chapter 1. thermodynamics
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Kinetic processes in materials
Chapter 1. Thermodynamics
Assoc.Prof. Nguyen Hong Hai
Hanoi University of Science and Technology
Textbook: Phase Transformations in Metals and Alloys. D.A. Porter and K.E. Easterling
Student DutiesClass attendance: ≥ 80%Homework: 100%Lab work: 100%
AssessmentMid-term grade: 0.3 (30%)Homework problemsIndependent laboratory project conducted by teamsFinal exam (multichoice and writing): 0.7 (70%)
Kinetic processes in Materials
Content
Rewiew of thermodynamics
Diffusion
Surfaces and interfaces
Phase transformation
Kinetic processes in Materials
Chapter 1. THERMODYNAMICS
The main use of thermodynamics in physical metallurgy is to
allow the prediction of whether an alloy is in equilibrium.
In considering phase transformation we are concerned with
changes towards equilibrium, and thermodynamics is
therefore very powerful tool.
However, the rate at which equilibrium is reached cannot be
determined by thermodynamics alone.
Kinetic processes in Materials
Kinetic processes in Materials
Figure 1a represents the physical space of materials.
The outermost circle is the performance circle,
which characterizes how a material behaves in
given environments.
The performance is realized by a set of properties,
in the second circle, relevant to those environments.
Those properties are achieved by an agglomeration
of phases with various structures and their
morphologies in the third circle.
The structures are obtained through proper processing in the fourth circle
with accurate chemical compositions in the fifth circle.
Kinetic processes in Materials
While the two outermost circles are environment-dependent, the three inner
circles can be directly related to the three key components in materials
science and engineering, crystallography, kinetics, and thermodynamics, as
shown in Figure 1b.
Thermodynamics and kinetics of materials are viewed as difficult subjects.
1.1. Equilibrum
System: mixture
of one or more
phases
Phase: a portion of the
system whose properties
and composition are
homogeneous and which
is physically distinct from
other part of the system.
A356
Kinetic processes in Materials
The main question of Thermodynamics is if a system is in equilibrium?
1.1. Equilibrum
Kinetic processes in Materials
Components: the
different elements
or chemical
compounds which
make up the system.
The composition of
a phase or the
system can be
described by giving
the relative amounts
of each component.
1.1. Equilibrum
Kinetic processes in Materials
T, 0C
1538
1394
G912
500
Fe 1 2 3 4 5 6 7
% C
l
l +
B 14870
l + l + Fe3C
11480
C
E F
+ Fe3C +
P S 7270 K
+ Fe3C
N
Fe3 C
Phase transformation: one or more phases in an alloy (the system)
change into a new phase or mixture of phases.
1.1. Equilibrum
-Al5FeSi - Al8Fe2Si
Stable Unstable
Desired
In reality
Kinetic processes in Materials
What determine the stability of phases?
At constant temperature and pressure the relative stability of a system is determined by its Gibbs free energy
G = H – TS (1.1)
H: enthalpy, the measure of the heat content of the system
H = E + PV (1.2)
E : internal energy of the system, P: the pressure and V: the volume.
For the condensed phases (solids and liquids) the PV term is usually very small in comparison with E, so H E.
T: absolute temperature
S: entropy of the system, the measure of the randomness of the system.
The intensive properties (independent on the size of the system): T and P
The extensive properties (directly proportional to the quantity of material in the system such as V,E,H,S,G)
Kinetic processes in Materials
1.1. Equilibrum
A system is said to be in equilibrum when
it is in the most stable state;
At constant temperature and pressure a
closed system (i.e. one of fixed mass and
composition) will be in stable equilibrum
if it has the lowest value of the Gibbs free
energy, or in mathematical term:
dG = 0 (1.3)
The state with the highest stability will be
that with low enthalpy and high entropy. Metastable (diamond)
Stable (graphite)
Unstable (dG 0)
dG = 0
dG = 0
AB
Arrangement of atoms
GibbsfreeenergyG
Kinetic processes in Materials
1.1. Equilibrum
Any transformation that results in a decrease in Gibbs free energy is
possible. Therefor a necessary criterion for any phase transformation is:
= G2 – G1 0 (1.4)
The answer to the question “How fast does a phase transformation
occur?” belong to the realm of kinetics
The usual way of measuring the size of the system is by the number of
moles of material it contains (the extensive properties are then molar
quantities, i.e. expressed in units per mole).
The number of moles of a given component in the system is given by the
mass of the component in grams divided by its atomic or molecular
weight.
The number of atoms or molecules within 1 mol of material is given by
Avogadro’s number (Na) and is 6.023x1023.
Kinetic processes in Materials
1.1. Equilibrum
Single component system: which contains a pure element or one
type of molecule that does not dissociate over the range of
temperature of interest.
In order to predict the phases that are stable or mixtures that are in
equilibrium at different temperatures it is necessary to calculate the
variation of G with T
1.2. Single component system
Kinetic processes in Materials
1.2.1. Gibbs Free Energy as a Function of Temperature
The variation of enthalpyH with T can be calculated by intergrating equation 1.5
The variation of entropywith temperature can also be derived from the specific heat CP, taking entropy at zero degrees Kelvin as zero
Specific heat: the quantity of heat (in jouls) required to raise the temperature of the substance by one degree Kelvin P
PT
HC
T
PdTCH298
T
P dTT
CS
0
(1.5)
Kinetic processes in Materials
Finally the variation of G with temperature (Fig 1.3) can be
obtained by combining Fig 1.2b and 1.2c using Equation 1.1
TS
H
G
H
T(K)
slope = -S
slope = CP
G
G = H – TS (1.1)
G decreases with increasing T at the rate given by –S
ST
G
P
Kinetic processes in Materials
1.2.1. Gibbs Free Energy as a Function of Temperature
Note:
1. At Tm both liquid and solid
phases have the same value of
G and can exist in equilibrium.
Tm is therefore the equilibrium
melting temperature at the
pressure concerned.
2. At Tm the heat supplied to the
system will not raise its
temperature but will be
supplying the latent heat of
melting (L) that required to
convert solid into liquid (line bc
in Fig. 1.4)
Fig. 1.4. Variation of entalpy (H) and free energy (G) with temperature for the solid and liquid phases of a pure metal.
Kinetic processes in Materials
1.2.1. Gibbs Free Energy as a Function of Temperature
The free energies of the liquid and solid at a temperature T are given by:
GL = HL - TSL
GS = HS - TSS
Therefore at a temperature T:
G = H - TS, [J.mol-1] (1.10)
H = HL – HS
S = SL - SS
If a liquid metal is undercooled by T below Tm before it solidifies,
solidification will be accompanied by a decrease of free energy G (J mol-1)
This free energy decrease provides the driving force for solidification.
Kinetic processes in Materials
1.2.1. Gibbs Free Energy as a Function of Temperature
Molar free energy
Temperature
GS
GL
G
T
TmT
At the equilibrium melting
temperature Tm the free
energies of solid and liquid are
equal, i.e. G = 0.
Consequently:
G = H - TmS = 0
And therefore at Tm
This is known as the entropy of fusion and for most metals is a constant R (8.3 J mol-1 K-1).
mm T
L
T
HS
(1.11)
Kinetic processes in Materials
1.2.1. Gibbs Free Energy as a Function of Temperature
Molar free energy
Temperature
GS
GL
G
T
TmT
Note: For small undercooling (T) the difference in the specific heats
of the liquid and solid (CLP – CS
P) can be ignored.
So, H and S therefore approximately independent of temperature.
Combining Equations 1.10 and 1.11 thus gives:
1.2.2. The driving Force for Solidification
Conclusion: G is proportional to the T (driving force for solidification
is proportional to the undercooling).
mT
LTLG
TST
TLG
m
G = H - TS (1.10)
mm T
L
T
HS
(1.11)
Kinetic processes in Materials
Molar free energy
Temperature
GS
GL
G
T
TmT
Metal Undercooling,
0 C
Metal Undercooling,
0 C
Metal Undercooling,
0 C
Hg 77 Sb 135 Mn 308
Ga 76 Ge 227 Ni 319
Sn 118 Ag 227 Co 330
Bi 90 Au 230 Fe 295
Pb 80 Cu 236 Pd 332
Al 195 Pb 80 Pt 370
The maximum undercooling, T, is obtained by metal atomisation,
which is needed for homogeneous nucleation. Tmax 1/3 Tm
Kinetic processes in Materials
1.2.2. The driving Force for Solidification
How much is maximum undercooling?
In practice the pure metals (single component system) are rarely applied.
In single component system all phases have the same composition, and
equilibrium simply involves pressure and temperature as variables
In alloys composition is also variable
Since pressure is usually fixed at 1atm, most attention will be given to
changes in composition and temperature
1.3. Binary Solutions
Kinetic processes in Materials
The Gibbs free energy of a binary solution of A and B atoms can be calculated from the energies of pure A and pure B
Assumed that A and B have the same crystal structure , can be mixed in any proportion to make a solid solution with the same crystal structure.
Assumed that 1 mol of homogeneous solid solution is made by mixing together XA mol of A vµ XB mol of B.
XA and XB are the mole fraction of A and B in the alloy.
Note: XA + XB = 1 (1.13)
1.3.1. The Gibbs Free Eneggy of Binary Solutions
The Gibbs free energy of the system before mixing is given by:
G1 = XAGA + XBGB J.mol-1 (1.14)0 XB 1A B
GA
GB
G1
Free energy per mole beformixing
Kinetic processes in Materials
After mixing the free energy of the
solid solution, G2 will be:
G2 = G1 + Gmix (1.15)
where Gmix is the change in Gibbs
energy caused by the mixing.
G1 = H1 - TS1
G2 = H2 - TS2
Hmix = H2 - H1
Smix = S2 - S1
Gmix = G2 – G1 = Hmix - TSmix (1.16)
Smix is the difference in entropy between the mixed and unmixed states.
Kinetic processes in Materials
1.3.1. The Gibbs Free Eneggy of Binary Solutions
where Hmix is the heat absorbed or evolved during mixing, representing the difference in internal energy (E) before and after mixing.
It can be called heat of mixing.
If Hmix = 0, the resultant solution is said to be ideal and the free energy
change on mixing is only due to the change in entropy:
Gmix = -T Smix (1.17)
Smix can be found as follows: S = k ln (1.18)
where k is Boltzmann’s constant
is a measure of randomness or the
number of distinguishable ways of arranging the atoms in
the solution.
NA = XA Na ; NB = XB Na
1.3.2. Ideal solution
NA and NB are number of A and B atoms, Na is Avogadro’s number
!!
!
BA
BA
NN
NN (1.19)
Kinetic processes in Materials
By substituting into Equations 1.18
and 1.19, using the Stirling’s
approximation (ln N! N lnN – N) and
the relationship Nak = R
where R is universal gas constant,
we have:
Smix = -R (XAlnXA + XBlnXB) (1.19’)
Gmix = RT (XAlnXA + XBlnXB) (1.20)
Note: Since XA and XB are less than unity, Smix is positive, i.e. there is an increase in entropy on mixing.
The free energy of mixing, Gmix :
Gmix is function of
composition and temperature
Kinetic processes in Materials
1.3.2. Ideal solution S = k ln(1.18)
!!
!
BA
BA
NN
NN (1.19)
G2 = G1 + Gmix
G2 = XAGA + XBGB + RT (XAlnXA + XBlnXB)
Note:
1. The actual free energy of the solution G will also depend on GA
and GB
2. As the temperature increases, GA
and GB decrease and the free energy curves assume a greater curvature
3. The free energy curves must end asymptotically at the vertical axes of the pure components
Kinetic processes in Materials
1.3.2. Ideal solution
For ideal solution a random arrangement of atoms is the equilibrum, or most stable arrangement
In alloy it is interest to know how the free energy of a given phase will
change when atoms are added or removed.
If a small quantity of A, dnA mol, is added at constant temperature and
pressure, the size of the system will increase by dnA and therefore the
total free energy of the system will also increase by a small amount dG’.
If dnA is small enough dG’ will be proportional to the amount of A added:
dG’ = A dnA (T, P, nB constant) (1.22)
The proportionality constant A is called the partial molar free energy of A
or the chemical potential of A in the phase:
1.3.3. Chemical Potential
Note: G’ refers to whole system, whereas
G denotes the molar free energy and is
therefore independent of the size of the
system.
BnPTA
An
G
,,
'
AnPTB
Bn
G
,,
'
Kinetic processes in Materials
Definition: chemical potential is the change of the free
energy of whole system (G’) while a very small amount of
a component is added.
Notes:
For binary solution at constant temperature and pressure
the separate contributions can be summed:
dG’ = A dnA + B dnB (1.24)
If the solution contains more than 2 components and T and P change Equation 1.24 must be added to give the general equation:
dG’ = -SdT + VdP + A dnA + B dnB + C dnC + …
o If A and B are added in the correct proportions (dnA : dnB = XA : XB, for example if XA = 2/3, XB = 1/3, dnA : dnB = 2) , the size of the system can be increased without changing A and B.
o So if Xamol A and XB mol B is added the free energy of the system will increase by the molar free energy. Therefore from (1.24):
G = A XA + B XB J mol-1 (1.25)
BnPTA
An
G
,,
'
AnPTB
Bn
G
,,
'
Kinetic processes in Materials
1.3.3. Chemical Potential
Comparison of Equation 1. 21 and 1.25
gives A and B for an ideal solution as:
A = GA + RT lnXA
B = GB + RT ln XB (1.26)
A and B can be obtained by extrapolating the tangent to the G curve to the sides of the molar free energy diagram
A
BA
XB
B
G
Molar free energy
G2 = XAGA + XBGB + RT (XAlnXA + XBlnXB) (1.21)
G = A XA + B XB (1.25)
Kinetic processes in Materials
1.3.3. Chemical Potential
In reality Hmix 0 (mixing is
endothermic (heat absorbed) or
exothermic (heat evolved).
It is assumed that the heat of mixing,
Hmix, is only due to the bond
energies between adjacent atoms.
Necessary conditions: volumes of
pure A and B are equal and do not
change during mixing so that the
interatomic distances and bond
energies are independent of
composition.
o The structure of a binary solid
solution is shown schematically in
Fig.
1.3.4. Regular Solutions
Three types of interatomic bonds
are present:
A-A bonds each with an energy AA
B-B bonds each with an energy BB
A-B bonds each with an energy AB
Kinetic processes in Materials
The internal energy of the solution, E, will
depend on the number of bonds of each type
PAA, PBB and PAB such that:
E = PAAAA + PBBBB + PABAB
Before mixing pure A and B contain only A-A and B-B bonds, so the change in internal energy on mixing is given by
Hmix = PAB (1.27)
where = AB – (AA + BB)/2 (1.28)
If = 0, Hmix = 0, the solution is ideal, the atoms are completely randomly arranged and
PAB = NazXAXB bonds mol-1 (1.29)
where Na: Avogadro’s number, z is the number of bonds per atom.
If < 0, the atoms in the solution will prefer to be surrounded by atoms of the opposite type and this will increase PAB: PAB > NazXAXB
If > 0, PAB will tend to be less than in a random solution: PAB < NazXAXB
Kinetic processes in Materials
1.3.4. Regular Solutions
In which case the following
equation is good approximation:
Hmix = NazXAXB = XAXB (1.30)
where = Na z (1.31)
Real solution that closely obey
Equation 1.30 are known as regular
solutions.
The variation of Hmix with
composition is parabolic and is
shown in Fig 1. 13 for > 0
A XB B
Hmix
per mol
Kinetic processes in Materials
1.3.4. Regular Solutions Hmix = PAB (1.27)
PAB = NazXAXB (1.29)
In alloys where the enthalpy of mixing is not zero ( and 0) the
assumption that a random arrangement of atoms is the equilibrum, or
most stable arrangement is not true, and the calculated value for Gmix
will not give the minimum free energy.
The actual arrangement of atoms will be a compromise that gives the
lowest internal energy consistent with sufficient entropy, or
randomness, to achieve the minimum free energy.
1.3.6. Real Solutions
= AB – (AA + BB)/2
Kinetic processes in Materials
Gmix = RT (XAlnXA + XBlnXB) (1.20)
In system with < 0 the
internal energy of system
is reduces by increasing
the number of A-B
bonds, i.e. by ordering
the atoms as shown in
figure a.
If > 0 the internal energy
can be reduced by
increasing the number of
A-A and B-B bonds, i.e. by
the clustering of the atoms
into A-rich and B-rich
groups (fig. b).
In systems where there is a size difference between the atoms the quasi-chemical model will underestimate the change in internal energy on mixing since no account is taken of the elastic strain fields which introduces a strain energy terminto Hmix. When the size difference is very large the interstitial solid solution are energrtically most favourable(Fig. c).
= AB – (AA + BB)/2
Kinetic processes in Materials
1.3.6. Real Solutions
If the atoms in a substitutional solid solution are completely randomly
arranged (each atom position is equivalent) PAB, the number of A-B bonds,
in such solution is given by Equation 1.29:
PAB = NazXAXB bonds mol-1
If (= Na z ) < 0 (or < 0) and the number of A-B bonds is greater than
this, the solution is said to contain short-range order (SRO).
The degree of the ordering can be quantified by defining a SRO
parameters such that:
where PAB (max) and PAB(random) refer to the maximum number of bonds
possible and the number of bonds for a random solution, respectively.
1.3.7. Ordered Phases
)((max)
)(
randomPP
randomPPs
ABAB
ABAB
Kinetic processes in Materials
a) Random A_B solution with a total of 100 atoms and XA = XB = 0.5, PAB ~ 100, S = 0.
b) Same alloy with short-range order PAB = 132, PAB(max) ~ 200, S = (132 -100)/(200 – 100) = 0.32
Example of orders
In solutions with compositions that are close to a simple ratio of A:B atoms another type of order as the long-range one (c) can be found. Now the atom sites are no longer equivalent but can be labelled as A-sites and B-sites.
Kinetic processes in Materials
1.3.7. Ordered Phases
Example: ordered substitutional structures in the Cu-Au system
a) high-temperature disordered structure; atoms Cu or Au can occupy any site.
b) At low temperature solution with XCu = XAu = 0,5, i.e. 50/50 Cu/Au mixture, form an ordered structure in which the Cu and Au are arranged in alternate layer; each atom position is no longer equivalent and the lattice is described as a CuAusuperlattice.
c) Alloy with the composition Cu3Au: another superlattice is found.
Kinetic processes in Materials
1.3.7. Ordered Phases
The entropy of mixing of structures with long – range order is extremly small and
with increasing temperature the degree of order decreases until some critical
temperature there is no long-range order at all.
This temperature is a maximum when the composition is ideal required for the
superlattice.
Note: The critical temperature for loss of long-range order increases with increasing
, or Hmix, and in many systems the ordered phase is stable up to the melting point
However, long-range order can still
be obtained when the composition
deviates from the ideal if some atom
sites are left vacant or if some atoms
sit on wrong sites.
In such cases it can be easier to
disrupt the order with increasing
temperature and the critical
temperature is lower.
Kinetic processes in Materials
1.3.7. Ordered Phases
Some most common ordered lattices
Kinetic processes in Materials
1.3.7. Ordered Phases
Often the configuration of atoms that has the minimum free energy after
mixing does not have the same crystal structure as either of the pure
components. In such cases the new structure is known as an intermediate
phase.
Intermediate phases are often based on an ideal atom ratio that results in a
minimum Gibss free energy.
For compositions that deviate from the ideal,
the free energy is higher giving a characteristic
“U” shape to the G curve.
The range of compositions over which the free
energy curve has a meaningful existence
depends on the structure of the phase and
the type of interatomic bonding.
1.3.8. Intermediate Phases
Kinetic processes in Materials
Ideal compositionA B
G X
1.3.8. Intermediate Phases
Gmix
Ideal compositiona)A B
G
GA
XB b)A B
G
GA
GB
In other structure fluctuations in
composition can be tolerated by
some atoms occupying “wrong”
positions or by atom sites being left
vacant, and in these cases the
curvature of the G curve is much less
(b).
When small composition deviations
cause a rapid rise in G the phase is
referred to as an intermetallic compound
and is usually stoichiometric, i.e. has a
formula AmBn, where m and n are
integers.
Kinetic processes in Materials
X
TEM picture showing a Si-rich zone arrounding in the presence of Sr
Kinetic processes in Materials
1.3.8. Intermediate Phases
Al5FeSi - phase, platelete form, very hamful, stable
-Al5FeSi
Al8Fe2Si - phase, Chinese script form, less hamful, but metastable
- Al8Fe2Si
1.4. Equilibrum in Heterogeneous Systems
o The stable forms of pure A and B at a given
temperature and pressure can be denoted as
and respectively
Assumed is fcc and is bcc. The molar free
energies of fcc A and bcc B are shown as point a
and b
The first step in drawing the free energy curve of
the fcc is, therefore, to convert the stable bcc
arrangement of B atoms into an unstable fcc
arrangement. This requires an increase in free
energy, bc
The free energy curve for the phase can now be
constructed as before by mixing fcc A and fcc B. -
Gmix for of composition X is given by the
distance de as usual
Note: A-rich alloys will have the lowest free energy as a homogeneous phase and B-rich alloys as phase.
Kinetic processes in Materials
It is usually the case that A and B do not have the
same crystal structure in their pure states at a given
temperature. In such cases two free energy curves
must be drawn, one for each structure.
Consider alloy X0
Futher reductions in free energy can be
achieved if the A and B atoms interchange
between the and phases until the
compositions e and e are reached (Fig b).
The free energy of the system Ge is now a
minimum and there is no desire for futher
change. Consequently the system is in
equilibrium and e and e are the equilibrium
compositions of the and phases.
Kinetic processes in Materials
1.4. Equilibrum in Heterogeneous Systems
If atoms are arranged as a homogeneous
phase, the free energy will be lowest as , i.e.
G0 per mole.
However, the system can lower its free
energy if the atoms separate into two phases
with compositons 1 and 1, for example (Fig.
a). The free energy of the system will then
reduced to G1.
Kinetic processes in Materials
1.4. Equilibrum in Heterogeneous Systems
This results is quite general and applies to
any alloy with an overal composition between
e and e: only the relative amounts of the
phases change, as given by the lever rule.
When the alloy composition lies outside this
range, however, the minimum free energy lies
on the G ang G curves and the equilibrium
state of the alloy is a homogeneous single
phase.
From Fig. b) it can be seen that equilibrium
between two phases requires that the
tangents to each G curve at equilibrium
compositions lie on a common line.
In the other words each component must
have the same chemical potential in the two
phases, i.e. for heterogeneous equilibrium:
A = A
, B = B
In the free energy curves that have been
drawn so far the surfaces, grain
boundaries and interphase interfaces have
been ignored.
Interphase interfaces can become extremly
important in the early stages of phase
transformations when one phase, , say,
can be present as very fine particles in the
other phase.
P
Atmos-phericpressure
If the phase is acted on by a pressure 1 atm, the phase is subjected
to an extra pressure P due to the curvature of the / interpace.
If is the / interfacial energy and the particles are spherical with
radius r, P is given appoximately by:
1.5. The Influence of Interfaces on Equilibrium
rP
2
Kinetic processes in Materials
P
Therefore the cuvre on the molar free energy-composition diagram in
Fig. 1.20b will be raised by an amount:
By definition, the Gibbs free energy contains a “PV” term: G = H – TS and H =
E + PV so an increase of pressure P therefore causes an increase in free
energy G:
G = V P
Vm is the molar volume of the phase
X
GGr
G
G
XXr XB
This free energy increase due
to interfacial energy is known
as a capillarity effect of the
Gibbs-Thomson effect.
r
VG m
2
Kinetic processes in Materials
1.5. The Influence of Interfaces on Equilibrium
An important practical
consequence of the Gibbs-
Thomson effect is that the
solubility of in is sentitive to the
size of the particles.
From the common tangent
construction it can be seen that the
concentration of solute B in in
equilibrium with across a curved
interface (Xr) is greater than X, the
equilibrium concentration for a
planar interface.
Taking the following typical values: = 200 mJ m2, Vm = 10-5 m3, R = 8.31 J mol-1 K-1, T = 500 K gives:
For r = 10 nm Xr / X 1.1
For particles visible in the light microscope
(r > 1 m) capillarity effects are very small
nmrX
X r 11
Kinetic processes in Materials
X
GGr
G
G
XXr XB
1.5. The Influence of Interfaces on Equilibrium
The thermodynamics can be used to
calculate the driving force for a
transformation: G = G2 – G1, but it can
not say how fast a transformation will
proceed.
The study of how fast processes occur
belongs to the science of kinetics.
1.6. The Kinetics of Phase Transformations
Before the free energy of the atom can
decrease from G1 to G2 the atom must
pass through a so-called transition or
activated states with a free energy Ga
above G1
Ga
G
Finalstate
Innitialstate
Activatedstate
G
G1
G2
Kinetic processes in Materials
As a result of the random thermal motion of the atoms the energy of
any particular atom will vary with time and occasionally it may be
sufficient for the atom to reach the activated state. This process is
known as thermal activation.
The rate at which a transformation
occurs will depend on the frequency with
which atoms reach the activated state.
Putting Ga = Ha - TSa and changing from
atomic to molar quantities enables this
equation to be wrotten as Arrhenius rate
equation
The probability of an atom reaching the activated state is given by
exp (-Ga/kT), where k is Boltzmann’s constant (R/Na) and Ga is
known as the activation free energy barrier.
kT
Grate
a
exp
Arrhenius rate equation
kT
Hrate
a
exp
Kinetic processes in Materials
1.6. The Kinetics of Phase Transformations
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