EQUILIBRIUM

1
EQUILIBRIUM
Introduction to Thermodynamics of
Materials David R Gaskell Taylor Francis,
New York (2003).
Thermodynamics and an Introduction to
Thermostatics Herbert B Callen John Wiley and
Sons, New York (2006).
Recommended website http//hyperphysics.phy-astr.
gsu.edu/hbase/heacon.htmlheacon
2
What will you learn in this chapter?
The fields of Thermodynamics and Kinetics are
vast oceans and Chapter 2 will introduce the bare
essentials required to understand the remaining
chapters.

• Stable, Metastable, Unstable Neutral
  equilibrium states
• Thermodynamic variables and potentials

In this text only some aspects will be dealt
with, readers may consult standard texts on
thermodynamics/thermostatics for a other
aspects/detailed account.
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Motivation
These slides are intended to set the stage for
understanding the purpose and power of
thermodynamics and its quantities. Gibbs Free
Energy (G) and Entropy (S) will be in special
focus.
1

• Let us start by performing the following
  (thought) experimentHeat a rod of Al from room
  temperature to 500?C ? As expected the rod will
  expand (A ? B in figure below).
• The expansion occurs because of two reasons 1?
  Vibration of atoms (leading to an increase in
  average spacing between atoms? the usual
  reason)(A ? M in figure below).
• 2? Increase in the concentration of vacancies (a
  vacancy is created when a Al atom goes to the
  surface and for every 4 vacancies created the
  volume equal to 1 unit cell is added). (M ? B in
  figure below).
• The 2nd reason is of subtler origin and must be
  surprising to many readers. Additionally, it is
  a smaller effect in terms of its contribution to
  the overall increase in length of the specimen
  (see solved example link below- it is about 1 in
  10000 effect).

Click here for solved example
Metal expands on heating due to 2 different
physical reasons!
It costs energy for the system to put vacancies
(broken bonds, distortion to the lattice)? then
why does the system tolerate vacancies?
4

• Now let us perform another (thought) experiment
  to put in perspective the previous
  experimentHeat a elastomer (cut rubber band)
  which has been stretched by a small weight by
  about 20?C (room temperature 20?C) ? the
  stretched rubber band will contract!
• The 2nd reason for the expansion of the Al rod is
  closely related to the contraction of the
  stretched rubber band! ? occurs because of
  thermodynamic reasons (quantities like Gibbs Free
  Energy (G) and Entropy (S)), which we shall learn
  in this chapter.
• In the case of the heating of the Al rod- how
  the vacancies form is an issue of kinetics.
  Kinetics will be dealt with in the topic of
  kinetics and chapter on Diffusion.

A stretched elastomer contracts on heating!
5
Motivation
2

• Let us next consider the melting of a pure metal
  at its melting point (MP) (at constant T and P) ?
  by supplying heat to the sample of metal (so that
  the metal sample is only partly molten). At the
  MP the liquid metal is in equilibrium with the
  solid metal.
• The liquid has higher potential energy as well as
  higher kinetic energy than the solid.
• Then why does the liquid co-exist with the solid?
• The answer to this question lies in the fact that
  internal energy is not the measure of stability
  of the system (under the circumstances).
• We will learn in this chapter that it is the
  Gibbs Free Energy (G). The molten metal has
  higher energy (internal energy and enthalpy), but
  also higher Entropy. So the melting is driven by
  an increase in Entropy of the system. The molten
  metal and the crystalline solid metal have the
  same G ? hence they co-exist in equilibrium.

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Stability and Equilibrium

• Equilibrium refers to a state ? wherein there is
  a balance of forces(as we shall see
  equilibrium points have zero slope in a
  energy-parameter plot)
• Stability relates to perturbations (usually small
  perturbations about an equilibrium state) (as
  we shall see stable relates to the curvature at
  the equilibrium points).

Force has been used here in a generalized sense
(as an agent which can cause changes)
Perturbation is usually a small
force/displacement imposed in a short span of
time.
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Equilibrium in a Mechanical System

• Let us start with a simple mechanical system ? a
  rectangular block (Figure in next slide) (under
  an uniform gravitational potential).
• The potential energy (PE) of the system depends
  on the height of the centre of gravity (CG).
• The system has higher PE when it rests on face-A,
  than when it rests on face-B.
• The PE of the system increases when one tilts it
  from C1 ? C2 configuration.
• In configurations such as C1,C2 C3 the system
  will be in equilibrium (i.e. will not change its
  configuration if there are no perturbations).
• In configuration C2 the system has the highest
  energy (point B) and any small perturbations to
  the system will take it downhill in energy ?
  Unstable state.
• Configuration C3 has the lowest energy (point C)
  and the system will return to this state if
  there are small perturbations ? the Stable state.
• Configuration C1 also lies in an energy well
  (like point C) and small perturbations will tend
  to bring back the system to state C1. However
  this state is not the global energy minimum and
  hence is called a Metastable state.
• Additionally, one can visualize a state of
  neutral equilibrium, like a ball on a plane
  (wherein the system is in a constant energy
  state with respect to configurations).

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Mechanical Equilibrium of a Rectangular Block
We start by considering the mechanical
equilibrium of a block- this is to get a first
feel- additional concepts will be required when
dealing with condensed matter systems.
A
Ball on a plane Neutral Equilibrium
B
Centre Of Gravity
C2
C3
C1
B
Unstable
Potential Energy f(height of CG)
Stable
A
Lowest CG of all possible states
Metastable state
C
Configuration
9

• Points to be noted? A system can exist in many
  states (as seen even for a simple mechanical
  system block on a plane)? These states could be
  stable, metastable or unstable? Using the
  relevant (thermodynamic) potential the stability
  of the system can be characterized (In the case
  of the block it is the potential energy, measured
  by the height of the CG for the case of the block
  on the plane)? System will evolve towards the
  stable state provided sufficient activation is
  provided (in the current example the system will
  go from C1 to C3 by sufficient jolting/shaking
  of the plane)

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Kinds of Stability (Equilibrium)

• Three kinds of equilibrium (with respect to
  energy)
• Global minimum ? STABLE STATE
• Local minimum ? METASTABLE STATE
• Maximum ? UNSTABLE STATE
• Constant energy ? Neutral State/Equilibrium

Also next slide

• Kind of equilibrium can be understood by making
  perturbations to the system
• For the mechanical system (block) this
  corresponds to tilting the block
• If the system changes its state after small
  perturbations then the system? is in an unstable
  state
• If the system returns to its original state after
  a small perturbation (tilt) then the system ? is
  in a stable or metastable state (lies in an
  energy minimum)
• If the system returns to its original position
  after small perturbations but does not do so for
  large perturbations then the system ? is in a
  metastable state (not in the global energy
  minimum)
• If there is no change in energy for any kind of
  perturbation then the system ? is in a state of
  neutral equilibrium (e.g. the case of the ball on
  a plane)
• In a 2D system where perturbations are possible
  in more than one direction (i.e. the energy
  landscape is a surface), perturbations in one
  direction may be stable and in another direction
  it may be unstable (like on a surface with
  negative Gaussian Curvature)

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• If the system tends to return to the original
  state after the perturbation ? Stable/Metastable
  state? In Metastable state the system goes to a
  new state if the amplitude of the perturbation is
  large.
• If the system goes to a different state on
  perturbation (the perturbation will tend to take
  the system to a new state very different from
  the original state) ? Unstable state
• If the system goes to the new perturbed state
  without a change in energy (the perturbation will
  tend to take the system to a new state close to
  the original state, as imposed by the
  perturbation) ? Neutral state

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Condensed Matter systems

• In Materials Science we are mainly interested
  with condensed matter systems (solids and
  liquids) (also sometimes with gases)
• The state of such a system is determined by
  Potentials analogous to the potential energy of
  the block (which is determined by the centre of
  gravity (CG) of the block).These potentials are
  the Thermodynamic Potentials (A thermodynamic
  potential is a Scalar Potential to represent the
  thermodynamic state of the system).
• The relevant potential depends on the
  parameters which are being held constant and
  the parameters which are allowed to change. More
  technically these are the State/Thermodynamic
  Variables (A state variable is a precisely
  measurable physical property which characterizes
  the state of the system- It does not matter as to
  how the system reached that state). Pressure (P),
  Volume (V), Temperature (T), Entropy (S) are
  examples of state variables.
• There are 4 important potentials (in some sense
  of equal stature). These are Internal Energy (U
  or E), Enthalpy (H), Gibbs Free Energy (G),
  Helmholtz Free Energy (A or F).? Of these
  internal energy can be conceived as a fundamental
  quantity, while enthalpy a quantity keep track of
  the available heat.? G and A arise due to the
  fact that the entropy of the universe
  increases.

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Intensive and Extensive Properties

• Intensive properties are those which are
  independent of the size of the system? P, T
• Extensive Properties are dependent on the
  quantity of material? V, E, H, S, G

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Thermodynamic potentials and the relation between
them

• There are 4 important potentials (in some sense
  of equal stature). These are Internal Energy,
  Enthalpy, Gibbs Free Energy, Helmholtz Free
  Energy
• The relation between these potentials and the
  state variables is as below.

? TS
U (or E) F (or A) U ? TS
H U PV G U PV ? TS
PV
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The relevant thermodynamic potential determining
the stability of the system

• In terms of the stimulus and response? P ? V
  (Pressure difference drives volume changes)? T
  ? S (Temperature difference drives entropy
  changes)? ? ? Ni (Chemical potential difference
  drives mass transfer) (subscript i stands for the
  ith species)
• The relevant thermodynamic potential which
  characterizes the stability of the system is
  dependent on the state variables which are held
  constant (as in the table below).
• The most conditions are one of Constant P and
  Constant T? Hence G (Gibbs Free Energy) is the
  most relevant thermodynamic potential.
• However, we should remember that depending on the
  state variables being held constant any one of
  the four potentials (U, H, F, G) could be the
  relevant potential. (i.e. all of the potentials
  have the same stature).

Constant V Constant P
Constant S U H
Constant T F G
The main response.
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Internal Energy (U)
1

• Internal Energy (U or E) Kinetic Energy (KE)
  Potential Energy (PE).
• The origin of Kinetic Energy ? Translations,
  Rotations, Vibrations.
• The origin of Potential Energy ? Bonding between
  atoms (interactions in the solid).
• Internal energy is the total of the energies of
  molecules (or atoms or ions..) and their
  interactions.
• The increase in internal energy on heating from 0
  to T Kelvin is given by the equation below where
  CV is the specific heat at constant volume and E0
  is the internal energy of the system at 0K.

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Enthalpy (H)
2
H U PV

• Enthalpy (H) Internal Energy PV (work done by
  the system).
• Measure of the heat content of the system.
• At constant pressure the heat absorbed or evolved
  is given by ?H.
• Transformation / reaction will lead to change of
  enthalpy of system.

• Enthalpy is also known as the heat content of
  the system.
• Why do we need a quantity like enthalpy?Suppose
  we burn a fuel (say petrol) in open air? all the
  energy of the combustion is not available to us,
  as some energy is consumed in making space for
  the products of the combustion. I.e. work is done
  by the system against the surrounding (atmosphere
  at 1atm pressure), to accommodate the products of
  combustion (and this work is not available to us
  for useful purposes).? If the combustion were
  to be carried out in a rigid (closed) container,
  then no work of expansion would have to be
  accounted for.
• So, if internal energy (U) is the quantity of
  energy in a system (and say ?U is being made
  available during an experiment), then H is a
  measure of the part available to us (some gets
  wasted in accommodating the products of the
  reaction? i.e. out of the ?U only ?H is available
  to us). This is true only if the entropy is
  constant (else we will have to get the amount
  available from the free energies (G or A)).

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• Gaseous state is considered as the reference
  state with no interactions.
• For condensed phases PV ltlt E ? H E.
• The increase in enthalpy on heating from 0 to T
  Kelvin is given by the equation below where CP
  is the specific heat at constant pressure and H0
  is the internal energy of the system at 0K (H0
  represents energy released when atoms are brought
  together from the gaseous state to form a solid
  at zero Kelvin).
• Enthalpy is usually measured by setting H 0 for
  a pure element in its stable state at 298 K (RT).

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Entropy (S)
What is entropy a measure of? (We will take this
up later).

• Entropy is perhaps one of the most profound and
  subtle concepts of nature.
• Entropy can be understood looking at a
  Macroscopic picture (interpretation) or a
  Microscopic picture (interpretation) (next
  slide).
• Though these are different approaches to
  understand entropy the result is the same
  entropy.
• In the Macroscopic view we work at the system
  level and worry about observable average
  quantities. In the Microscopic view we go into
  all the details about the system.
• The entropy of an isolated system will increase
  in a spontaneous process (cannot spontaneously
  decrease)? II law.
• Since, the above can be applied to the entire
  universe (a large isolated system!)? the
  entropy of the Universe will always increase
  (here we are not using the word spontaneous as
  the Universe cannot be coaxed into doing a
  non-spontaneous processes!).
• The microscopic interpretation (view) is the
  Statistical Physics/Mechanics picture, which is
  valid for large systems (i.e. systems with a
  large collections of atoms, molecules etc.).
• Entropy is times arrow ? time increases in the
  direction of increasing entropy.
• For a system the Internal energy (U) measures the
  quantity of the energy, while Entropy (S) can be
  thought of a measure of the quality of that
  energy (lower value of entropy will imply a
  better quality? as the entropy of the universe
  always increases).

The universe is in a expanding phase now. If it
were to stop expanding and start contracting ?
entropy of the universe is expected to decrease
in the contracting phase. We are ignoring
wormholes, which connect our universe to other
parallel universes for now (or take those
universes part of ours)!!
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• Due to Ludwig Boltzmann
• Microscopic view
• Statistical Physics Definition
• Theoretical understanding

Warning! Busy slide ? take time to understand

• Due to Rudolf Clausius
• Macroscopic view
• Classical Thermodynamic definition
• Usually what is experimentally measured

Entropy
Two views and not two types
Thermal
Configurational other
The entropy change of a system at temperature T
absorbing an infinitesimal amount of heat ?Q in
a reversible way, is
Zero or ve
? if heat input is at lower T (as compared to
higher T) then for the same heat input (?Q),
there will be a higher increase in entropy.
Boltzmann constant (1.38 ? 10?23 J/K)
No. of different configurations the system
For a system at Constant Energy, many
microscopic states can give a macroscopic state
of identical energy. These microscopic states
could originate from various sources like
Configurational, Electronic, Vibrational and
Rotational (etc.) states. ? In many cases the
configurational term may be the predominant one
considered.

• These two descriptions are equivalent.
• Sometimes 0K there are some degenerate states ?
  in which case the configurational entropy is non
  zero (which can be confirmed by experiments,
  wherein one measures the macroscopic entropy).
  This is called residual entropy.

If heat is desorbed then entropy change will be
negative.
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A note on the equation

• We had noted that the Boltzmann constant arises
  because we try to make a unit Kelvin equal to a
  unit ?C.
• Suppose we measured temperature in terms of ?,
  then entropy would be a pure number.

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Entropy (S)
A simple understanding of entropy

• One way of simply stating the concept behind
  entropy is? A system will, more often than not,
  be found in states with higher
  probability. (This is nothing but the statement
  of the obvious!)
• However, the implications of the above are
  profound. This can be best understood by
  considering the mixing of two ideal gases (or in
  the toy model below as the mixing of 6 circles-
  3 red and 3 blue, on 6 fixed lattice sites).
• Assuming that red and blue circles can move about
  randomly on the fixed sites and also assuming
  that the probability of the occurrence of each
  state is identical (i.e. no state is preferred
  over any other state) there are 20 possible
  configurations as shown in the next slide.
• As seen (from the figure in the next slide) the
  majority of the states (18/20) are mixed states
  and only two are the totally unmixed ones.
• Hence, purely from a probabilistic point of view,
  mixed states occur more often than the unmixed
  ones.
• This implies, if we start with a unmixed
  configuration as in the figure below and the
  system can access all possible states with equal
  probability ? the system will go from a unmixed
  state (of low entropy) to a mixed state (of
  higher entropy).

Two boxes separated by a barrier initially
A
B
Unmixed state
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Unmixed state
Mixed states with various degrees of
mixing

• In the case of two gases initially separated by a
  barrier, which are allowed to mix on the removal
  of the barrier the number of mixed states are
  very large compared to the unmixed states. Hence,
  if all configurational states are accessible, the
  system will more likely found in the mixed state.
  I.e. the system will go from a unmixed state to a
  mixed state (worded differently the system will
  go from order to disorder).
• On the other hand it is unlikely (improbable)
  that the system will go from mixed state to a
  unmixed state. (Though this is not impossible ?
  i.e. a mixed system can spontaneously get unmix
  itself!!)

Note the profoundness of the concept of entropy
comes from its connection to free-energy (via T)
18 mixed states 2 unmixed states
We assume that all states have equal
probability of occurring and are all accessible
Unmixed state
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What is entropy a measure of?

• In literature entropy is considered to be a
  measure of? the degree of randomness in a
  system? the uncertainty about the system? the
  degeneracy of the system.
• The terms randomness, uncertainty, etc. have
  other connotations in other contexts, which can
  lead to confusion to the reader. To avoid this we
  simply state that entropy is a measure based on
  the multiplicity of the system (of microstates of
  a system, corresponding to a macrostate).
• If a system can exist in multiple states
  (corresponding to a particular macrostate), then
  this leads to an increased entropy. For a gas
  (with non-interacting atoms) some of these states
  could be ordered as below (State-A) but more
  likely these will be some random disordered
  state (State-B). Note that the number of
  disordered states is overwhelmingly large.

State-A
State-B
Gas atoms in a box arriving at a regular
configuration by chance.
Gas atoms in a box? usual (disordered) state we
expect.
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Calculating Configurational Entropy

• In the microscopic (statistical mechanics)
  interpretation of Entropy, we take into account
  the multiplicity of microstates which give rise
  to a macrostate
• For example for a constant energy system (with a
  constant number of particles/species and volume),
  the total energy of the system may be obtained
  due to a multiplicity of microstates originating
  from various sources like Configurational,
  Electronic, Vibrational and Rotational states.
• If we consider only configurational entropy for
  now- the multiplicity of states can arise from
  various configurations of the atomic species.
  E.g. If we are talking about a pure crystal of A
  with just one B atom, this B atom could be in any
  one of the lattice positions (all of identical
  energy) giving rise to a multiplicity in the
  microstates. (for now we ignore the surface
  states)
• Note again that for statistical mechanics to be
  valid we have to deal with large systems (for
  illustration purposes we often draw small
  systems!)

Then the configurational entropy is given by
Of course there are many more such
configurations- 3 are drawn for illustration
3 configurations of equal energy
Zero or ve
Boltzmann constant (1.38 ? 10?23 J/K)
No. of different configurations of equal energy
for a constant energy system
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Entropy change due to mixing of two pure elements

• Let us consider the entropy change due to mixing
  of two pure crystalline elements A B (a simple
  case for illustration of the concept of entropy).
• The unmixed state is two pure elements held
  separately. The mixed state (for now assuming
  that the enthalpy of mixing is negative- i.e. the
  elements want to mix) represents an atomic level
  mixing of the two elements.
• Let the total number of lattice sites (all
  equivalent) be N.
• The Entropy of the unmixed state is zero (as in
  pure crystalline elements atoms are
  indistinguishable and hence represent one state).
  Spure A Spure B k ln(1) 0
• In the mixed state the entropy of the system
  increases (Smixed state)
• The number of permutations possible in the mixed
  system is ?

Zero
An useful formula for evaluating ln(factorials)
is the Stirlings approximation
? asymptotically equal, e 2.718
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Entropy change during melting

• At the melting point of a material when heat is
  supplied (?Q) to the material it does not lead to
  an increase in the temperature. Instead, the
  absorbed heat leads to melting? i.e. the energy
  goes into breaking of bonds in the solid and
  consequently a transformation in the state of the
  material (solid ? liquid). The entire process of
  melting takes place at a constant temperature
  (Tm). The heat absorbed is called the Latent Heat
  of Fusion (?Hfusion).
• Suppose we take a mole of Al atoms melt then the
  change in entropy can be calculated as below.
• In the solid state the atoms are fixed on a
  lattice (of course with vibrations!) and this
  represents a low entropy state. On melting the
  entropy of the system increases as the atoms are
  free to move around and many configurations are
  possible. From this point of view often Entropy
  is considered as a measure of disorder (however,
  it must be clear that the phrase measure of
  disorder is used with the understanding of the
  context)

Data Enthalpy of fusion (?Hf) 10.67 kJ/mole,
Melting Point (Tm) 933.4 K (660.25?C)
28
What is the entropy of mixing 0.5 mole of A with
0.5 mole of B on a mole of lattice sites (N0
sites)?
Solved Example
Data
Using
29
Free Energies

• We will take up a brief discussion about free
  energies before considering G and A.
• We all know that there is no free lunch? so
  what is meant by free energy?? It is the
  energy which is available to do work.From the
  second law we know that all the available energy
  cannot be converted into work ? some has to be
  lost as heat (to the surrounding/sink). You
  cannot keep all your earnings? some has to be
  paid as tax (and we are not even including
  hafta here!!
• We have noted that all the internal energy change
  may not be available as heat for us (some may be
  used up to do work to accommodate the products?
  of say a combustion process) and we introduced
  the concept of enthalpy for this.
• ?Needless to say, only a spontaneous process can
  be used for extracting work out of a system. (For
  a driven process we have to supply work!).
• ?In a spontaneous process the entropy of the
  universe has to increase. So
  (?Ssystem?Ssurrounding) gt 0. (E.g. If heat
  eneters the system from the surrounding then
  ?Ssystem will be positive, while ?Ssurrounding
  will be negative).
• To decide the direction of spontaneity of a
  process statement ? is enough ? but then we have
  to do two separate computations (one for the
  system and one for the surrounding).
• If we could come up with a quantity, which helps
  us do calculations only on the system this will
  help. Further if the quantity is a state
  function, then I do not have to worry about the
  reversibility of the heat transfer (remember ?S
  Qreverisble/T).

30
Free Energies cotd

• If entropy is constant in a process, then the
  direction (condition) for spontaneity is obvious
  ? it is the direction in which the internal
  energy (U) or enthalpy (H) decreases. It is the
  entropy tax which is making things difficult.
• Under two conditions this is possible.? If the
  process is carried out under constant T, V.? If
  the process is carried out under constant T, P.
• ? At constant T,V (let Q enter the
  system)Condition of spontaneity At constant
  V there can be no expansion work and if there is
  no other form of work (magnetic, electrical etc.)
  then this energy will lead to an increase in the
  internal energy of the system (?U).
• Note that we do not have to write ?Ureversible as
  U is a state function (depends only on the
  initial and final states).
• Suppose we introduce a symbol A for U?TS.
• For a spontaneous process at constant T, V
• A is called the Helmholtz Free Energy ? it is a
  thermodynamic potential (and a state function as
  U and S are state functions). Hence, if we know
  the initial and final values of A ? we can
  conclude if the process is spontaneous (at
  constant T,V).

Rigid wallsV constant
Heat flow direction
T
Q
T
or equivalently
Zero at constant T
31

• ? At constant T,P (let Q enter the
  system)Condition of spontaneity At constant
  P, if there is only expansion work (no other form
  of work magnetic, electrical etc.) then this
  energy will lead to an increase in the internal
  energy of the system (?H).
• Note that we do not have to write ?Hreversible as
  H is a state function (depends only on the
  initial and final states).
• Suppose we introduce a symbol G for H?TS.
• For a spontaneous process at constant T, H
• A is called the Gibbs Free Energy ? it is a
  thermodynamic potential (and a state function as
  H and S are state functions). Hence, if we know
  the initial and final values of G ? we can
  conclude if the process is spontaneous (at
  constant T,P).

or equivalently
Zero at constant T
32
Helmholtz Free Energy (A)
3

• Helmholtz Free Energy (A or F) E ? T.S
• S is the entropy of the system
• At constant V T, for a process/reaction to
  take place spontaneously the system has to reduce
  its Helmholtz Free Energy. For a system to go
  from state 1 ? 2 the change in F would beF2
  ? F1 ?F (E2 ? E1) ? T (S2 ? S1) ?E ?
  T?SThis change of state would take place
  spontaneously if ?F is Negative
• This implies that reactions which lead to an
  increase in the internal energy (E) are allowed
  (at a sufficiently high temperature) if there
  is a Entropic benefit for the process to
  occur(the concept of entropy will be dealt with
  in the context of Gibbs Free Energy)

?A ?E ? T ?S
used in the general sense
33
Gibbs Free Energy (G)
4
And the concept of Entropy

• Gibbs Free Energy (G) H ? T.S? S is the
  entropy of the system
• For a process/reaction to take place
  spontaneously the system has to reduce its Gibbs
  Free Energy (at constant P T). For a system to
  go from state 1 ? 2 the change in G would
  beG2 ? G1 ?G (H2 ? H1) ? T (S2 ? S1) ?H ?
  T?SThis change of state would take place
  spontaneously if ?G is Negative
• This implies that even Endothermic reactions are
  allowed (at a sufficiently high temperature) if
  there is a Entropic benefit (weighed in with T)
  for the process to occur
• An example of the above is the presence of
  (equilibrium concentration of) vacancies in a
  crystal (more about this later)
• Many a times we are concerned with the relative
  stability of two phases at a given T and P. We
  asks questions such as? at 1 atm. pressure
  50?C, which of the phases is stable ice, water
  or steam? (Answer can be found later in the
  slides).

G H ? T.S
?G ?H ? T ?S
used in the general sense
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• The Slope of the G vs T curve at 0K is zero.
• The curvature is always negative.

Melting transition
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Q A
At 1 atm. pressure 50?C, which of the phases is
stable ice, water or steam?