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Thermodynamics in Materials Engineering Mat E
212 - Course Notes R. E. Napolitano Department
of Materials Science Engineering Iowa State
University Auxiliary Functions
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Review Entropy of Mixing
Generally, we are interested in ?Smix as a
function of composition.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-20
3
Review Entropy of Mixing
(Entropy of mixing - per mole of the mixture)
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-20
4
Review Entropy of Mixing
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-20
5
Recap
6
Recap
(a) Rev. Constant Volume Process
d
P
a
b
c
V
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Recap
(b) Reversible Adiabatic Process
d
P
a
b
c
V
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Recap
(c) Reversible Isothermal Process
d
P
a
b
c
V
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Recap
(d) Reversible Constant P Process
d
P
a
b
c
V
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The overall framework
Our construction has thus far been based on
UU(S,V). However, these are not generally the
most convenient (useful) variables.
U
We are interested in developing useful criteria
for equilibrium and for comparisons of stability
between different states.
The volume is a measurable and/or controllable
quantity.
S
The entropy, however, is generally not a quantity
that can be directly measured or controlled.
V
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The Combined 1st and 2nd Law
We consider the combined 1st and 2nd laws
graphically by examining a differential element
of the internal energy surface, UU(S,V).
U
S
V
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The Combined 1st and 2nd Law
We consider the combined 1st and 2nd laws
graphically by examining a differential element
of the internal energy surface, UU(S,V).
U
S
V
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Geometric relations
We examine the U(S) at constant V to obtain
further relationships.
We write U(S) using the slope-intercept method.
We solve for the intercept, ?.
U
From the combined 1st and 2nd laws
Substituting, we obtain
S
NOTE Specifying a particular value of V and a
particular value of T defines a value for A for
the equilibrium surface. (V defines the plane and
T defines the slope.)
and we define the Helmholtz free energy
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
14
Geometric relations
We examine the U(V) at constant S to obtain
further relationships.
We write U(V) using the slope-intercept method.
We solve for the intercept
?
From the combined 1st and 2nd laws.
U
Substituting, we obtain
V
and we define the Enthalpy
NOTE Specifying a particular value of S and a
particular value of P defines a value for H
(for the equilibrium surface).
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
15
Geometric relations
We examine the U at constant S/V to obtain
further relationships.
We write U(S,V) using the slope-intercept method.
We solve for the intercept, Q
From the combined 1st and 2nd laws.
Substituting, we obtain
V
NOTE Specifying a particular value of T and a
particular value of P defines a value for G
(for the equilibrium surface).
and we define the Gibbs free energy
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
16
Auxiliary Functions
In summary, the auxiliary functions express the
intercepts as functions of P,V,S, and T.
U
S
V
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
17
Auxiliary Functions
In summary, the auxiliary functions express the
intercepts as functions of P,V,S, and T.
U
S
V
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
18
Auxiliary Functions
In summary, the auxiliary functions express the
intercepts as functions of P,V,S, and T.
U
S
V
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
19
Auxiliary Functions
In summary, the auxiliary functions express the
intercepts as functions of P,V,S, and T.
U
S
V
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
20
Example Helmholtz free energy
Lets consider the change in Helmholz free energy
for a constant volume - constant temperature
process.
Suppose we have a quantity of liquid contained
isothermally in a closed vessel of fixed volume
(State 1).
State 1
State 2
OBSERVATION At some fixed temperature, the
liquid begins to evaporate, with the vapor
uniformly filling the available volume. After a
fraction of the liquid has evaporated, the
process stops and the system remains in State 2.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
21
Example Helmholtz free energy
Lets consider the change in Helmholz free energy
for a constant volume - constant temperature
process.
Suppose we have a quantity of liquid contained
isothermally in a closed vessel of fixed volume
(State 1).
DU(nV)
State 1
State 2
OBSERVATION At some fixed temperature, the
liquid begins to evaporate, with the vapor
uniformly filling the available volume. After a
fraction of the liquid has evaporated, the
process stops and the system remains in State 2.
-TDS(nV)
nV (EQ)
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
22
Auxiliary Functions
These functions provide useful conditions for
equilibrium under certain constraints.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
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Relations from the auxiliary functions
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
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Maxwell Relations
Lets consider an arbitrary state function of two
variables,
and, as we have done several times already, lets
write the total differential of that function.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
25
Maxwell Relations
f(x,y)
Lets consider an arbitrary state function of two
variables,
df
2
and, as we have done several times already, lets
write the total differential of that function.
y
1
dy
x
dx
Now, lets assume that the partial derivatives
themselves are state functions of x and y, as
well, where
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
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Maxwell Relations
We now apply this analysis to our thermodynamic
state functions
and obtain the well known Maxwell relations.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
27
Another Relation
f(x,y)
Lets consider an arbitrary state function of two
variables.
df
2
y
1
dy
x
For any incremental change of state at constant f
dx
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
28
Other sources of energy
Realizing that there may be other sources of
energy
where these may be electrical, magnetic,
chemical, etc.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
29
Chemical Potential
Considering the chemical energy, we realize that
there will be an energy contribution from each
component in the material.
where C is the number of components in the
system.
We now define the chemical potential for the
ith component as
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
30
A preview of our strategy
We are interested in G(T,P,n1,n2,nC). In
Materials Science, we are often interested in
constant pressure (i.e. 1 atm). Therefore, the
problem is reduced to quantifying the Gibbs free
energy as a function of temperature and
composition.
Recall that the Gibbs free energy is related to
the experimentally measureable quantity of heat
capacity
The problem is that the above relationship gives
G for a particular composition. We could choose
to measure the heat capacity for many
compositions, and compute G(T,ni) using the above
relation. A more reasonable (and more useful)
approach is to use reference compositions for
which we measure the heat capacity and compute a
reference value of Gº(T). We then compute the
free energy for a mixture or solution of the
reference components by combining the appropriate
reference free energies and then adding the
energy associated with mixing. This energy of
mixing must include all enthalpic and entropic
contributions
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12
31
Reminder
HW 4 is due on Friday at the START of class.
Mat E 212 - Thermodynamics in Materials
Engineering - R.E. Napolitano
4-12