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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