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 = LkiRead More

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