3.012 Fundamentals of Materials Science
Fall 2003
Lecture 11: 10.10.03 Chemical potentials and the Gibbs free energy
Today:
LAST TIME ............................................................................................................................................................ 2
DESCRIBING MULTI-PHASE/MULTI-COMPONENT SYSTEMS .................................................................................... 3
Molar and Partial molar quantities ................................................................................................................................................... 3
The partial molar free energy ............................................................................................................................................................ 5
CHEMICAL POTENTIALS IN MULTI-PHASE MATERIALS AT EQUILIBRIUM ............................................................... 6
Materials that rearrange their components to reach equilibrium...................................................................................................... 6
The Gibbs-Duhem Relation................................................................................................................................................................ 8
REFERENCES ....................................................................................................................................................... 10
Reading:
Partial Molar Quantities: Dill and Bromberg Ch. 9 pp. 163-168
Supplementary Reading:
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Lecture 11 – chemical potentials at equilibrium
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Last time
Lecture 11 – chemical potentials at equilibrium
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Describing multi-phase/multi-component systems

Before discussing chemical equilibrium in multi-phase/multi-component systems, we need to introduce some
nomenclature used to describe such systems
Molar and Partial molar quantities
Molar quantities

Rather than calculating total amounts for extensive quantities like internal energy and entropy, it is often
convenient to define molar quantities for single-component systems- the amount of Y per mole of material. The
molar value of some extensive quantity Y is defined as:

The units of molar quantities are thus the same as the parent variable per mole- internal energy is J/mole, entropy
is J/mole•K.
Molar quantities for composition

consider a two-component system (A and B atoms) that has 2 phases present,  and :







The composition of each phase is:
 phase:
total moles of atoms in  phase:
moles of A atoms in  phase:
moles of B atoms in  phase:



mole fraction of A atoms in  phase:

N total
 N A  N B
N A
N B
N
X A  A
N total

Lecture 11 – chemical potentials at equilibrium
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Fall 2003
mole fraction of B atoms in  phase:
 phase:
X B 
N B
 1 X A

N total
…
In these quantities, we’ve introduced the mole 
fraction- which is simply a number fraction of each
component present. This is a useful quantity since it provides a measure of the amount of a given
component in a phase independent of the total system size.
How do we describe the composition of the system as a whole? The convention is to use the average
composition: The average composition is expressed as total mole fractions of A and B in the system.
o

Average mole fraction of A:
Average mole fraction of B:

N A,total
N A  N A
 
N total
N A  N A  N B  N B
N
N B  N B
X A  B,total  
 1 X B
N total
N A  N A  N B  N B
XA 

The phase fractions f and f are the mole fraction of total atoms residing in each phase:


N total
N A  N B
 
N total N A  N B  N A  N B

N total
N A  N B

f 
 
 1 f 



N total N A  N B  N A  N B
f 

o
The last relationship is required since the sum of the phase fractions must be 1.

Partial molar quantities in multi-component systems

Molar quantities are more complicated for a multi-component system. When multiple components are present,
the question becomes, Y per mole of what? To handle this situation, we define partial molar quantities that define
extensive variables per mole of each component. A general partial molar quantity is defined as:
Y 
Yi   
n i T ,P,n j i
(Eqn 3)
o

Because Y depends on the amount nA, nB, nC, … of moles of each component present, we define it as a
partial derivative with respect to one of the components, while the numbers of all other components are
kept constant. The partial molar quantity is thus, in words, ‘the amount Y changes upon addition of a
 mole of component i, while all other components are kept constant (and at constant temperature and
pressure).’ Note this last phrase which is part of the definition of partial molar quantities- the derivative is
taken at constant temperature and pressure.
Partial molar quantities
Yi are related to the total quantity Y by:

Lecture 11 – chemical potentials at equilibrium
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
Fall 2003
The usefulness of partial molar quantities can be seen if we consider the example of determining the partial molar
volume of a multicomponent system. If we have a single-component system, the molar volume is just:
o
But suppose we have a two-component system (with components A and B). A and B may have very
different molecular volumes- thus addition of a mole of A to the system may have a very different effect
on the total volume than addition of a mole of B. ILLUSTRATION? The partial molar volumes allow us to
quantify these differences:
V 
VA   
nA T ,P,n B
(Eqn 6)

V 
VB   
nB T ,P,n A
o
This can be of course generalized to any number of components.
o
LINKAGE BETWEEN
PARTIAL MOLAR QUANTITIES- TIME FOR THIS?
The partial molar free energy

A very useful partial molar quantity is the partial molar free energy:
G 
Gi   
n i T ,P,n j i
(Eqn 7)

What is this derivative equivalent to? Comparing the differential expression for the Gibbs free energy with the
algebraic definition of its differential:

o
We can match the terms and see that:
G 
Gi   
 i
N i T ,P,N j i
(Eqn 10)
o
Thus, the partial molar free energy is equal to the chemical potential. In words, the partial molar free
energy is the change in total free energy that occurs per mole of component i added, with all other
components constant (at constant T and P). The chemical potential of i measures the change in free
 energy caused by adding more i to the system.
Lecture 11 – chemical potentials at equilibrium
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Chemical Potentials in Multi-phase materials at equilibrium
Materials that rearrange their components to reach equilibrium

We’ve discussed chemical work in Lecture 7- and how a multi-phase material that cannot exchange molecules
with its surroundings can still alter its internal energy (or free energy) by rearranging the molecules within its
phases, creating new phases, or eliminating phases that were present initially.
o

These rearrangements are driven by the chemical potential, but we haven’t yet examined what happens
to the chemical potentials of different components at equilibrium, and we haven’t yet shown where
molecules move in response to chemical potentials. Do molecules move from high chemical potential to
low chemical potential, or vice versa? When is there no driving force for molecules to exchange between
phases?
We can answer these questions by combining the equilibrium condition dG = 0 with the conservation of matter.
o If we have a closed multi-phase system (as illustrated above) and no chemical reactions can occur that
change one component into another, then we must have a constant total number of moles of each
component after any change in the system:
(Eqn 11)
n Atotal  constan t  n A  n A  ... n AP
(Eqn 12)
dn Atotal  0  dn A  dn A  ... dn AP
(Eqn 13)
dnA  dnA  ... dnAP 


dnA  dnA  dnA ... dnAP 
…
 dnAP  dnA  ... dnAP1

condition at constant T and P is dG = 0. If we combine this equilibrium criterion with the
Our equilibrium
conservation of matter equations above, we can write the following:

P
C
   dn
k
j
(Eqn 14)
k
j
0
equilibrium condition at constant temperature and pressure
k1 j1
Lecture 11 – chemical potentials at equilibrium
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or
P1 C

(Eqn 15)
( kj   Pj )dn kj  0
k1 j1
o
(Eqn 15) expanded for the case illustrated above, of a two-component, two-phase system would be:

o
(Eqn 15) indicates how chemical potential differences move molecules in a material. Let’s go through an
example. Suppose we have a material comprised of two phases  and  as illustrated below. If the initial
distribution of molecules in these phases was such that µ A > µA, then a driving force exists to move A
molecules from the  phase to the  phase. Diffusion will occur to move A molecules into the  phase
until the chemical potential of A molecules in each phase is the same.

We can see this mathematically because:
P1 C
dG     kj   Pj dN kj  A  A dn A  B  B dn B  A  A dn A
(Eqn 17)
k1 j1


(Eqn 18)
And for a spontaneous process:
dG  A  A dnA  0
Since (µA - µA) > 0, dnA must be < 0: the system can only spontaneously move toward
equilibrium by processes where dG < 0 (i.e. the free energy is decreasing!).
What about the value of the chemical potentials in a system at equilibrium? Suppose we have a material

that contains
two phases  and  and 3 components A, B, and C.

o
P
C
dG  SdT  VdP     kj dn kj
(Eqn 19)
k1 j1
 SdT  VdP  A dn A  B dn B  C dnC  A dn A  B dn B  C dnC
o
At constant temperature and pressure, at equilibrium:


Lecture 11 – chemical potentials at equilibrium
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0  A dn A  B dn B  C dnC  A dn A  B dn B  C dnC
(Eqn 20)
o
The most common experimental condition is that our multi-phase material constitute a closed system- no
atoms can leave or enter the material. This provides an additional set of constraints:

o
Putting together our constraints in (Eqn 21) with the criterion for equilibrium (Eqn 20), we get the follwoing
conditions on the chemical potentials at equilibrium:
o
At equilibrium, the chemical potential of each component in the system must be equal in every
phase.
The Gibbs-Duhem Relation

We previously derived a differential expression for the Gibbs free energy:
C
dG    j dn j  VdP  SdT
(Eqn 22)
j1

Let’s compare this to the integrated expression for G. The definition of partial molar free energy gives us:

o
o
(written here for the simplified case of a single-phase material)
If we write a differential expression for G using (Eqn 23)…
C
C
i1
i1
dG   N i di   i dN i
(Eqn 24)
o
Combining (Eqn 22) and (Eqn 24):

o
This is the Gibbs-Duhem equation. DISCUSS MEANING.
Example applications of the Gibbs-Duhem equation
what
Lecture 11 – chemical potentials at equilibrium
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Lecture 11 – chemical potentials at equilibrium
Fall 2003
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Fall 2003
References
Lecture 11 – chemical potentials at equilibrium
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