1 Basics of chemical thermodynamics 2 Methods of thermodynamic value measurements Materials Chemistry Thermodynamic Modeling Phase diagram Microstructure Properties Mechanical properties ID: 814856
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Slide1
Introduction to thermodynamics
1. Basics of chemical thermodynamics2. Methods of thermodynamic value measurements
Slide2Materials 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
Slide3Materials 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
Slide4Basic
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.
Slide5A 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
Slide6The
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.
Slide7Second 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.
Slide8Statistical thermodynamics approach
Physical meaning : S=kBlnWkB – 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!)
Nln
(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.
Slide9Third law of thermodynamics
Theorem of Nernst: When T 0 K, heat capacity CP0, thermal expansion coefficient
aV 0 and entropy S0.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
Slide10Functions 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):
Slide11Thermodynamic 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.
Slide12Conditions 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
Slide13Heat 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
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
Slide15Partial
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
)
Slide16Gibbs-
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
Slide17Raoult‘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
Slide18Activity, 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
Slide19System
Cu-Ni
When x
i
1
g
i
1
and solution is close to ideal
Slide20Ideal 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
Slide21Henry‘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
Slide22Models 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
Slide23Chemical 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
Slide24Examples: 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
Slide25The 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
Slide26D
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
Slide27Gibbs-
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.
Slide28Gibbs-
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
Slide29Heterogeneous
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
Slide30Methods 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
Slide31Solution
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+yBAxByReagents 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
Slide32Combustion 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)
Slide33Enthalpy of transformation
DTA – differential thermal analysis
DSC – differential scanning calorimetry
D
H=KA
K
is constant,
A area under the curve
Enthalpy of transformation
Slide34Heat 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
Slide35The 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
)