J. Phys. Chem. 1996, 100, 9193-9195
9193
van’t Hoff’s Equation for Endoreversible Chemical Reactions F. Angulo-Brown* and L. A. Arias-Herna´ ndez Departamento de Fı´sica, Escuela Superior de Fı´sica y Matema´ ticas, Instituto Polite´ cnico Nacional, Edificio # 9, UP Zacatenco, 07738, Me´ xico, DF, Me´ xico ReceiVed: October 9, 1995; In Final Form: February 25, 1996X
We propose that the so-called van’t Hoff’s equation for the enthalpy changes in a chemical reaction may be modified in the context of endoreversible thermodynamics. This modification takes into account the role played by the heat supplied by the thermal bath for maintaining constant the temperature of the chemical system. We analyze the reaction of hydrogen iodide synthesis by means of the modified van’t Hoff’s equation.
1. Introduction In the last two decades an endoreversible thermodynamics (ET) has been developed.1-5 This discipline started from the study of Carnot-like models for thermal engines operating in finite time. These models consist of a class of irreversible heat engines where their whole entropy production is ascribed only to the coupling between the working substance and its surroundings and it is permitted that the working fluid undergoes only reversible transformations. By means of this approach it has been possible to obtain successful models and optimizations of real heat engines.6-9 The ET scope has been extended beyond the ordinary heat engines, such as occurred with the early thermodynamics in the past century.10-15 In particular, Ondrechen et al.,16,17 De Vos,18,19 and Gordon et al.20,21 extended the endoreversibility concept toward chemical engines, i.e., engines that convert differences in chemical potential into work. These engines are the analogues of heat engines driving heat transfer for producing work from temperature differences. In ref 10 another ET extension was introduced. In that paper the superconducting transition was analyzed by means of the socalled method of Carnot cycles (MCC)11 in a finite-time thermodynamics context, obtaining a generalized Rutgers’ relation for the finite discontinuity of the specific heat at the superconducting critical temperature. In the present paper following a method of endoreversible Carnot cycles (MECC) applied to the van’t Hoff equilibrium box, we obtain a generalized van’t Hoff’s equation for gas reactions. The new equation is used for analyzing experimental data of heat of reaction for the formation of hydrogen iodide.
Figure 1. Pressure-volume diagram of an infinitesimal Carnot cycle.
efficiency ηC of the cycle is
ηC )
dA T - dT dT )1) |Q| T T
(1)
In a second step, dA is evaluated in terms of the variables of the thermodynamic space considered for the working substance, and so a relationship among the variables defining dA, the heat involved in the process, and the temperature is obtained. Naturally, all the relations obtained with this method can also be derived by using thermodynamic potentials. As is well-known, the reversible Carnot efficiency, given by
ηC ) 1 -
T2 T1
(2)
2. Method of Endoreversible Carnot Cycles By means of the second law of thermodynamics in its original form, Becker22 and other authors23,24 have obtained several important thermodynamic equilibrium relations, for example, the Clausius-Clapeyron equation, the Rutgers relation, the law of mass action, and the Stefan-Boltzmann law. This is accomplished by applying the so-called method of Carnot cycles (MCC). This method takes into account the constraints of reversibility and equilibrium of the classical thermodynamics through the properties of the Carnot cycle. Briefly, the method consists in letting the system of interest undergo a virtual infinitesimal Carnot cycle between two very close isotherm branches at temperatures T and T - dT, as shown in Figure 1. If the area of the cycle is denoted as dA and the absorbed heat in the isothermic branch at temperature T is Q, then the Carnot X
Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(95)03017-6 CCC: $12.00
with T2 and T1 the absolute temperatures of the cold and hot reservoirs, respectively, is an upper bound for the efficiency of real heat engines. This maximum efficiency limit is only reachable in the completely reversible regime with no delivered power and with vanishing entropy production. Thus, eq 1 has the same restraints as eq 2. Within the finite-time thermodynamics context it has been possible to build endoreversible heat engine models which produce power output and entropy. For example, for the endoreversible Curzon and Ahlborn (CA) engine,25 where the working fluid exchanges heat with its reservoirs by means of the so-called Newton’s law of heat conduction, there exists a maximum power regime with efficiency given by
ηCA ) 1 -
x
© 1996 American Chemical Society
T2 T1
(3)
9194 J. Phys. Chem., Vol. 100, No. 21, 1996
Angulo-Brown and Arias-Herna´ndez of this device and the MCC, obtained the so-called van’t Hoff’s equation, given by
d ln KC Q ) 2 dT RT
(9)
where Q is the molar heat of reaction, R is the gas constant, and KC is the constant of mass action, for example, given by
CACB CAB
(10)
A + B ) AB
(11)
KC(T) ) for the reaction
Figure 2. Temperature-entropy diagram of an endoreversible Carnot cycle: (a) with thermal resistances r1 and r2 at both couplings; (b) with thermal resistance r2 at the cold coupling and thermal equilibrium at the hot coupling; (c) with thermal resistance r1 at the hot coupling and thermal equilibrium at the cold coupling.
being CA,B,AB the molar concentrations in the equilibrium box (moles per unit volume). In Becker’s derivation of eq 9 by means of the reaction A + B f AB in a box at temperature T and the inverse reaction AB f A + B in a box at temperature T - dT, the following equation is obtained22
which is between zero and ηC. If the CA engine works in a regime maximizing the function E, given by
dA h
E ) P - T2σ
dT
(4)
where P is the power output of the cycle, σ the total entropy production (system plus surroundings) per cycle, and T2 is the temperature of the cold reservoir, then ref 26 shows that the cycle’s efficiency will be
ηE ≈ 1/2(ηC + ηCA)
(5)
This optimization criterion leads to a configuration of the CA cycle such that for maximum E, it produces around 80% of the maximum power and only about 30% of the entropy produced in the maximum power regime. De Vos27 showed that eq 3 holds for the three cycle configurations depicted in Figure 2. Thus, we can use the case b in such a figure for applying results as eqs 3 and 5 to the infinitesimal Carnot cycle of Figure 1. For the endoreversible maximum power regime we have,
dA )1ηCA ) Q
x
T - dT dT ≈ T 2T
(6)
(
dA 1 T - dT ≈ 1+1Q 2 T
xT -T dT) ≈ dT/ T 4
(7)
3
In both cases we cut off a Taylor expansion at first order. In an arbitrary finite-time regime we have
dA dT )R ηFT ≡ Q T
T
)
A h -Q (12)
T
where A h is the total gain in work, Q h is the heat supplied by the heat bath used to keep T constant, and Q is the molar heat of reaction. From eq 12 and using the expression for A h , given by22
[
A h ) RT ln
CA0CB0 0 CAB
]
- ln KC - 1
(13)
0 with CA,B,AB the molar concentrations of substances A, B, and AB in the auxiliary containers of the van’t Hoff’s box, Becker immediately obtains eq 9. If we follow step by step Becker’s procedure but using eq 8 instead eq 1, we obtain
dA h dT
)
RQ h T
-
R(A h - Q) T
(14)
Using eq 13 to obtain the derivative of A h and substituting it into eq 14, we obtain
Q h Q + (1 - R)Q h d ln KC Q (15) ) 2 + (1 - R) 2 ) 2 dT RT RT RT
and for the maximum E regime we obtain
ηE )
Q h
)
(8)
with 0 e R e 1, where for example, R ) 1/2 for the maximum power operation, R ) 3/4 for the maximum-E case, and R ) 1 for the maximum efficiency regime. The coefficient R will have implications for every result obtainable from the MCC. Thus, R in eq 8 is not a fitting parameter but a consequence of a second-law analysis. Equation 8 is the basis of a method of endoreversible Carnot cycles (MECC). 3. Endoreversible van’t Hoff Isotherm The van’t Hoff equilibrium box is a device for performing a gas chemical reaction in a reversible way. Becker,22 by means
which is a generalized van’t Hoff’s equation for endoreversible conditions. For the full reversible case R ) 1 w (1 - R) ) 0 and eq 9 is recovered. 4. Application to the Hydrogen Iodide Case We shall now apply eq 15 to the case of formation of hydrogen iodide. The experimental data we shall use for the heat of reaction are due to Bodestein28 and Taylor and Crist.28 The studied reaction is
H2 + I2 ) 2HI
(16)
If we assume that the true equilibrium heat of reaction (for this chemical transformation) is that obtained by means of equilibrium statistical mechanics,29 applied over a mixture of ideal gases, which is given [for the temperature interval 666.6781.25 K] by
Q ) ∆HreV ) -13.3 kJ/mol
(17)
Endoreversible Chemical Reactions
J. Phys. Chem., Vol. 100, No. 21, 1996 9195
we can estimate the deviations of the ∆H experimental data respect to this value (H is the enthalpy). For example, Taylor and Crist (TC)28 reported for the chemical equation (16) a heat of reaction
∆HTC ) -12.6 ( 0.076 kJ/mol
(18)
[for the temperature interval 666.8-763.8 K]. As it can be seen the experimental error is very small (in the order of 0.077 kJ/ mol). These authors showed in their paper Bodenstein’s experimental points, in KC Vs 1/T, for the same reaction. A least-squares fitting of these data leads to
∆HB ) -15.7 kJ/mol
(19)
[for the temperature interval 666.66-781.25 K]. Both experimental values of the heat of reaction were obtained by integration of equation (9), given by
ln KC ) -
∆H +b RT
(20)
believe that this operation was the cause of an overestimation of ∆HB beyond the values permitted by a second law analysis. 5. Conclusions Endoreversible thermodynamics has shown that it may be useful to treat problems beyond thermal engines. In this paper we propose that by means of ET ideas it is possible to modify the expression of the so-called van’t Hoff’s isotherm for the case where finite exchanges of heat between the system and its reservoirs are considered. These exchanges unavoidably generate entropy and this fact must be expressed in a modification of equilibrium relations. For the reaction studied here, we propose that experimental deviations from theoretical equilibrium values of ∆H are influenced for irreversible heat flows. In fact, the endoreversible contributions (1 - R) Q h are considerably larger than experimental errors in ∆Hexp. Thus, our treatment gives some insights over the analysis of experimental heat reactions. Our approach permits reactions to occur endoreversibly, and all irreversibility sources are located at the links between system and surroundings.
where ∆H/R is the slope of this straight line in the plane ln KC - 1/T and b is a constant. If we assume that experimental ∆H values mentioned here correspond to endoreversible values, given by eq 15
Acknowledgment. This work was partially supported by COFAA-IPN (FAB) and CONACYT (L.A.A.H.).
h ) ∆Hrev + (1 - R) Q h ∆Hexp ) ∆Her ) Q + (1 - R) Q (21)
(1) Sieniutycz, S., Salamon, P., Eds. Finite Time Thermodynamics and Thermoeconomics; Taylor and Francis: New York, 1990. (2) De Vos, A. EndoreVersible Thermodynamics of Solar Energy ConVersion; Oxford University Press: Oxford, U.K., 1992. (3) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; Wiley: New York, 1985. (4) Andresen, B.; Salamon, P.; Berry, R. S. Phys. Today 1984, 37, 62. (5) Bejan, A. AdVanced Engineering Thermodynamics; Wiley: New York, 1988. (6) Mozurkewich, M.; Berry, R. S. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 1986. (7) Mozurkewich, M.; Berry, R. S. J. Appl. Phys. 1982, 53, 34. (8) Hoffmann, K. H.; Watowich, S. J.; Berry, R. S. J. Appl. Phys. 1985, 58, 2125. (9) Angulo-Brown, F.; Ferna´ndez-Betanzos, J.; Dı´az-Pico, C. A. Eur. J. Phys. 1994, 15, 38. (10) Angulo-Brown, F.; Ye´pez, E.; Zamorano-Ulloa, R. Phys. Lett. A 1993, 183, 431. (11) Torres, J. L. Il NuoVo Cimento 1991, 13D, 177. (12) Watowich, S. J.; Hoffmann, K. H.; Berry, R. S. J. Appl. Phys. 1985, 58, 2893. (13) Watowich, S. J.; Hoffmann, K. H.; Berry, R. S. Il NuoVo Cimento 1989, 104B, 131. (14) Brown, G. R.; Snow, S.; Andresen, B.; Salamon, P. Phys. ReV. A 1986, 34, 4370. (15) Gordon, J. M.; Zarmi, Z. Am. J. Phys. 1989, 57, 995. (16) Ondrechen, M.; Berry, R. S.; Andresen, B. J. Chem. Phys. 1980, 72, 5118. (17) Ondrechen, M.; Berry, R. S.; Andresen, B. J. Chem. Phys. 1980, 73, 5838. (18) De Vos, A. J. Phys. Chem. 1991, 95, 4534. (19) De Vos, A. Solar Cells 1991, 31, 181. (20) Gordon, J. M.; Orlov, V. N. J. Appl. Phys. 1993, 74, 5303. (21) Gordon, J. M. J. Appl. Phys. 1993, 73, 8. (22) Becker, R. Theory of Heat; Springer: Berlin, 1967; Chapter 17. (23) Pippard, A. B. Elements of Classical Thermodynamics; Cambridge University Press: Cambridge, U.K., 1981. (24) Sears, F. W. An Introduction to Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics; Addison-Wesley: New York, 1966. (25) Curzon, F. L.; Ahlborn, B. Am. J. Phys. 1975, 43, 22. (26) Angulo-Brown, F. J. Appl. Phys. 1991, 69, 7465. Arias-Herna´ndez, L. A.; Angulo-Brown, F. ReV. Mex. Fı´s. 1994, 40, 866. (27) De Vos, A. Am. J. Phys. 1985, 53, 570. (28) Taylor, A. H., Jr.; Crist, R. H. J. Am. Chem. Soc. 1941, 63, 1377. (29) McQuarrie, D. A. Statistical Thermodynamics; Harper and Row: New York, 1973; Chapter 9.
then we can make comparisons between the theoretical equilibrium value ∆Hrev, and the experimental ones given by Bodenstein and Taylor-Crist. For the TC case we have
h (22) ∆HTC ) -12.6 kJ/mol ) -13.3 kJ/mol + (1 - R) Q and therefore
(1 - R) Q h ) 0.7 kJ/mol
(23)
which is a quantity considerably larger than the experimental error in eq 18. Thus, since (1 - R) g 0, the heat supplied by the heat bath Q h must be a positive quantity, just as expected for a endothermic formation reaction. In the case of Bodenstein’s experiment, we have
h ∆HB ) -15.7 kJ/mol ) -13.3 kJ/mol + (1 - R) Q
(24)
and
(1 - R) Q h ) -2.4 kJ/mol
(25)
which implies (1 - R) e 0, since Q h (the heat supplied by the bath) is necessarily positive. Unfortunately, Bodenstein’s experimental errors are not available to us, since his original paper was published on 1897.28 Thus, in this case, the experimental ∆HB is not consistent with the Carnot theorem. Therefore, Bodenstein’s data violate the second law of thermodynamics and it is necessary to provide a possible explanation for this fact. As we said, Bodenstein’s data were taken from the TC paper,28 and these authors assert that “due to the plot of ln KC against 1/T of Bodenstein’s experimental data shows considerable deviation from a straight line and a proper estimation of the slope can not be made without some means of weighting the high and low temperature values...”. Apparently, Taylor and Crist made such weighting for fitting Bodenstein’s data to a straight line with a well-defined slope. We
References and Notes
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