Lecture 11 - Irreversible Processes Notes | EduRev

: Lecture 11 - Irreversible Processes Notes | EduRev

 Page 1


Materials Science & Metallurgy Part III Course M16
Materials Modelling H. K. D. H. Bhadeshia
Lecture 11: Irreversible Processes
Thermodynamics generally deals with measurable properties of materials, formulated on the
basis of equilibrium. Thus, properties such as entropy and free energy are, on an appropriate
scale, static and time–invariant during equilibrium. There are other parameters not relevant
to the discussion of equilibrium: thermal conductivity, di?usivity and viscosity, but which are
interesting because they can describe a second kind of time–independence, that of the steady–
state (Denbigh, 1955). Thus, the concentration pro?le does not change during steady–state
di?usion, even though energy is being dissipated by the di?usion.
The thermodynamics of irreversible processes deals with systems which are not at equilibrium
but are nevertheless stationary. The theory in e?ect uses thermodynamics to deal with kinetic
phenomena. There is nevertheless, a distinction between the thermodynamics of irreversible
processes and kinetics (Denbigh). The former applies strictly to the steady–state, whereas
there is no such restriction on kinetic theory.
Reversibility
A process whose direction can be changed by an in?nitesimal alteration in the external con-
ditions is called reversible. Consider the example illustrated in Fig. 1, which deals with the
response of an ideal gas contained at uniform pressure within a cylinder, any change being
achieved by the motion of the piston. For any starting point on the P/V curve, if the applica-
tion of an in?nitesimal force causes the piston to move slowly to an adjacent position still on
the curve, then the process is reversible since energy has not been dissipated. The removal of
the in?nitesimal force will cause the system to revert to its original state.
On the other hand, if there is friction during the motion of the piston, then deviations occur
from the P/V curve as illustrated by the cycle in Fig. 1. An in?nitesimal force cannot move
the piston because energy is dissipated due to friction (as given by the area within the cycle).
Suchaprocess, which involves the dissipation of energy, isclassi?edasirreversiblewithrespect
to an in?nitesimal change in the external conditions.
More generally, reversibility means that it is possible to pass from one state to another with-
out appreciable deviation from equilibrium. Real processes are not reversible so equilibrium
thermodynamics can only be used approximately, though the same thermodynamics de?nes
whether or not a process can occur spontaneously without ambiguity.
For irreversible processes the equations of classical thermodynamics become inequalities. For
example, at the equilibrium melting temperature, the free energies of the liquid and solid
are identical (G
liquid
= G
solid
) but not so below that temperature (G
liquid
> G
solid
). Such
inequalities are much more di?cult to deal with though they indicate the natural direction
of change. For steady–state processes however, the thermodynamic framework for irreversible
processes as developed by Onsager is particularly useful in approximating relationships even
though the system is not at equilibrium.
The Linear Laws
At equilibrium there is no change in entropy or free energy. An irreversible process dissipates
energyandentropyiscreatedcontinuously. IntheexampleillustratedinFig.1, thedissipation
Page 2


Materials Science & Metallurgy Part III Course M16
Materials Modelling H. K. D. H. Bhadeshia
Lecture 11: Irreversible Processes
Thermodynamics generally deals with measurable properties of materials, formulated on the
basis of equilibrium. Thus, properties such as entropy and free energy are, on an appropriate
scale, static and time–invariant during equilibrium. There are other parameters not relevant
to the discussion of equilibrium: thermal conductivity, di?usivity and viscosity, but which are
interesting because they can describe a second kind of time–independence, that of the steady–
state (Denbigh, 1955). Thus, the concentration pro?le does not change during steady–state
di?usion, even though energy is being dissipated by the di?usion.
The thermodynamics of irreversible processes deals with systems which are not at equilibrium
but are nevertheless stationary. The theory in e?ect uses thermodynamics to deal with kinetic
phenomena. There is nevertheless, a distinction between the thermodynamics of irreversible
processes and kinetics (Denbigh). The former applies strictly to the steady–state, whereas
there is no such restriction on kinetic theory.
Reversibility
A process whose direction can be changed by an in?nitesimal alteration in the external con-
ditions is called reversible. Consider the example illustrated in Fig. 1, which deals with the
response of an ideal gas contained at uniform pressure within a cylinder, any change being
achieved by the motion of the piston. For any starting point on the P/V curve, if the applica-
tion of an in?nitesimal force causes the piston to move slowly to an adjacent position still on
the curve, then the process is reversible since energy has not been dissipated. The removal of
the in?nitesimal force will cause the system to revert to its original state.
On the other hand, if there is friction during the motion of the piston, then deviations occur
from the P/V curve as illustrated by the cycle in Fig. 1. An in?nitesimal force cannot move
the piston because energy is dissipated due to friction (as given by the area within the cycle).
Suchaprocess, which involves the dissipation of energy, isclassi?edasirreversiblewithrespect
to an in?nitesimal change in the external conditions.
More generally, reversibility means that it is possible to pass from one state to another with-
out appreciable deviation from equilibrium. Real processes are not reversible so equilibrium
thermodynamics can only be used approximately, though the same thermodynamics de?nes
whether or not a process can occur spontaneously without ambiguity.
For irreversible processes the equations of classical thermodynamics become inequalities. For
example, at the equilibrium melting temperature, the free energies of the liquid and solid
are identical (G
liquid
= G
solid
) but not so below that temperature (G
liquid
> G
solid
). Such
inequalities are much more di?cult to deal with though they indicate the natural direction
of change. For steady–state processes however, the thermodynamic framework for irreversible
processes as developed by Onsager is particularly useful in approximating relationships even
though the system is not at equilibrium.
The Linear Laws
At equilibrium there is no change in entropy or free energy. An irreversible process dissipates
energyandentropyiscreatedcontinuously. IntheexampleillustratedinFig.1, thedissipation
Fig. 1: The curve represents the variation in pressure within the cylinder as
the volume of the ideal gas is altered by positioning the frictionless piston.
The cycle represents the dissipation of energy when the motion of the piston
causes friction.
was due to friction; di?usion ahead of a moving interface is dissipative. The rate at which
energy is dissipated is the product of the temperature and the rate of entropy production (i.e.
Ts) with:
Ts =JX (1)
whereJ isageneralised?uxofsomekind,andX ageneralisedforce. Inthecaseofanelectrical
current, the heat dissipation is the product of the current (J) and the electromotive force (X).
As long as the ?ux–force sets can be expressed as in equation 1, the ?ux must naturally
depend in some way on the force. It may then be written as a function J{X} of the force X.
At equilibrium, the force is zero. If J{X} is expanded in a Taylor series about equilibrium
(X = 0), we get
J{X} =
8
X
0
a
n
X
n
=
8
X
0
f
(n)
{0}
n!
X
n
=J{0}+J
0
{0}
X
1!
+J
00
{0}
X
2
2!
...
Note that J{0} = 0 since that represents equilibrium. If the high order terms are neglected
then we see that
J ?X.
This is a key result from the theory, that the forces and their conjugate ?uxes are linearly
related (J ? X) whenever the dissipation can be written as in equation 1, at least when the
deviations from equilibrium are not large. Some examples of forces and ?uxes in the context
of the present theory are given in Table 1.
Page 3


Materials Science & Metallurgy Part III Course M16
Materials Modelling H. K. D. H. Bhadeshia
Lecture 11: Irreversible Processes
Thermodynamics generally deals with measurable properties of materials, formulated on the
basis of equilibrium. Thus, properties such as entropy and free energy are, on an appropriate
scale, static and time–invariant during equilibrium. There are other parameters not relevant
to the discussion of equilibrium: thermal conductivity, di?usivity and viscosity, but which are
interesting because they can describe a second kind of time–independence, that of the steady–
state (Denbigh, 1955). Thus, the concentration pro?le does not change during steady–state
di?usion, even though energy is being dissipated by the di?usion.
The thermodynamics of irreversible processes deals with systems which are not at equilibrium
but are nevertheless stationary. The theory in e?ect uses thermodynamics to deal with kinetic
phenomena. There is nevertheless, a distinction between the thermodynamics of irreversible
processes and kinetics (Denbigh). The former applies strictly to the steady–state, whereas
there is no such restriction on kinetic theory.
Reversibility
A process whose direction can be changed by an in?nitesimal alteration in the external con-
ditions is called reversible. Consider the example illustrated in Fig. 1, which deals with the
response of an ideal gas contained at uniform pressure within a cylinder, any change being
achieved by the motion of the piston. For any starting point on the P/V curve, if the applica-
tion of an in?nitesimal force causes the piston to move slowly to an adjacent position still on
the curve, then the process is reversible since energy has not been dissipated. The removal of
the in?nitesimal force will cause the system to revert to its original state.
On the other hand, if there is friction during the motion of the piston, then deviations occur
from the P/V curve as illustrated by the cycle in Fig. 1. An in?nitesimal force cannot move
the piston because energy is dissipated due to friction (as given by the area within the cycle).
Suchaprocess, which involves the dissipation of energy, isclassi?edasirreversiblewithrespect
to an in?nitesimal change in the external conditions.
More generally, reversibility means that it is possible to pass from one state to another with-
out appreciable deviation from equilibrium. Real processes are not reversible so equilibrium
thermodynamics can only be used approximately, though the same thermodynamics de?nes
whether or not a process can occur spontaneously without ambiguity.
For irreversible processes the equations of classical thermodynamics become inequalities. For
example, at the equilibrium melting temperature, the free energies of the liquid and solid
are identical (G
liquid
= G
solid
) but not so below that temperature (G
liquid
> G
solid
). Such
inequalities are much more di?cult to deal with though they indicate the natural direction
of change. For steady–state processes however, the thermodynamic framework for irreversible
processes as developed by Onsager is particularly useful in approximating relationships even
though the system is not at equilibrium.
The Linear Laws
At equilibrium there is no change in entropy or free energy. An irreversible process dissipates
energyandentropyiscreatedcontinuously. IntheexampleillustratedinFig.1, thedissipation
Fig. 1: The curve represents the variation in pressure within the cylinder as
the volume of the ideal gas is altered by positioning the frictionless piston.
The cycle represents the dissipation of energy when the motion of the piston
causes friction.
was due to friction; di?usion ahead of a moving interface is dissipative. The rate at which
energy is dissipated is the product of the temperature and the rate of entropy production (i.e.
Ts) with:
Ts =JX (1)
whereJ isageneralised?uxofsomekind,andX ageneralisedforce. Inthecaseofanelectrical
current, the heat dissipation is the product of the current (J) and the electromotive force (X).
As long as the ?ux–force sets can be expressed as in equation 1, the ?ux must naturally
depend in some way on the force. It may then be written as a function J{X} of the force X.
At equilibrium, the force is zero. If J{X} is expanded in a Taylor series about equilibrium
(X = 0), we get
J{X} =
8
X
0
a
n
X
n
=
8
X
0
f
(n)
{0}
n!
X
n
=J{0}+J
0
{0}
X
1!
+J
00
{0}
X
2
2!
...
Note that J{0} = 0 since that represents equilibrium. If the high order terms are neglected
then we see that
J ?X.
This is a key result from the theory, that the forces and their conjugate ?uxes are linearly
related (J ? X) whenever the dissipation can be written as in equation 1, at least when the
deviations from equilibrium are not large. Some examples of forces and ?uxes in the context
of the present theory are given in Table 1.
Force Flux
e.m.f. =
?f
?z
Electrical Current
-
1
T
?T
?z
Heat ?ux
-
?µ
i
?z
Di?usion ?ux
Stress Strain rate
Table 1: Examples of forces and their conjugate ?uxes. z is distance, f is the
electrical potential in Volts, and µ is a chemical potential. “e.m.f.” stands for
electromotive force.
Multiple Irreversible Processes
There are many circumstances in which a number of irreversible processes occur together. In
a ternary Fe–Mn–C alloy, the di?usion ?ux of carbon depends not only on the gradient of
carbon, but also on that of manganese. Thus, a uniform distribution of carbon will tend to
become inhomogeneous in the presence of a manganese concentration gradient. Similarly, the
?ux of heat may not depend on the temperature gradient alone; heat can be driven also by
an electromotive force (Peltier e?ect)†. Electromigration involves di?usion driven by an elec-
tromotive force. When there is more then one dissipative process, the total energy dissipation
rate can still be written
Ts =
X
i
J
i
X
i
. (2)
In general, if there is more than one irreversible process occurring, it is found experimentally
that each ?ow J
i
is related not only to its conjugate force X
i
, but also is related linearly to all
other forces present. Thus,
J
i
=M
ij
X
j
(3)
with i, j = 1,2,3.... Therefore, a given ?ux depends on all the forces causing the dissipation
of energy.
Onsager Reciprocal Relations
Equilibrium in real systems is always dynamic on a microscopic scale. It seems obvious that
to maintain equilibrium under these dynamic conditions, a process and its reverse must occur
at the same rate on the microscopic scale. The consequence is that provided the forces and
?uxes are chosen from the dissipation equation and are independent, M
ij
= M
ji
. This is
known as the Onsager theorem, or the Onsager reciprocal relations. It applies to systems near
equilibrium when the properties of interest have even parity, and assuming that the ?uxes and
their corresponding forces are independent. An exception occurs with magnetic ?elds in which
case there is a sign di?erence M
ij
=-M
ji
(Miller, 1960).
† In the Peltier e?ect, the two junctions of a thermocouple are kept at the same temperature
but the passage of an electrical current causes one of the junctions to absorb heat and the
other to liberate the same quantity of heat. This Peltier heat is found to be proportional to
the current.
Page 4


Materials Science & Metallurgy Part III Course M16
Materials Modelling H. K. D. H. Bhadeshia
Lecture 11: Irreversible Processes
Thermodynamics generally deals with measurable properties of materials, formulated on the
basis of equilibrium. Thus, properties such as entropy and free energy are, on an appropriate
scale, static and time–invariant during equilibrium. There are other parameters not relevant
to the discussion of equilibrium: thermal conductivity, di?usivity and viscosity, but which are
interesting because they can describe a second kind of time–independence, that of the steady–
state (Denbigh, 1955). Thus, the concentration pro?le does not change during steady–state
di?usion, even though energy is being dissipated by the di?usion.
The thermodynamics of irreversible processes deals with systems which are not at equilibrium
but are nevertheless stationary. The theory in e?ect uses thermodynamics to deal with kinetic
phenomena. There is nevertheless, a distinction between the thermodynamics of irreversible
processes and kinetics (Denbigh). The former applies strictly to the steady–state, whereas
there is no such restriction on kinetic theory.
Reversibility
A process whose direction can be changed by an in?nitesimal alteration in the external con-
ditions is called reversible. Consider the example illustrated in Fig. 1, which deals with the
response of an ideal gas contained at uniform pressure within a cylinder, any change being
achieved by the motion of the piston. For any starting point on the P/V curve, if the applica-
tion of an in?nitesimal force causes the piston to move slowly to an adjacent position still on
the curve, then the process is reversible since energy has not been dissipated. The removal of
the in?nitesimal force will cause the system to revert to its original state.
On the other hand, if there is friction during the motion of the piston, then deviations occur
from the P/V curve as illustrated by the cycle in Fig. 1. An in?nitesimal force cannot move
the piston because energy is dissipated due to friction (as given by the area within the cycle).
Suchaprocess, which involves the dissipation of energy, isclassi?edasirreversiblewithrespect
to an in?nitesimal change in the external conditions.
More generally, reversibility means that it is possible to pass from one state to another with-
out appreciable deviation from equilibrium. Real processes are not reversible so equilibrium
thermodynamics can only be used approximately, though the same thermodynamics de?nes
whether or not a process can occur spontaneously without ambiguity.
For irreversible processes the equations of classical thermodynamics become inequalities. For
example, at the equilibrium melting temperature, the free energies of the liquid and solid
are identical (G
liquid
= G
solid
) but not so below that temperature (G
liquid
> G
solid
). Such
inequalities are much more di?cult to deal with though they indicate the natural direction
of change. For steady–state processes however, the thermodynamic framework for irreversible
processes as developed by Onsager is particularly useful in approximating relationships even
though the system is not at equilibrium.
The Linear Laws
At equilibrium there is no change in entropy or free energy. An irreversible process dissipates
energyandentropyiscreatedcontinuously. IntheexampleillustratedinFig.1, thedissipation
Fig. 1: The curve represents the variation in pressure within the cylinder as
the volume of the ideal gas is altered by positioning the frictionless piston.
The cycle represents the dissipation of energy when the motion of the piston
causes friction.
was due to friction; di?usion ahead of a moving interface is dissipative. The rate at which
energy is dissipated is the product of the temperature and the rate of entropy production (i.e.
Ts) with:
Ts =JX (1)
whereJ isageneralised?uxofsomekind,andX ageneralisedforce. Inthecaseofanelectrical
current, the heat dissipation is the product of the current (J) and the electromotive force (X).
As long as the ?ux–force sets can be expressed as in equation 1, the ?ux must naturally
depend in some way on the force. It may then be written as a function J{X} of the force X.
At equilibrium, the force is zero. If J{X} is expanded in a Taylor series about equilibrium
(X = 0), we get
J{X} =
8
X
0
a
n
X
n
=
8
X
0
f
(n)
{0}
n!
X
n
=J{0}+J
0
{0}
X
1!
+J
00
{0}
X
2
2!
...
Note that J{0} = 0 since that represents equilibrium. If the high order terms are neglected
then we see that
J ?X.
This is a key result from the theory, that the forces and their conjugate ?uxes are linearly
related (J ? X) whenever the dissipation can be written as in equation 1, at least when the
deviations from equilibrium are not large. Some examples of forces and ?uxes in the context
of the present theory are given in Table 1.
Force Flux
e.m.f. =
?f
?z
Electrical Current
-
1
T
?T
?z
Heat ?ux
-
?µ
i
?z
Di?usion ?ux
Stress Strain rate
Table 1: Examples of forces and their conjugate ?uxes. z is distance, f is the
electrical potential in Volts, and µ is a chemical potential. “e.m.f.” stands for
electromotive force.
Multiple Irreversible Processes
There are many circumstances in which a number of irreversible processes occur together. In
a ternary Fe–Mn–C alloy, the di?usion ?ux of carbon depends not only on the gradient of
carbon, but also on that of manganese. Thus, a uniform distribution of carbon will tend to
become inhomogeneous in the presence of a manganese concentration gradient. Similarly, the
?ux of heat may not depend on the temperature gradient alone; heat can be driven also by
an electromotive force (Peltier e?ect)†. Electromigration involves di?usion driven by an elec-
tromotive force. When there is more then one dissipative process, the total energy dissipation
rate can still be written
Ts =
X
i
J
i
X
i
. (2)
In general, if there is more than one irreversible process occurring, it is found experimentally
that each ?ow J
i
is related not only to its conjugate force X
i
, but also is related linearly to all
other forces present. Thus,
J
i
=M
ij
X
j
(3)
with i, j = 1,2,3.... Therefore, a given ?ux depends on all the forces causing the dissipation
of energy.
Onsager Reciprocal Relations
Equilibrium in real systems is always dynamic on a microscopic scale. It seems obvious that
to maintain equilibrium under these dynamic conditions, a process and its reverse must occur
at the same rate on the microscopic scale. The consequence is that provided the forces and
?uxes are chosen from the dissipation equation and are independent, M
ij
= M
ji
. This is
known as the Onsager theorem, or the Onsager reciprocal relations. It applies to systems near
equilibrium when the properties of interest have even parity, and assuming that the ?uxes and
their corresponding forces are independent. An exception occurs with magnetic ?elds in which
case there is a sign di?erence M
ij
=-M
ji
(Miller, 1960).
† In the Peltier e?ect, the two junctions of a thermocouple are kept at the same temperature
but the passage of an electrical current causes one of the junctions to absorb heat and the
other to liberate the same quantity of heat. This Peltier heat is found to be proportional to
the current.
References
Christian, J. W., (1975) Theory of Transformations in Metals and Alloys, 2nd ed., Pt.1, Perg-
amon Press, Oxford.
Cook, H. E. and Hilliard, J., (1969) Journal of Applied Physics, 40, 2191.
Darken, L. S., (1949) TMS–AIME, 180, 430.
Denbigh, K. G., (1955) Thermodynamics of the Steady State, John Wiley and Sons, Inc., New
York, U.S.A.
Einstein, A., (1905) Ann. Phys., 17, 549.
Hartley, G. S., (1931) Transactions Faraday Soc., 27, 10.
Kirkaldy, J. S., (1970) Advances in Materials Research, 4, 55.
Miller, D. G., (1960) Chem. Rev., 60, 15.
Onsager, L., (1931) Physical Review, 37, 405.
Onsager, L., (1931) Physical Review, 38, 2265.
Onsager, L., (1945–46) Ann. N.Y. Acad. Sci., 46, 241.
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