+
+
2814
Ind. Eng. Chem. Res. 1996, 35, 2814-2815
Reply to the Comments by V. Brandani on “Size-Dependent Adsorption Models in Microporous Materials. 1. Thermodynamic Consistency and Theoretical Analysis” M. Giona† and M. Giustiniani*,‡ Dipartimento di Ingegneria Chimica, Universita´ di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy, and Dipartimento di Ingegneria Chimica, Universita´ di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy
Sir: The comments by V. Brandani refer to the definition of Gibbs free energy and to the concept of thermodynamic consistency in adsorption. Unnecessary though this will be for the average reader familiar with thermodynamics and with manyvariable calculus (change of order of differentiation)1 applied in order to derive Maxwell equations, we are pleased to remind V. Brandani of the following elementary points: 1. Definition of Gibbs Free Energy As pointed out in ref 2, the Gibbs free energy for an adsorbed phase has been historically defined in two ways. One form is taken directly from the solution thermodynamics, i.e.
Ga ) Ea + PVa - TSa
(1)
or Ga ) Fa + PVa, where Fa ) Ea - TSa is the Helmholtz free energy. Starting from the analogy with solution thermodynamics, for an adsorbed phase we have3 N
dEa ) T dSa - P dVa - π dA +
µia dni ∑ i)1
N
a
a
E ) TS - PV - πA +
∑ i)1
µai ni
N
µai ni ∑ i)1
(4)
This is given not only by Ruthven4 on p 66 (eq 3.24) but also by many other authors, e.g., by Adamson5 on p 560 (eq XVI-110), by Rudzinski and Everett6 on p 4 (eq 1.2.18), and by Rudisill and LeVan2 (eq 13) and used by Sircar and Myers7 and Suwanayuen and Danner.8 Note that the equation reported by Ruthven,
Gs ) -Φna + µsns differs from eq 4 only in that it is written for the onecomponent case, the “surface work term” being expressed indifferently by means of different couples of intensive/extensive variables, i.e., Φna ) πA (see ref 4, †
Universita´ di Cagliari. ‡ Universita ´ di Roma “La Sapienza”.
µai dni ∑ i)1
(5)
by keeping the intensive variables T, π, and µai constant. Unfortunately, because of a typographical error, the sign of the π dA term in the paper appears as positive. It is in any case clear that the the term -πA appearing in eq 4 must be obtained by integration of the term -π dA at constant π. The misprint is evident. The meaning and the thermodynamic validity of the equation are therefore unquestionable. Equation 5 is reported by Adamson5 on p 560 (eq XVI-110) and, with the volume term Va dP added, by Ruthven4 (p 66, eq 3.23) and Rudzinski and Everett6 on p 4. The second definition of the Gibbs free energy, as reported by Hill,3 Adamson,5 and Rudisill and LeVan,2 is in the form
G a ) Ea - TSa + PVa + πA
(6)
and gives N
Ga)
µai ni ∑ i)1
(7)
(3)
The substitution of eq 3 into eq 1 gives the definition of the Gibbs free energy for an adsorbed phase quoted in our paper:
Ga ) -πA +
N
dGa ) -Sa dT - π dA +
(2)
The integration of eq 2, keeping the intensive quantities T, P, π, and µai constant, gives a
eq 3.21, p 65). Therefore, eq 4 is a Gibbs free energy and is clearly reported as such by Ruthven.4 Equation 4 is, of course, obtained as usual by integrating its corresponding differential definition, as quoted in our paper:
This form is reported in our paper (see eq 2) and can be found in Hill,3 Van Ness,9 Myers and Prausnitz,10 and Rudisill and LeVan.2 It is obvious that the two definitions to Gibbs free energy differ for the term πA, i.e., G a ) Ga + πA. By neglecting the volume term, we can also write G a ) Fa + πA. Therefore, this definition of Gibbs free energy leads to the conclusion that N µai ni - πA is the Helmholtz free energy, as reported ∑i)1 by Van Ness.9 The two definitions of Gibbs free energy were already quoted and commented in a paper in ref 11. 2. Thermodynamic Consistency Equation 5 written by V. Brandani is not at all equivalent to eq 5 in the paper but clearly incorrect, as well as its consequent eq 6. The difference between the two expressions stems from the erroneous calculation of second-order derivatives in the Maxwell equations. It is well-known that the Maxwell equations are obtained by means of Schwartz’s theorem on the change of order in derivation, i.e., by imposing that
( ) ( )
∂ ∂Ga ∂ ∂Ga ) ∂Pi ∂Pj ∂Pj ∂Pi
(8)
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2815
This is eq 4 in the paper. It is obvious that the derivative with respect to Pi is obtained by keeping Pk*i constant. The variables specifying the gas-phase composition are Pk, k ) 1, ..., N, N being the number of components. The set of Pk with k * i is indicated in the paper by means of the bold symbol P. This notation has also been adopted by Keller, see eq 3.23 in ref 12, and further in ref 11 and cannot lead to any kind of confusion, since all the derivation of the Maxwell equations is a purely mathematical result given by the change of order in the second derivatives. No further comment is required as the derivation of the Maxwell equations is such a well-established topic. It is evident that the example quoted by V. Brandani should read
( ) ∂Pi ∂Pi
( ) ∂Pj ∂Pi
)1 P
)0
P
so that the Maxwell equations are obtained as reported in our paper and not as he writes, his eq 6 being clearly obtained by assuming incorrectly that
( ) ∂Pj ∂Pi
) -1
P
i.e., obtaining these derivatives by keeping the total pressure constant, instead of the set of partial pressures P ) Pk*i.
The criterion to test experimental adsorption data proposed by Van Ness9 and suggested by V. Brandani lies beyond the scope of the paper. We analyzed the thermodynamic consistency of “semiempirical models” of adsorption isotherms in the generic multicomponent case, in particular of the Keller model, and pointed out that it is thermodynamically well posed because it satisfies the Maxwell equations. Incidentally, it should be observed that answers have already been given to the criticisms made by V. Brandani in the literature quoted, especially in Keller.12 Literature Cited (1) Courant, R.; John, F. Introduction to Calculus and AnalysissVolume 2; John Wiley & Sons: New York, 1974. (2) Rudisill, E. N.; LeVan, M. D. Chem. Eng. Sci. 1992, 47, 1239. (3) Hill, T. L. Adv. Catal. 1952, 4, 211. (4) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley & Sons: New York, 1984. (5) Adamson, A. W. Physical Chemistry of Surfaces; John Wiley & Sons: New York, 1982. (6) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (7) Sircar, S.; Myers, A. L. Chem. Eng. Sci. 1973, 28, 489. (8) Suwanayuen, S.; Danner, R. P. AIChE J. 1980, 26, 68. (9) Van Ness, H. C. Ind. Eng. Chem. Fundam. 1969, 8, 464. (10) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (11) Giustiniani, M.; Giona, M.; Marrelli, L.; Viola, A. In Chaos and Fractals in Chemical Engineering; Biardi, G., Giona, M., Giona, A. R., Eds.; World Scientific Publishing: Singapore, 1995. (12) Keller, J. U. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1510.
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