Idea Transcript
Thermal Coefficients and Heat Capacities in Systems with Chemical Reaction The Le Chatelier-Braun Principle J. Giiemez
Departamento de Fisica Aplicada, Universidad de Cantabria, E-39005 Santander, Spain S. Velasco Departamento de Fisica Aplicada, Universidad de Salamanca, E-37008 Salamanca. Spain
M. A. Matias Departamento de Quimica Fisica, Universidad de Salamanca, E-37008 Salamanca, Spain The historical development of the study of the equilibrium states in chemical reactions is based much on the formulation of empirical laws or principles, extracted only from t h e empirical evidence. Among these i s t h e Le ChHtelier principle ( I ) : a system at equilibrium resists attempts to change its temperature, pressure, or concentration of a reagent. The standard presentation of the principle has been the center of some criticism for being sometimes inaccurate or, at least, too vague (2).Nowadays, thermodynamics offers a solid and rigorous basis for the effect of external perturbations on a chemical reaction. Namely, the basis of the Le ChHtelier principle is in the stability conditions of thermodynamic equilibrium states against external perturbations. Although considerable attention has been dedicated to the discussion and applications of the Le Chgtelier's principle (3-101, no similar study has appeared in the peda-
gogical literature dedicated to the so-called Le ChHtelierBraun principle, which allows a more subtle interpretation of the way in which a system returns to the equilibrium state (11, 12): while a system tries to reduce the external perturbation, other processes are also induced which lead to the same result. In this article we illustrate the way in which the occurrence of a chemical reaction affects the response of a system against external mechanical or thermal perturbations, i n light of the Le ChHtelier-Braun principle. To illustrate an application consider the possibility of switching on or off the occurrence of a chemical reaction using a suitable catalyst, and then analyzing in both cases the response of the system to the external perturbation. Some Formal Results In this section we shall develop some general thermodynamic relationships to be used later on. Consider a general
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thcmnodynamw potential, that ii, tht, intcrnal cnwgy l.', a function of the extcnsiw variables S, 1: N , ... or one of its Legendre transforms (II), i n which one or more extensive variables are replaced by their intensive conjugate variables. Thus, one has the thermodynamic potential WXz, X2, ..., X,,Ir+,. ..., In),function of r extensive variables Xi and n - r intensive variables I,, for which the total differential takes the form, d 0 = X1;dK- xqdIj i=l
j=,+l
(1)
rather than the integral quantity implied by the difference between the final and initial states. Thermodynamic Coefficients Response of a System to External Perturbations 'Ib study the response of a system to external perturhations we shall consider the thermal coefficients, namely thermal expansion, isothermal compressibility, and specific heat a t constant pressure and volume in the case where a chemical reaction may occur. First we shall focus on the volume, considering a s a function V = K'I: 1: 51, to obtain for a n infinitesimal process,
The stability criteria of thermodynamics require that this potential have a minimum for the equilibrium state, and thus, that the second derivative of 0 he positive. From this condition we get (11, 121, Variation of Volume
From the previous expression one may get the following for the variation of volume with P and T, However, be careful when the intensive variable is defined with a minus sign, as for pressure P. If we consider a multicomponent closed system in which a chemical reaction may occur, then i t is useful to switch from the mole numbers as the measure of chemical composit,ion to a new variable 5 called extent of the reaction (or degree of advancement of the reaction). This variable can he written in terms of the mole numbers N, and stoichiometric coefficients v;, taken as positive for products and negative for reactants, in the form d< = dN,/v;. In terms of this new extensive variable, the terms representing chemical work, that is, ZpidN; can be written in the convenient form -sld€,, where the affinity sl, introduced by De Donder as
and also,
where one gets these changes as a sum of two contributions, the first coming from the variation i n the absence of a chemical reaction, and the second coming from the presence of a chemical reaction. Change in \
Now we must calculate the change of using eqs 5-7,
5 with T and P,
is the conjugate work variable of 5. In the case of the Gibbs potential G, eq 1takes the form N
dG = S d T + VdP +
piWi = =SdT + VdP - .!ad\ i=l
(4)
This equation shows that in the G representation the natural variables in a closed system with chemical reaction are ('I: 1: 6). Thus, from the condition of eq 2 relative to the conjugate variables 91 and 5, i t takes the form
From the equalization of the second mixed partial derivatives of G, we get the results,
(12)
giving t h e usual mathematical formulation of t h e Le Chatelier principle in thermodynamics textbooks (11, 12) for the effect of perturbations of the intensive variables on the extent of the reaction. The effect of T and P will depend only on the sign of A,H and ApV, a s G" z 0 by eq 5. Thermal Coefficients in the Presence of Chemical Reaction
Placing eq 11in eq 9 and eq 12 in eq 10, we get the following expressions for the thermal coefficients KT and a in the presence of chemical reaction, TVG"
which are the analogue of the Maxwell relations (11)for the variables d and 5, written in terms of the heat of reaction A,H and reaction volume A7V. A, does not imply a difference between final and initial states, but rather the operator ala 0 and KT.; > 0. The physical content of these inequalities is known as Le Chstelier's principle. I t establishes that, for example, if some system has two subsystems separated by a movable wall, then if a t any moment the pressure in the two subsystems becomes unequal, the wall moves from the region of high pressure to the region of low pressure. This is true when the whole system is isolated, separated from the outer world by adiabatic walls (as KS.;> 01, or surrounded by diathermal walls, and when heat can be exchanged a t a constant T ( a s KT,: > 0). If K ~ .or, KT,: could take a negative value, then the wall would move from the region of low pressure to the region of high pressure and the system would never return to the equilibrium state. The same words apply to heat capacities a t constant pressure and volume: CP,: > 0 and Cv 0. Thus, if temperature rises in a part of a system, heat is transferred towards the part with a lower temperature.
KT,:
1191
Regarding eqs 18-19 we must point out that no such relationship holds for ~ p (KS,:) , and CP,:, CV:, when a chemical reaction takes place. Obtaining the Thermal Coefficients The thermal coefficients for a chemical reaction occurring can he obtained, in practice, from the the state equation for the system and the form of the chemical potential. The easiest case is that of a mixture of ideal gases, where one has
for the state equation, and
for the chemical potential. From eq 20 one obtains
A System That May Exchange Heat
This interpretation of the stability criteria in terms of the Le Chhtelier principle appears to be quite straightforward. Nevertheless, a more subtle interpretation is possible, whose physical content is known as the Le ChhtelierBraun principle. This interpretation is closely related to the fact that one also has KT; > K- - > 0. and that can be interpreted in the following \;;ay. '$stem that is adiabaticallv isolated from the surronndines can onlv have the orimary response regulated by the L; Chstelier principle by opposing the change of pressure. Instead, a system that may exchange heat, like a system separated from a reservoir by a diathermal wall, can react in two ways. It will have the same primary response as the adiabatically isolated system, hut will also show a secondary response because the exchange of heat will occur in the direction of changing pressure even more than with the primary. In other words, the Le Chstelier-Braun principle says that all the available terms of generalized work I;dX;, like those in eq 1,will act so that the external perturbation is opposed, in addition to that term affected by such perturbation, which is the one that gives the primary effect. Isothermal Compressibility
where Av = Zv;.On the other hand, from eq 21 we get
Then, G for this example can he obtained from eqs 5 and 3, and considering eq 23,
In this vein we can interpret eq 14 by noting that KT ,, the isothermal compressibility in the presence of a chemical reaction, appears as the sum of two contributions. The first one K~,,representsthe compressibility in the absence of chemical reaction, that is, keeping constant the mole number of every component, and the second contribution represents the effect of the chemical reaction in KT, This chemical reaction can also be interpreted as coming from
,.
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a n additional term of work-ddk. So the availability of this additional work to oppose the external perturbation implies K~~ t K T , ~ which , can be easily confirmed by inspec~ K T , ~differ by a tion of eq 14, that is, by noting that K Z and positive quantity a s G > 0 by eq 5. Regarding the possibility of switching off or on the term representing chemical work, one could imagine a chemical reaction in which the reaction rate is slow enough to ignore it. I n the presence of a n appropriate catalyst (or by application of a flame) the system could react, and then one could consider the cases of constant 5 (no chemical reaction occurs) or constant d (a chemical process takes place). Adiabatic Compressibility By similar reasoning, one could also obtain the adiabatic compressibility Q in the case where a chemical reaction occurs, Q , ~I.t is left a s a n exercise to the reader to show that one obtains the result ~ , d K>S , ~This exercise can be done using a thermodynamic potential more adequate to this physical situation: the enthalpy H. Thermal Capacities Something similar can be said about the relative values of the thermal capacities. The previous arguments show that CPr > Cvg > 0 (see also eq 18) because when a system is heated a t constant pressure it has a n additional term of work (-PdW to convert heat into work (or vice versa). A system that is heated (cooled)a t constant pressure when a chemical reaction can also occur will show a higher heat capacity than if the system is unable to react Le., has no catalyst): Ced 2 Ceg > 0.This is again the Le ChatelierBraun principle in action, as easily shown by recalling eq 17, and noticing that the term due to the action of the chemical reaction is again always positive. I t is left as a n exercise to the reader to show that the thermal capacity a t constant volume in the presence of a
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Journal of Chemical Education
2 chemical reaction Cva also obeys the relationship Cv,g. The proof is similar to the constant pressure case, noticing that one now needs to use the Helmholtz potential A, appropriate to the situation of constant V(one will get A ) . The result will be that A,U will replace A,H in the numerator.
Thermal Expansion Coefficient The thermal expansion coefficient a, as it is well-known, takes a positive value in most cases, with the well-known exceptions of water between 0 "C and 4 "C and the rubber band (14). Anyway, the sign of this coefficient cannot be determined from the stability conditions of thermodynamics because the variables appearing are not conjugate to one another. So the primary response of a system to a temperature jump will be a n increase in volume, except in the above-mentioned cases. The presence of a chemical reaction (compare with eq 13) will also bring a contribution whose sign will not be definite as in the previous case but will depend on the signs of 4 V a n d A P . I t is possible that new examples of negative thermal expansion coefficients could be found when a chemical reaction occurs if the chemical contribution is negative and dominates the positive contribution of q. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11 12.
. R. S. J. Chem. Edue. 1980.57.417-420. iimJ. J cham. ~ d wi s. s 7 , 3 i , 3 i - 3 3 9 .
Treotow
D.
Mellon, E. K J. Chom. Educ. 1979.56. 380481. Bodner, C. M J Chem.Educ. 1980.57.117-119. FemAnderPnni,R. J . Ch~m.Edue.1982.59.550-553, Btiee, L. K. J Chem. Edue. 1983.60.387-389. Treptaw, R. S. J Chem. Educ 19ffl. 61,499-502. Campbell, J.A. J Cham. Educ 1985,62,231-232. He1fferieh.F 0.J. Chem. Edue. 1985.62,30%308. Kemp. H. R. J. Chcm. Ed-. 1987,64,482484. Callsn, H. B. Tholmodynomics;Wileley:New York, 1960. Munster. A. c1ossieai Thermadvnamies: Wile": New YWk. 1970