3.012 Fundamentals of Materials Science
Fall 2003
Lecture 14: 10.27.03 Gibbs free energy: Shapes of things, and stability
Today:
LAST TIME .............................................................................................................................................................................................. 2
STABLE, METASTABLE, AND UNSTABLE EQUILIBRIA ............................................................................................................................... 3
Constraints on thermodynamic properties ensure stability ............................................................................................................... 3
Le Chatelier’s Principle: Materials at equilibrium want to stay there .............................................................................................. 5
REQUIREMENTS FOR THE SHAPE OF FREE ENERGY CURVES1 .................................................................................................................... 6
REFERENCES ........................................................................................................................................................................................... 8
Reading:
Supplementary Reading:
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Lecture 14 – Gibbs free energy & stability
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3.012 Fundamentals of Materials Science
Fall 2003
Last time
Lecture 14 – Gibbs free energy & stability
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2/15/16
3.012 Fundamentals of Materials Science
Fall 2003
Stable, metastable, and unstable equilibria

The condition of equilibrium and constant temperature and pressure is summed up by the equation:
GT  0,P 0  0
(Eqn 1)
…any small change in the system must cause the free energy to rise, if the system is in stable
equilibrium.
The total free energy is:
o


G  H  TS  U  T0S  P0V
(Eqn 2)

If the temperature and pressure are held constant at (T0,P0), then stable equilibrium is identified by finding the
minimum value of G in the landscape of possible U, S, and V values taken by the system. Using a figure from our
first lecture:

o
Thermodynamic equilibrium conditions for the Gibbs free energy in constant temperature and pressure
materials are similar to the conditions for mechanical equilibrium of the potential energy of a ball held at
different heights: Materials can exist in stable, metastable, or unstable equilibrium states. A small change
in the conditions of the system in an unstable state will cause a spontaneous change toward a new
metastable or stable equilibrium.
Constraints on thermodynamic properties ensure stability

How do we define conditions for a stable equilibrium? We have made use of setting the differential of
thermodynamic functions to zero, e.g. dU = 0. However, this only defines an extremum in U- it could be a
maximum or a minimum. A maximum in U represents an unstable equilibrium- any small fluctuation can push the
system in a new direction. So what additional conditions can we prescribe that guarantee the extremum is a
minimum- a stable equilibrium? The equilibrium condition is:
U  0
(Eqn 3)

Lecture 14 – Gibbs free energy & stability
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Fall 2003

f  f (x)  f (x o ) 
129
121
105
113
97
V
df
dx
x o dx 
d2 f
dx 2
dx 
2
xo
...
Our function U is multivariate, so we will make use of the multivariate form of the Taylor expansion:

(Eqn 5)

S
We can use a Taylor expansion for U(S,V) near the extremum point in S,V space to write an expression for U.
As a reminder, a Taylor expansion for a point f(x) near the known point f(xo) is given by:
(Eqn 4)

89
73
S
81
65
49
57
33
41
17
25
1
9
50
45
40
35
30
U 25
20
15
10
5
0

f 
f 
1  2 f
2 f
2 f
2
2
f x, y     dx    dy   2 dx   2 dy   2
dxdy ...
x y
2 x
y
xy
y x

This expansion applied to U is:

(Eqn 6)

U 
U 
1  2U
 2U
 2U
2
2
U    dS    dV   2 dS   2 dV  
(dSdV ) ...  0
S V
V S
2 S
V
VS


The inequality at the end is required for a stable equilibrium. If we are examining a small perturbation near an
extremum, the equation is evaluated at the extremum point- the first derivatives are 0.

FILL IN DISCUSSION OF POSITIVE DEFINITE HESSIAN?
o DIAGONALS >= 0; DETERMINANT >=0 (2X2 MATRIX)

The first requirement to fall out of this analysis is:

Lecture 14 – Gibbs free energy & stability
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3.012 Fundamentals of Materials Science
Fall 2003
 2U 
 2   0
S V
T 
   0
S V
(Eqn 7)
T
0
CV
o

This is the requirement for thermal stability. Since the absolute temperature must be > 0 by the third law,
the heat capacity must also be greater than 0. In addition, this says that the entropy must increase if the
 temperature increases, for stability.
Finally, we arrive at the requirement for mechanical stability:
P
0
V
(Eqn 8)
Le Chatelier’s Principle: Materials at equilibrium want to stay there


Le Chatelier’s principle describes how a system responds to a perturbation away from its equilibrium state. One
way to state it is:1
o

If an attempt is made ot change the pressure, temperature, or concentration of a system in equilibrium,
then the equilibrium will shift in such a manner as to diminish the magnitude of the alteration in the factor
which is varied.
LE CHATELIER’S PRINCIPLE

Usually, you first encounter this principle in the context of basic chemistry:
A BC  D
(Eqn 9)
o


If this chemical reaction comes to equilbrium, and I then add more molecules of D to the reaction, the
system will respond by a slight reversal of the A + B -> C + D transformation to compensate.

This tendency of systems to act against perturbations that move them away from their equilibrium state can be
understood by simply considering what happens to the free energy of the system when it is perturbed.
Suppose I have a two-phase system of two components. At equilibrium, the chemical potentials of component A
in each phase are equal:
(Eqn 10)
A  A



If the chemical potential should fluctuate so that A  A , then the free energy will rise from its minimum point
slightly.
 Now, from this point in the free energy landscape, how can the system get back to equilibrium? The only
spontaneous process is one where dG < 0, and such a process will be achieved if A molecules are transported
from the  phase to the  phase.

Look for a better description if possible.
Lecture 14 – Gibbs free energy & stability

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Requirements for the shape of free energy curves2

We’ve discussed conditions for spontaneous processes and equilibrium- the Gibbs free energy must decrease in
a spontaneous process at constant temperature and pressure, and the free energy must reach a minimum to be
at equilibrium. But what does the free energy as a function of typical experimental parameters look like- for
example, what does a plot of G vs. T look like? Do we know anything about how it must look? The answer is
yes: the relationships between the Gibbs free energy and other thermodynamic parameters tell us numerous
things about how plots of G vs. various thermodynamic variables must behave:
o
Consider a plot of G vs. temperature:
G
?
T
o
What should the slope of this curve be?
 Using our differential expression for G:
P
C
dG  VdP  SdT    ij dN ij
(Eqn 11)
j1 i1
…and combining this with the algebraic definition of the differential for G(T,P,N):


(Eqn 12)
P C 
G 
G 
G 
dG    dT    dP     j 
dN ij
T P,N
P T ,N
N i T ,N
j1 i1 
j i
  …thus, the slope of the curve is –S. Which means…
o
what is the curvature of G vs.T?
 The curvature is given by the second derivative of G vs. T- it is the rate of change of the slope:
 2G 
 2   0 or
T P,N

Lecture 14 – Gibbs free energy & stability
 2G 
 2   0
T P,N

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 2G 
 2   0
T P, N
G
 2G 
 2   0
T P, N
G
m << 0


m<0
m~0
T
o
T
The curvature is also related to know thermodynamic quantities:
 2G 
S 
CP
 2      
T P,N
T
T P,N
(Eqn 14)


The heat capacity and the absolute temperature must always be ≥ 0, thus the curvature must be
< 0. Thus plots of the free energy vs. temperature at constant pressure must always have
 the general shape shown at left above.
Similar analyses may be applied to predict the shape of other free energy curves of interest.
Lecture 14 – Gibbs free energy & stability
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References
1.
2.
Bergeron, C. G. & Risbud, S. H. Introduction to Phase Equilibria in Ceramics (American Ceramic Society,
Westerville, OH, 1984) 158 pp.
Carter, W. C. 3.00 Thermodynamics of Materials Lecture Notes http://pruffle.mit.edu/3.00/ (2002).
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