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# Chapter 16 Irreversible Thermodynamics Chemical Engineering Notes | EduRev

## Chemical Engineering : Chapter 16 Irreversible Thermodynamics Chemical Engineering Notes | EduRev

``` Page 1

16
I r reve rsi bl e The rmod yna m ics
16.1 Introduction
Classical thermodynamics deals with transitions from one equilibrium state to another and
since it does not analyse the changes between state points it could be called thermostatics.
The term thermodynamics will be reserved, in this chapter, for dynamic non-equilibrium
processes.
In previous work, phenomenological laws have been given which describe irreversible
processes in the form of proportionalities, e.g. Fourier’s law of heat conduction, Ohm’s law
relating electrical current and potential gradient, Fick’s law relating flow of matter and
concentration gradient etc. When two of these phenomena occur simultaneously they
interfere, or couple, and give rise to new effects. One such cross-coupling is the reciprocal
effect of thermoelectricity and electrical conduction: the Peltier effect (evolution or
absorption of heat at a junction due to the flow of electrical current) and thermoelectric
force (due to maintenance of the junctions at different temperatures). It is necessary to
formulate coupled equations to deal with these phenomena, which are ‘phenomenological’
inasmuch as they are experimentally verified laws but are not a part of the comprehensive
theory of irreversible processes.
It is possible to examine irreversible phenomena by statistical mechanics and the
kinetic theory but these methods are on a molecular scale and do not give a good
macroscopic theory of the processes. Another method of considering non-equilibrium
processes is based on ‘pseudo-thermostatic theories’. Here, the laws of thermostatics
are applied to a part of the irreversible process that is considered to be reversible and
the rest of the process is considered as irreversible and not taken into account.
Thomson applied the second law of thermostatics to thermoelectricity by considering
the Thomson and Peltier effects to be reversible and the conduction effects to be
irreversible. The method was successful as the predictions were confirmed by
experiment but it has not been possible to justify Thomson’s hypothesis from general
considerations.
Systematic macroscopic and general thermodynamics of irreversible processes can be
obtained from a theorem published by Onsager (1931a,b). This was developed from
statistical mechanics and the derivation will not be shown but the results will be used. The
theory, based on Onsager’s theorem, also shows why the incorrect thermostatic methods
give correct results in a number of cases.
Page 2

16
I r reve rsi bl e The rmod yna m ics
16.1 Introduction
Classical thermodynamics deals with transitions from one equilibrium state to another and
since it does not analyse the changes between state points it could be called thermostatics.
The term thermodynamics will be reserved, in this chapter, for dynamic non-equilibrium
processes.
In previous work, phenomenological laws have been given which describe irreversible
processes in the form of proportionalities, e.g. Fourier’s law of heat conduction, Ohm’s law
relating electrical current and potential gradient, Fick’s law relating flow of matter and
concentration gradient etc. When two of these phenomena occur simultaneously they
interfere, or couple, and give rise to new effects. One such cross-coupling is the reciprocal
effect of thermoelectricity and electrical conduction: the Peltier effect (evolution or
absorption of heat at a junction due to the flow of electrical current) and thermoelectric
force (due to maintenance of the junctions at different temperatures). It is necessary to
formulate coupled equations to deal with these phenomena, which are ‘phenomenological’
inasmuch as they are experimentally verified laws but are not a part of the comprehensive
theory of irreversible processes.
It is possible to examine irreversible phenomena by statistical mechanics and the
kinetic theory but these methods are on a molecular scale and do not give a good
macroscopic theory of the processes. Another method of considering non-equilibrium
processes is based on ‘pseudo-thermostatic theories’. Here, the laws of thermostatics
are applied to a part of the irreversible process that is considered to be reversible and
the rest of the process is considered as irreversible and not taken into account.
Thomson applied the second law of thermostatics to thermoelectricity by considering
the Thomson and Peltier effects to be reversible and the conduction effects to be
irreversible. The method was successful as the predictions were confirmed by
experiment but it has not been possible to justify Thomson’s hypothesis from general
considerations.
Systematic macroscopic and general thermodynamics of irreversible processes can be
obtained from a theorem published by Onsager (1931a,b). This was developed from
statistical mechanics and the derivation will not be shown but the results will be used. The
theory, based on Onsager’s theorem, also shows why the incorrect thermostatic methods
give correct results in a number of cases.
Entropy flow and entropy production 3 17
16.2 Definition of irreversible or steady state thermodynamics
All previous work on macroscopic ‘thermodynamics’ has been related to equilibrium. A
system was said to be in equilibrium when no spontaneous process took place and all the
thermodynamic properties remained unchanged. The macroscopic properties of the system
were spatially and temporally invariant.
System 2
: System boundary
Distance
Fig. 16.1 Steady state conduction of heat along a bar
Consider the system shown in Fig 16.1, in which a thermally insulated rod connects two
reservoirs at temperatures T2 and T, respectively. Heat flows between the two reservoirs by
conduction along the rod. If the reservoirs are very large the conduction of energy out of,
or into, them can be considered not to affect them. A state will be achieved when the rate
of heat flow, dQ/dt, entering the rod equals the rate of heat flow, -dQ/dt, leaving the rod.
If a thermometer were inserted at any point in the rod the reading would not change with
time, but it would be dependent on position. If the rod were of uniform cross-section the
temperature gradient would be linear. (This is the basis of the Searle’s bar conduction
experiment. )
The temperature in the bar is therefore a function of position but is independent of time.
The overall system is in a ‘stationary’ or ‘steady’ state but not in ‘equilibrium’, for that
requires that the temperature be uniform throughout the system. If the metal bar were
isolated from all the influences of the surroundings and from the heat sources, i.e. if it is
made an isolated system, the difference between the steady state and equilibrium becomes
obvious. In the case where the system was in steady state, processes would occur after the
isolation (equalisation of temperature throughout the bar); where the system was already
in equilibrium, they would not.
16.3 Entropy flow and entropy production
Still considering the conduction example given above. If the heat flows into the left-hand
end of the bar due to an infinitesimal temperature difference, i.e. the process is reversible,
the left-hand reservoir loses entropy at the rate
(16.1)
Page 3

16
I r reve rsi bl e The rmod yna m ics
16.1 Introduction
Classical thermodynamics deals with transitions from one equilibrium state to another and
since it does not analyse the changes between state points it could be called thermostatics.
The term thermodynamics will be reserved, in this chapter, for dynamic non-equilibrium
processes.
In previous work, phenomenological laws have been given which describe irreversible
processes in the form of proportionalities, e.g. Fourier’s law of heat conduction, Ohm’s law
relating electrical current and potential gradient, Fick’s law relating flow of matter and
concentration gradient etc. When two of these phenomena occur simultaneously they
interfere, or couple, and give rise to new effects. One such cross-coupling is the reciprocal
effect of thermoelectricity and electrical conduction: the Peltier effect (evolution or
absorption of heat at a junction due to the flow of electrical current) and thermoelectric
force (due to maintenance of the junctions at different temperatures). It is necessary to
formulate coupled equations to deal with these phenomena, which are ‘phenomenological’
inasmuch as they are experimentally verified laws but are not a part of the comprehensive
theory of irreversible processes.
It is possible to examine irreversible phenomena by statistical mechanics and the
kinetic theory but these methods are on a molecular scale and do not give a good
macroscopic theory of the processes. Another method of considering non-equilibrium
processes is based on ‘pseudo-thermostatic theories’. Here, the laws of thermostatics
are applied to a part of the irreversible process that is considered to be reversible and
the rest of the process is considered as irreversible and not taken into account.
Thomson applied the second law of thermostatics to thermoelectricity by considering
the Thomson and Peltier effects to be reversible and the conduction effects to be
irreversible. The method was successful as the predictions were confirmed by
experiment but it has not been possible to justify Thomson’s hypothesis from general
considerations.
Systematic macroscopic and general thermodynamics of irreversible processes can be
obtained from a theorem published by Onsager (1931a,b). This was developed from
statistical mechanics and the derivation will not be shown but the results will be used. The
theory, based on Onsager’s theorem, also shows why the incorrect thermostatic methods
give correct results in a number of cases.
Entropy flow and entropy production 3 17
16.2 Definition of irreversible or steady state thermodynamics
All previous work on macroscopic ‘thermodynamics’ has been related to equilibrium. A
system was said to be in equilibrium when no spontaneous process took place and all the
thermodynamic properties remained unchanged. The macroscopic properties of the system
were spatially and temporally invariant.
System 2
: System boundary
Distance
Fig. 16.1 Steady state conduction of heat along a bar
Consider the system shown in Fig 16.1, in which a thermally insulated rod connects two
reservoirs at temperatures T2 and T, respectively. Heat flows between the two reservoirs by
conduction along the rod. If the reservoirs are very large the conduction of energy out of,
or into, them can be considered not to affect them. A state will be achieved when the rate
of heat flow, dQ/dt, entering the rod equals the rate of heat flow, -dQ/dt, leaving the rod.
If a thermometer were inserted at any point in the rod the reading would not change with
time, but it would be dependent on position. If the rod were of uniform cross-section the
temperature gradient would be linear. (This is the basis of the Searle’s bar conduction
experiment. )
The temperature in the bar is therefore a function of position but is independent of time.
The overall system is in a ‘stationary’ or ‘steady’ state but not in ‘equilibrium’, for that
requires that the temperature be uniform throughout the system. If the metal bar were
isolated from all the influences of the surroundings and from the heat sources, i.e. if it is
made an isolated system, the difference between the steady state and equilibrium becomes
obvious. In the case where the system was in steady state, processes would occur after the
isolation (equalisation of temperature throughout the bar); where the system was already
in equilibrium, they would not.
16.3 Entropy flow and entropy production
Still considering the conduction example given above. If the heat flows into the left-hand
end of the bar due to an infinitesimal temperature difference, i.e. the process is reversible,
the left-hand reservoir loses entropy at the rate
(16.1)
3 18 Irreversible thermodynamics
Similarly the right-hand reservoir gains entropy at the rate
1 dQ
-- - -- -
dt i’; dt
Thus, the total change of entropy for the whole system is
(16.2)
(16.3)
Now T2 > T, and therefore the rate of change of entropy, dS/dt > 0.
To understand the meaning of this result it is necessary to consider a point in the bar.
At the point I from the left-hand end the thermometer reading is T. This reading is
independent of time and is the reading obtained on the thermometer in equilibrium with
the particular volume of the rod in contact with it. Hence, the thermometer indicates
the ‘temperature’ of that volume of the rod. Since the temperature is constant the
system is in a ‘steady state’, and at each point in the rod the entropy is invariant with
time. However, there is a net transfer of entropy from the left-hand reservoir to the
right-hand reservoir, i.e. entropy is ‘flowing’ along the rod. The total entropy of the
composite system is increasing with time and this phenomenon is known as ‘entropy
production’.
16.4 Thermodynamic forces and thermodynamic velocities
It has been suggested that for systems not far removed from equilibrium the development
of the relations used in the thermodynamics of the steady state should proceed along
analogous lines to the study of the dynamics of particles, i.e. the laws should be of the
form
J=LX (16.4)
X is the thermodynamic force;
L is a coefficient independent of X and J and is scalar in form, while both J and X
are vector quantities.
where J is the thermodynamic velocity or flow;
The following simple relationships illustrate how this law may be applied.
Fourier’s equation for one-dimensional conduction of heat along a bar is
(16.5)
where Q = quantity of energy (heat); T = temperature; A = area of cross-section; 1 = length;
k = thermal conductivity.
Ohm’s Law for flow of electricity along a wire, which is also one-dimensional, is
(16.6)
where I = current; qr = charge (coulomb); e = potential difference (voltage); A = area of
cross-section of wire; I = length; A = electrical conductivity.
Page 4

16
I r reve rsi bl e The rmod yna m ics
16.1 Introduction
Classical thermodynamics deals with transitions from one equilibrium state to another and
since it does not analyse the changes between state points it could be called thermostatics.
The term thermodynamics will be reserved, in this chapter, for dynamic non-equilibrium
processes.
In previous work, phenomenological laws have been given which describe irreversible
processes in the form of proportionalities, e.g. Fourier’s law of heat conduction, Ohm’s law
relating electrical current and potential gradient, Fick’s law relating flow of matter and
concentration gradient etc. When two of these phenomena occur simultaneously they
interfere, or couple, and give rise to new effects. One such cross-coupling is the reciprocal
effect of thermoelectricity and electrical conduction: the Peltier effect (evolution or
absorption of heat at a junction due to the flow of electrical current) and thermoelectric
force (due to maintenance of the junctions at different temperatures). It is necessary to
formulate coupled equations to deal with these phenomena, which are ‘phenomenological’
inasmuch as they are experimentally verified laws but are not a part of the comprehensive
theory of irreversible processes.
It is possible to examine irreversible phenomena by statistical mechanics and the
kinetic theory but these methods are on a molecular scale and do not give a good
macroscopic theory of the processes. Another method of considering non-equilibrium
processes is based on ‘pseudo-thermostatic theories’. Here, the laws of thermostatics
are applied to a part of the irreversible process that is considered to be reversible and
the rest of the process is considered as irreversible and not taken into account.
Thomson applied the second law of thermostatics to thermoelectricity by considering
the Thomson and Peltier effects to be reversible and the conduction effects to be
irreversible. The method was successful as the predictions were confirmed by
experiment but it has not been possible to justify Thomson’s hypothesis from general
considerations.
Systematic macroscopic and general thermodynamics of irreversible processes can be
obtained from a theorem published by Onsager (1931a,b). This was developed from
statistical mechanics and the derivation will not be shown but the results will be used. The
theory, based on Onsager’s theorem, also shows why the incorrect thermostatic methods
give correct results in a number of cases.
Entropy flow and entropy production 3 17
16.2 Definition of irreversible or steady state thermodynamics
All previous work on macroscopic ‘thermodynamics’ has been related to equilibrium. A
system was said to be in equilibrium when no spontaneous process took place and all the
thermodynamic properties remained unchanged. The macroscopic properties of the system
were spatially and temporally invariant.
System 2
: System boundary
Distance
Fig. 16.1 Steady state conduction of heat along a bar
Consider the system shown in Fig 16.1, in which a thermally insulated rod connects two
reservoirs at temperatures T2 and T, respectively. Heat flows between the two reservoirs by
conduction along the rod. If the reservoirs are very large the conduction of energy out of,
or into, them can be considered not to affect them. A state will be achieved when the rate
of heat flow, dQ/dt, entering the rod equals the rate of heat flow, -dQ/dt, leaving the rod.
If a thermometer were inserted at any point in the rod the reading would not change with
time, but it would be dependent on position. If the rod were of uniform cross-section the
temperature gradient would be linear. (This is the basis of the Searle’s bar conduction
experiment. )
The temperature in the bar is therefore a function of position but is independent of time.
The overall system is in a ‘stationary’ or ‘steady’ state but not in ‘equilibrium’, for that
requires that the temperature be uniform throughout the system. If the metal bar were
isolated from all the influences of the surroundings and from the heat sources, i.e. if it is
made an isolated system, the difference between the steady state and equilibrium becomes
obvious. In the case where the system was in steady state, processes would occur after the
isolation (equalisation of temperature throughout the bar); where the system was already
in equilibrium, they would not.
16.3 Entropy flow and entropy production
Still considering the conduction example given above. If the heat flows into the left-hand
end of the bar due to an infinitesimal temperature difference, i.e. the process is reversible,
the left-hand reservoir loses entropy at the rate
(16.1)
3 18 Irreversible thermodynamics
Similarly the right-hand reservoir gains entropy at the rate
1 dQ
-- - -- -
dt i’; dt
Thus, the total change of entropy for the whole system is
(16.2)
(16.3)
Now T2 > T, and therefore the rate of change of entropy, dS/dt > 0.
To understand the meaning of this result it is necessary to consider a point in the bar.
At the point I from the left-hand end the thermometer reading is T. This reading is
independent of time and is the reading obtained on the thermometer in equilibrium with
the particular volume of the rod in contact with it. Hence, the thermometer indicates
the ‘temperature’ of that volume of the rod. Since the temperature is constant the
system is in a ‘steady state’, and at each point in the rod the entropy is invariant with
time. However, there is a net transfer of entropy from the left-hand reservoir to the
right-hand reservoir, i.e. entropy is ‘flowing’ along the rod. The total entropy of the
composite system is increasing with time and this phenomenon is known as ‘entropy
production’.
16.4 Thermodynamic forces and thermodynamic velocities
It has been suggested that for systems not far removed from equilibrium the development
of the relations used in the thermodynamics of the steady state should proceed along
analogous lines to the study of the dynamics of particles, i.e. the laws should be of the
form
J=LX (16.4)
X is the thermodynamic force;
L is a coefficient independent of X and J and is scalar in form, while both J and X
are vector quantities.
where J is the thermodynamic velocity or flow;
The following simple relationships illustrate how this law may be applied.
Fourier’s equation for one-dimensional conduction of heat along a bar is
(16.5)
where Q = quantity of energy (heat); T = temperature; A = area of cross-section; 1 = length;
k = thermal conductivity.
Ohm’s Law for flow of electricity along a wire, which is also one-dimensional, is
(16.6)
where I = current; qr = charge (coulomb); e = potential difference (voltage); A = area of
cross-section of wire; I = length; A = electrical conductivity.
Onsager's reciprocal relation 319
Fick's Law for the diffusion due to a concentration gradient is, in one dimension
dCi
dn,
dt dl
-=-k-
(16.7)
where ni = amount of substance i, Ci = concentration of component i, and k = diffusion
coefficient. This law was derived in relation to biophysics by analogy with the laws of
thermal conduction to explain the flow of matter in living organisms. It will be shown later
that this equation is not as accurate as one proposed by Hartley, in which the gradient of
the ratio of chemical potential to temperature is used as the driving potential.
Other similar relationships occur in physics and chemistry but will not be given here.
The three equations given above relate the flow of one quantity to a difference in potential:
hence, there is a flow term and a force term as suggested by eqn (16.4). It will be shown
that although eqns (16.5), (16.6) and (16.7) appear to have the correct form, they are not
the most appropriate relationships for some problems. Equations (16.5), (16.6) and (16.7)
also define the relationship between individual fluxes and potentials, whereas in many
situations the effects can be coupled.
16.5 Onsager's reciprocal relation
If two transport processes are such that one has an effect on the other, e.g. heat conduction
and electricity in thermoelectricity, heat conduction and diffusion of gases, etc., then the
two processes are said to be coupled. The equations of coupled processes may be written
Equation (16.8) may also be written in matrix form as
(16.8)
(16.8a)
It is obvious that in this equation the basic processes are defined by the diagonal
coefficients in the matrix, while the other processes are defined by the off-diagonal terms.
Consider the situation where diffusion of matter is occurring with a simultaneous
conduction of heat. Both of these processes are capable of transferring energy through
a system. The diffusion process achieves this by mass transfer, i.e. each molecule of
matter carries some energy with it. The thermal conductivity process achieves the
transfer of heat by the molecular vibration of the matter transmitting energy through
the system. Both achieve a similar result of redistributing energy but by different
methods. The diffusion process also has the effect of redistributing the matter
throughout the system, in an attempt to achieve the equilibrium state in which the
matter is evenly distributed with the minimum of order (i.e. the maximum entropy or
minimum chemical potential). It can be shown, by a more complex argument, that
thermal conduction will also have an effect on diffusion. First, if each process is
considered in isolation the equations can be written
J, = L,, X,
J2 = L2,X2
the equation of conduction without any effect due to diffusion
the equation of diffusion without any effect due to conduction
Page 5

16
I r reve rsi bl e The rmod yna m ics
16.1 Introduction
Classical thermodynamics deals with transitions from one equilibrium state to another and
since it does not analyse the changes between state points it could be called thermostatics.
The term thermodynamics will be reserved, in this chapter, for dynamic non-equilibrium
processes.
In previous work, phenomenological laws have been given which describe irreversible
processes in the form of proportionalities, e.g. Fourier’s law of heat conduction, Ohm’s law
relating electrical current and potential gradient, Fick’s law relating flow of matter and
concentration gradient etc. When two of these phenomena occur simultaneously they
interfere, or couple, and give rise to new effects. One such cross-coupling is the reciprocal
effect of thermoelectricity and electrical conduction: the Peltier effect (evolution or
absorption of heat at a junction due to the flow of electrical current) and thermoelectric
force (due to maintenance of the junctions at different temperatures). It is necessary to
formulate coupled equations to deal with these phenomena, which are ‘phenomenological’
inasmuch as they are experimentally verified laws but are not a part of the comprehensive
theory of irreversible processes.
It is possible to examine irreversible phenomena by statistical mechanics and the
kinetic theory but these methods are on a molecular scale and do not give a good
macroscopic theory of the processes. Another method of considering non-equilibrium
processes is based on ‘pseudo-thermostatic theories’. Here, the laws of thermostatics
are applied to a part of the irreversible process that is considered to be reversible and
the rest of the process is considered as irreversible and not taken into account.
Thomson applied the second law of thermostatics to thermoelectricity by considering
the Thomson and Peltier effects to be reversible and the conduction effects to be
irreversible. The method was successful as the predictions were confirmed by
experiment but it has not been possible to justify Thomson’s hypothesis from general
considerations.
Systematic macroscopic and general thermodynamics of irreversible processes can be
obtained from a theorem published by Onsager (1931a,b). This was developed from
statistical mechanics and the derivation will not be shown but the results will be used. The
theory, based on Onsager’s theorem, also shows why the incorrect thermostatic methods
give correct results in a number of cases.
Entropy flow and entropy production 3 17
16.2 Definition of irreversible or steady state thermodynamics
All previous work on macroscopic ‘thermodynamics’ has been related to equilibrium. A
system was said to be in equilibrium when no spontaneous process took place and all the
thermodynamic properties remained unchanged. The macroscopic properties of the system
were spatially and temporally invariant.
System 2
: System boundary
Distance
Fig. 16.1 Steady state conduction of heat along a bar
Consider the system shown in Fig 16.1, in which a thermally insulated rod connects two
reservoirs at temperatures T2 and T, respectively. Heat flows between the two reservoirs by
conduction along the rod. If the reservoirs are very large the conduction of energy out of,
or into, them can be considered not to affect them. A state will be achieved when the rate
of heat flow, dQ/dt, entering the rod equals the rate of heat flow, -dQ/dt, leaving the rod.
If a thermometer were inserted at any point in the rod the reading would not change with
time, but it would be dependent on position. If the rod were of uniform cross-section the
temperature gradient would be linear. (This is the basis of the Searle’s bar conduction
experiment. )
The temperature in the bar is therefore a function of position but is independent of time.
The overall system is in a ‘stationary’ or ‘steady’ state but not in ‘equilibrium’, for that
requires that the temperature be uniform throughout the system. If the metal bar were
isolated from all the influences of the surroundings and from the heat sources, i.e. if it is
made an isolated system, the difference between the steady state and equilibrium becomes
obvious. In the case where the system was in steady state, processes would occur after the
isolation (equalisation of temperature throughout the bar); where the system was already
in equilibrium, they would not.
16.3 Entropy flow and entropy production
Still considering the conduction example given above. If the heat flows into the left-hand
end of the bar due to an infinitesimal temperature difference, i.e. the process is reversible,
the left-hand reservoir loses entropy at the rate
(16.1)
3 18 Irreversible thermodynamics
Similarly the right-hand reservoir gains entropy at the rate
1 dQ
-- - -- -
dt i’; dt
Thus, the total change of entropy for the whole system is
(16.2)
(16.3)
Now T2 > T, and therefore the rate of change of entropy, dS/dt > 0.
To understand the meaning of this result it is necessary to consider a point in the bar.
At the point I from the left-hand end the thermometer reading is T. This reading is
independent of time and is the reading obtained on the thermometer in equilibrium with
the particular volume of the rod in contact with it. Hence, the thermometer indicates
the ‘temperature’ of that volume of the rod. Since the temperature is constant the
system is in a ‘steady state’, and at each point in the rod the entropy is invariant with
time. However, there is a net transfer of entropy from the left-hand reservoir to the
right-hand reservoir, i.e. entropy is ‘flowing’ along the rod. The total entropy of the
composite system is increasing with time and this phenomenon is known as ‘entropy
production’.
16.4 Thermodynamic forces and thermodynamic velocities
It has been suggested that for systems not far removed from equilibrium the development
of the relations used in the thermodynamics of the steady state should proceed along
analogous lines to the study of the dynamics of particles, i.e. the laws should be of the
form
J=LX (16.4)
X is the thermodynamic force;
L is a coefficient independent of X and J and is scalar in form, while both J and X
are vector quantities.
where J is the thermodynamic velocity or flow;
The following simple relationships illustrate how this law may be applied.
Fourier’s equation for one-dimensional conduction of heat along a bar is
(16.5)
where Q = quantity of energy (heat); T = temperature; A = area of cross-section; 1 = length;
k = thermal conductivity.
Ohm’s Law for flow of electricity along a wire, which is also one-dimensional, is
(16.6)
where I = current; qr = charge (coulomb); e = potential difference (voltage); A = area of
cross-section of wire; I = length; A = electrical conductivity.
Onsager's reciprocal relation 319
Fick's Law for the diffusion due to a concentration gradient is, in one dimension
dCi
dn,
dt dl
-=-k-
(16.7)
where ni = amount of substance i, Ci = concentration of component i, and k = diffusion
coefficient. This law was derived in relation to biophysics by analogy with the laws of
thermal conduction to explain the flow of matter in living organisms. It will be shown later
that this equation is not as accurate as one proposed by Hartley, in which the gradient of
the ratio of chemical potential to temperature is used as the driving potential.
Other similar relationships occur in physics and chemistry but will not be given here.
The three equations given above relate the flow of one quantity to a difference in potential:
hence, there is a flow term and a force term as suggested by eqn (16.4). It will be shown
that although eqns (16.5), (16.6) and (16.7) appear to have the correct form, they are not
the most appropriate relationships for some problems. Equations (16.5), (16.6) and (16.7)
also define the relationship between individual fluxes and potentials, whereas in many
situations the effects can be coupled.
16.5 Onsager's reciprocal relation
If two transport processes are such that one has an effect on the other, e.g. heat conduction
and electricity in thermoelectricity, heat conduction and diffusion of gases, etc., then the
two processes are said to be coupled. The equations of coupled processes may be written
Equation (16.8) may also be written in matrix form as
(16.8)
(16.8a)
It is obvious that in this equation the basic processes are defined by the diagonal
coefficients in the matrix, while the other processes are defined by the off-diagonal terms.
Consider the situation where diffusion of matter is occurring with a simultaneous
conduction of heat. Both of these processes are capable of transferring energy through
a system. The diffusion process achieves this by mass transfer, i.e. each molecule of
matter carries some energy with it. The thermal conductivity process achieves the
transfer of heat by the molecular vibration of the matter transmitting energy through
the system. Both achieve a similar result of redistributing energy but by different
methods. The diffusion process also has the effect of redistributing the matter
throughout the system, in an attempt to achieve the equilibrium state in which the
matter is evenly distributed with the minimum of order (i.e. the maximum entropy or
minimum chemical potential). It can be shown, by a more complex argument, that
thermal conduction will also have an effect on diffusion. First, if each process is
considered in isolation the equations can be written
J, = L,, X,
J2 = L2,X2
the equation of conduction without any effect due to diffusion
the equation of diffusion without any effect due to conduction
320 Irreversible thermodynamics
Now, the diffusion of matter has an effect on the flow of energy because the individual,
diffusing, molecules carry energy with them, and hence an effect for diffusion must be
included in the term for thermal flux, J, . Hence
J1= LIlX, + LI2X2 (16.9)
where LIZ is the coupling coeficient showing the effect of mass transfer (diffusion) on
energy transfer.
In a similar manner, because conduction has an effect on diffusion, the equation for
mass transfer can be written
(16.10)
52 = L21 Xl + L22X2
where L,, is the coupling coefficient for these phenomena.
A general set of coupled linear equations is
(16.11)
k
The equations are of little use unless more is known about the forces X, and the
coefficients Lip This information can be obtained from Onsager’s reciprocal relation.
There is considerable latitude in the choice of the forces X, but Onsager’s relation chooses
the forces in such a way that when each flow Ji is multiplied by the appropriate force Xi,
the sum of these products is equal to the rate of creation of entropy per unit volume of the
system, 8, multiplied by the temperature, T. Thus
TO = JIXl + J2X2 + ... = CAXi (16.12)
i
Equation (16.12) may be rewritten
(16.13)
where
Onsager further showed that if the above mentioned condition was obeyed, then, in
( 1 6.14)
This means that the coupling marrix in eqn (16.8a) is symmetric, Le. L,, = L,, for the
particular case given above. The significance of this is that the effects of parameters on
each other are equivalent irrespective of which is judged to be the most, or least,
significant parameter. Consideration will show that if this were not true then it would be
possible to construct a system which disobeyed the laws of thermodynamics. It is not
proposed to derive Onsager’s relation, which is obtained from molecular considerations -
it will be assumed to be true.
In summary, the thermodynamic theory of an irreversible process consists of first
finding the conjugate fluxes and forces, Ji and xi, from eqn (16.13) by calculating the
entropy production. Then a study is made of the phenomenological equations (16.11) and
general
Lik = Lki
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