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On the thermodynamic efficiency of galvanic cells
Electrochimica
Acta.
1966. Vol.
11. pp. 125 to 127.
Pergamon
Press Ltd.
Printed
in Northern
I&and
SHORT COMMUNICATION
ON THE THERMODYNAMIC EFFICIENCY OF GALVANIC CELLS* P. VAN RYSSELBERGHE Departmentsof Chemistryand ChemicalEngineering,Stanford University, California, U.S.A. CURRENTdiscussions of the thermodynamic efficiency of galvanic cells (particularly fuel cells) as transformers of chemical into electric energy usually remain remote from the type of reasoning provided by the thermodynamics of irreversible processes. The customary analysis of the problem starts, and very nearly finishes, with the use of the classical formula AG = AH - TAS,
(1)
connecting the changes in the free enthalpy G, the enthalpy H and the entropy S due to a cell reaction taking place at the constant temperature T. This formula is readily obtained from the definition G = H -TS. Under reversible conditions the external electric work performed by the cell is, at constant T and constant pressure p, W rev.=
-AG,
(2)
and the efficiency in terms of AG is wrev,/(- AG) = 1, while that in terms of AH is w,,,./(-AH)
= 1 - T. (AS/AH).
(3)
Under irreversible conditions the electric work w and the corresponding efficiencies are smaller on account of polarization effects at the electrodes and of the ohmic drop within the cell, a statement which we can render quantitative by making use of the irreversible thermodynamic point of view. This we shall do in two stages: we shall first pursue the reasoning in terms of the finite differences considered above, and, second we shall examine an infinitesimal process and treat it by the De Donder method.l,2 The entropy change AS consists in general of an external contribution Aext. S and of an internal entropy production hint. S 2 0. Whether we regard the cell as a closed or as an open system the Aext.S corresponding to the AS of (1) is equal to the heat Q received by the system. We thus have AG=
AH-
From the first law of thermodynamics
Q - T. Ain,S. we have, at constant
AH=Q-w * Manuscriptreceived 3 May 1965
125
(4) p:
(9
126
Short communication
and formula (4) becomes AG = -w - T. Ain,,S.
(6)
The efficiency in terms of AG is then w/(-AG)
= 1 - (-T.
Ai,t.S/AG).
(7)
In order to express Aint,S in terms of overtensions and of the ohmic drop, the more precise reasoning involving an infinitesimal process is needed. This is also the case for Q. The efficiency in terms of AH is obtained from (5), w/(-AH)
= 1 - (-Q/AH).
(8)
Let us note in passing that, for one molar occurrence of the cell reaction, we have3n4 AG = -w,,,.
= --zFE = zFU,,,,,
(9)
in which z is the charge number, F the faraday, E the electromotive force and U,,,. the reversible electric tension of the cell. In order for (9) to be exact, the cell must be regarded as being of infinite size, the molar occurrence of the cell reaction leaving then the physical chemical state of the cell unaltered. This is another of the many reasons which make the reasoning by infinitesimals imperative. We shall now consider the infinitesimal process corresponding to the increase d5 of the degree of advancement 5 of the cell reaction. We have in general, whether Tand p vary or not: Vdp-dw
dH=dQ+
(10)
and Vdp-dw.
dG=dQ-TdS-SST+
(11)
If we introduce the uncompensated heat dQ’ and the entropy production according to the De Donder statement of the second law of thermodynamics
dint,S
dQ’ = T dS - dQ = T di,t.S > 0,
(12)
dG=
(13)
(11) becomes -SdT+
Vdp--w--Q’.
On the other hand we have the two exact differentials5 dH=CdT+[V-T.(W/aT)].dp-rd&
(14)
in which C is the instantaneous heat capacity (abbreviated notation for C,, and r is the instantaneous heat of reaction (abbreviated notation for rTv), and dG=
-SdT+
Vdp-Ad&
in which A is the instantaneous chemical affinity of the cell reaction. with (14) and (13) with (15) we find dw=rdE+dQ-CdT+T.(aV/aT).dp
(15)
Identifying (10)
(16)
and dw = A d5 - de’.
(17)
Short communication
127
We have shown elsewhere3*s that, in the case of the irreversible behaviour of a galvanic cell dQ’=zF.(ya+
b.4 +JW.d5,
(18)
in which qs, is the positive anodic overtension, qc is the negative cathodic overtension, R is the internal resistance of the cell and I is the positive electric current strength. We have here z and d5 positive because we consider the cell reaction as written and as occurring in its spontaneous direction. Making dE correspond to the lapse of time dt, we have I = zF. (dE/dt). (19) From (17) and (18) we obtain the instantaneous efficiency in terms of A dE, dw/Adt = I - dQ’/AdS = 1 - zF. (~a + lqcl + RO/A
(20)
and that in terms of r dl, dw/rd5 = A/r - zF. (+ + 1~1 + RZ)/r.
(21)
these quantities are obtainable Since r = -(i3H/a&, and A = -(aG/a&,, from enthalpy and free enthalpy data (sometimes available themselves from electrochemical measurements at equilibrium), while the overtensions and the ohmic drop result from non-equilibrium measurements. The strict foundation provided by the thermodynamics of irreversible processes should be the point of departure for any meaningful practical discussion of the efficiency of galvanic cells, of that of fuel cells in particular. REFERENCES 1. 2.
TH. DE DONDER, L’A@itL Gauthier-Villars, Paris (1927). TH. DE DONDER and P. VAN RYSSELBERGHE,Thermodynamic
Theory of Afinity. Stanford University Press(l936); L’ AfinitL Gauthier-Villars, Paris (1936). 3. P. VAN RYSSELEERGHE,EIectrochemicaf AJinity. Hermann, Paris (1955). 4. P. VAN RYSSELBERGHE,.I. Chem. Educ. 41,486 (1964). 5. P. VAN RYSSELBERGHE, Thermodynamics of Irreversible Processes. Hermann, Paris and Blaisdell, New York (1963).
Acta.
1966. Vol.
11. pp. 125 to 127.
Pergamon
Press Ltd.
Printed
in Northern
I&and
SHORT COMMUNICATION
ON THE THERMODYNAMIC EFFICIENCY OF GALVANIC CELLS* P. VAN RYSSELBERGHE Departmentsof Chemistryand ChemicalEngineering,Stanford University, California, U.S.A. CURRENTdiscussions of the thermodynamic efficiency of galvanic cells (particularly fuel cells) as transformers of chemical into electric energy usually remain remote from the type of reasoning provided by the thermodynamics of irreversible processes. The customary analysis of the problem starts, and very nearly finishes, with the use of the classical formula AG = AH - TAS,
(1)
connecting the changes in the free enthalpy G, the enthalpy H and the entropy S due to a cell reaction taking place at the constant temperature T. This formula is readily obtained from the definition G = H -TS. Under reversible conditions the external electric work performed by the cell is, at constant T and constant pressure p, W rev.=
-AG,
(2)
and the efficiency in terms of AG is wrev,/(- AG) = 1, while that in terms of AH is w,,,./(-AH)
= 1 - T. (AS/AH).
(3)
Under irreversible conditions the electric work w and the corresponding efficiencies are smaller on account of polarization effects at the electrodes and of the ohmic drop within the cell, a statement which we can render quantitative by making use of the irreversible thermodynamic point of view. This we shall do in two stages: we shall first pursue the reasoning in terms of the finite differences considered above, and, second we shall examine an infinitesimal process and treat it by the De Donder method.l,2 The entropy change AS consists in general of an external contribution Aext. S and of an internal entropy production hint. S 2 0. Whether we regard the cell as a closed or as an open system the Aext.S corresponding to the AS of (1) is equal to the heat Q received by the system. We thus have AG=
AH-
From the first law of thermodynamics
Q - T. Ain,S. we have, at constant
AH=Q-w * Manuscriptreceived 3 May 1965
125
(4) p:
(9
126
Short communication
and formula (4) becomes AG = -w - T. Ain,,S.
(6)
The efficiency in terms of AG is then w/(-AG)
= 1 - (-T.
Ai,t.S/AG).
(7)
In order to express Aint,S in terms of overtensions and of the ohmic drop, the more precise reasoning involving an infinitesimal process is needed. This is also the case for Q. The efficiency in terms of AH is obtained from (5), w/(-AH)
= 1 - (-Q/AH).
(8)
Let us note in passing that, for one molar occurrence of the cell reaction, we have3n4 AG = -w,,,.
= --zFE = zFU,,,,,
(9)
in which z is the charge number, F the faraday, E the electromotive force and U,,,. the reversible electric tension of the cell. In order for (9) to be exact, the cell must be regarded as being of infinite size, the molar occurrence of the cell reaction leaving then the physical chemical state of the cell unaltered. This is another of the many reasons which make the reasoning by infinitesimals imperative. We shall now consider the infinitesimal process corresponding to the increase d5 of the degree of advancement 5 of the cell reaction. We have in general, whether Tand p vary or not: Vdp-dw
dH=dQ+
(10)
and Vdp-dw.
dG=dQ-TdS-SST+
(11)
If we introduce the uncompensated heat dQ’ and the entropy production according to the De Donder statement of the second law of thermodynamics
dint,S
dQ’ = T dS - dQ = T di,t.S > 0,
(12)
dG=
(13)
(11) becomes -SdT+
Vdp--w--Q’.
On the other hand we have the two exact differentials5 dH=CdT+[V-T.(W/aT)].dp-rd&
(14)
in which C is the instantaneous heat capacity (abbreviated notation for C,, and r is the instantaneous heat of reaction (abbreviated notation for rTv), and dG=
-SdT+
Vdp-Ad&
in which A is the instantaneous chemical affinity of the cell reaction. with (14) and (13) with (15) we find dw=rdE+dQ-CdT+T.(aV/aT).dp
(15)
Identifying (10)
(16)
and dw = A d5 - de’.
(17)
Short communication
127
We have shown elsewhere3*s that, in the case of the irreversible behaviour of a galvanic cell dQ’=zF.(ya+
b.4 +JW.d5,
(18)
in which qs, is the positive anodic overtension, qc is the negative cathodic overtension, R is the internal resistance of the cell and I is the positive electric current strength. We have here z and d5 positive because we consider the cell reaction as written and as occurring in its spontaneous direction. Making dE correspond to the lapse of time dt, we have I = zF. (dE/dt). (19) From (17) and (18) we obtain the instantaneous efficiency in terms of A dE, dw/Adt = I - dQ’/AdS = 1 - zF. (~a + lqcl + RO/A
(20)
and that in terms of r dl, dw/rd5 = A/r - zF. (+ + 1~1 + RZ)/r.
(21)
these quantities are obtainable Since r = -(i3H/a&, and A = -(aG/a&,, from enthalpy and free enthalpy data (sometimes available themselves from electrochemical measurements at equilibrium), while the overtensions and the ohmic drop result from non-equilibrium measurements. The strict foundation provided by the thermodynamics of irreversible processes should be the point of departure for any meaningful practical discussion of the efficiency of galvanic cells, of that of fuel cells in particular. REFERENCES 1. 2.
TH. DE DONDER, L’A@itL Gauthier-Villars, Paris (1927). TH. DE DONDER and P. VAN RYSSELBERGHE,Thermodynamic
Theory of Afinity. Stanford University Press(l936); L’ AfinitL Gauthier-Villars, Paris (1936). 3. P. VAN RYSSELEERGHE,EIectrochemicaf AJinity. Hermann, Paris (1955). 4. P. VAN RYSSELBERGHE,.I. Chem. Educ. 41,486 (1964). 5. P. VAN RYSSELBERGHE, Thermodynamics of Irreversible Processes. Hermann, Paris and Blaisdell, New York (1963).