, 973 K) La2O3(s, 973 K)→dilute solution

in 2PbO.B2O3 (973 K)Y2O3(s, 298.15 K)→Y2O3(s, 973K)Y2O3(s, 973 K)→ dilute solution in 2PbO

.B2O3 (973 K)

LaYO3(s,298.15 K)→ dilute solution in 2PbO.B2O3 (973 K)

83.83 [31]-126.0±4.4 [29]81.94 [31]

-61.7±1.1 [30]39.9±2.4 *

 

(6) 0.5La

2

O

3

(s,298.15 K)+0.5 Y

2

O

3

(s,298.15 K) = LaYO

3

(s,298.15 K)

D

r(6)

H = 0.5

D

r(1)

H + 0.5

D

r(2)

H + 0.5

D

r(3)

H + 0.5

D

r(4)

H -

D

r(5)

H -50.87±4.0

Enthalpy

of

formation

of

LaYO

3

by drop-solution calorimetryxA+yBAxByReagents A and B mixed in ratio x/y in crucible are dropped into calorimeter at TcIf reaction is complete, the total amount of heat is measured:Q1=DH(C,Tc-Tr)+DH(A,Tc-Tr)+DH(B,Tc-Tr)+DfH(AxBy,Tc)The second drop already reacted sampleQ2=DH(C,Tc-Tr)+DH(AxBy,Tc-Tr)Q1-Q2=DfH(AxBy) at 298 K

Slide32

Combustion bomb

calorimetry: measure of combustion reaction at constant volume xM+y/2O2=

MxOyPartial

enthalpy measurements:

D

U

comb

=C

V

D

T,CV

is heat capacity of calorimeter and contentDHcomb=

DUcomb+DPV

DHcomb=DUcomb+RTcomb

Dngas

Small amounts of metal

i

are dropped into calorimeter containing melted element

j, Fi –area under measured temperature-time curveDH298T0 is enthalpy of heating from 298 to T0 (T0 is initial temperature of calorimeter at x=0, W – thermal equivalent of calorimeter, DT(x)=T(x)-T0,

DHTr,i – enthalpy of fusion if T<Tfus,i

, CP,i – heat capacityCalibration using substance with known DHcomb

(benzoic acid)

Slide33

Enthalpy of transformation

DTA – differential thermal analysis

DSC – differential scanning calorimetry

D

H=KA

K

is constant,

A area under the curve

Enthalpy of transformation

Slide34

Heat capacity/enthalpy increment measurements at elevated temperatures

Drop calorimetry: sample heated up to measured temperature is dropped into calorimeter at room temperature and temperature rise is measured to derive enthalpy

Inverse drop calorimetry sample with room temperature is dropped into hot chamber with measured temperatureDifferential scanning

calorimetry

HF

sample

,

HF

St

and

Hf

blank

are heat flow through sample, standard material and empty crucible,

M

St

–mass of standard material,

M

sample – mass of sample

Slide35

The Gibbs

energy and activity measurementsVapour Pressure data

Electrochemical methods

P

0

i

– partial pressure of pure component

i

, P

i

–

partial pressure of solution

Knudsen effusion cell:

Sample is evaporated under vacuum, equilibrium is attained, molecular beam is formed from effusing

vapour

and directed to ionization chamber of mass-spectrometer for measurement of species intensity

k is

instrument

constant, Ij is ion intensity, sj

is ionisation cross section of molecular species jStatic methods, the dew-point and non-isothermal isopiestic methods, the Knudsen effusion and Langmuir free-evaporation method

F- Faraday

constant, n – charge

, E – EMF

Zn

(

liq

)|ZnCl

2

in (

LiCl-KCl

)|

Zn-Pb

(

liq

)

Zn-2e→Zn

+2

Cu

+2

+2e →

Cu

=-

nEF

 

Zn

(pure)



Zn

(in

Pb

solution

)