- Email: [email protected]
On the reality of the residual entropy of glasses and disordered crystals: The entropy of mixing
Journal of Non-Crystalline Solids 357 (2011) 463–465
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
On the reality of the residual entropy of glasses and disordered crystals: The entropy of mixing Martin Goldstein 299 Riverside Drive, No. 7A, New York, NY 10025, United States
a r t i c l e
i n f o
Available online 26 August 2010 Keywords: Residual entropy; Thermodynamics; Glasses; Disordered crystals
a b s t r a c t We have previously shown that the assumption that the configurational entropy of a supercooled liquid vanishes at Tg leads to a non-trivial violation of the second law. Here we consider the example of the entropy of mixing. We use as a model system two similar chemical substances which form an ideal solution in a mixed phase. We apply the reasoning of our earlier paper to show that this vanishing would lead to a dilemma; either it violates the second law of thermodynamics, or else it cannot be demonstrated by any conceivable experiment. We show further that the vanishing of the entropy of mixing on kinetic arrest leads to the counter-intuitive result that the chemical potential of each component in an infinitely dilute kinetically arrested (or glassy) solution can equal or the chemical potential of the pure component. The most parsimonious conclusion from these results is that residual entropies are real. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The idea that when certain degrees of freedom become kinetically arrested the entropy associated with them vanishes has been debated for the better part of a century, beginning when the quantum mechanical basis of the third law of thermodynamics was recognized [1]. Those on the affirmative side can be divided into two camps; one asserting that the “residual entropy” determined by calorimetric measurements is entirely an artifact of performing such measurements on a system out of equilibrium [2–4], and the other that the residual entropy has the physical meaning usually attributed to it even though in reality it is not the true entropy at 0 K [5]. The experimental evidence provided by usually excellent agreement between apparent residual entropies and statistical calculations for those disordered crystals for which such a calculation is possible [6] would appear to count strongly against the first view. Those who believe in the reality of residual entropy argue that the vanishing of entropy on kinetic arrest, claimed by its proponents to be a clear requirement of the statistical mechanical definition of entropy, would imply a contradiction between statistical mechanics and classical thermodynamics, and they prefer to retain the latter in unmodified form [7,8]. In particular, they argue that while the existence of residual entropy is based on calorimetric measurements performed in part on systems out of equilibrium, classical thermodynamics permits such measurements, or at least provides limits on the error in entropy deduced from them. Direct calculation of the entropy errors has shown that they are small
E-mail address: [email protected]. 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.05.089
compared to the residual entropy itself [8–11]. Langer et al. have proposed a statistical ensemble in which each system in the ensemble is subjected to an identical thermal history on cooling into the glassy state, resulting in a non-zero residual entropy [11]. In an earlier paper, we reviewed attempts to answer the question experimentally, and concluded that by and large they supported the reality of residual entropy [12]. Further, we have argued that a spontaneous vanishing of entropy on cooling a substance through a kinetic transition between ergodic and non-ergodic conditions would not only represent a violation of the second law of thermodynamics, but that this violation is not a trivial one correctible by a minor reformulation of that law, but instead leads directly to the possibility of construction of a perpetual motion machine of the second kind [12–14]. In particular we showed that if the configurational entropy of a glass vanished at the glass transition temperature Tg, the solubility of the glass in a solvent would be greatly increased. This would permit a cyclic process in which osmotic work in diluting this solution to the concentration corresponding to the solubility of the supercooled liquid or crystal phase could be performed in gross violation of the predicted Carnot efficiency of the cycle. In this paper we apply this reasoning to the case of the entropy of mixing. We have several reasons for considering this case: first, because it has been used as an example to argue for the vanishing of residual entropy [3]; second because in many systems it is an entropy easy to calculate and visualize, unlike the configurational entropy of a supercooled liquid; third, because in many such systems there is no hysteresis or other indications of irreversibility on going through the temperature of kinetic arrest; and finally, such a vanishing of the entropy of mixing has striking and quite counter-intuitive consequences.
464
M. Goldstein / Journal of Non-Crystalline Solids 357 (2011) 463–465
temperature range between Tt and Tb. This provides the approximate equations:
2. Description of the model We consider two chemically similar substances A and B, which are capable of forming an ideal solution in a mixed phase. The molecules in both the pure phases and the mixed phase will be assumed to be capable of diffusion, with an exponential temperature dependence reflecting the existence of a potential barrier to diffusive motion. It follows that there will be some narrow temperature range in which diffusive motion undergoes kinetic arrest, at least on any conceivable experimental time scale. Let us denote a temperature at the top of this range as Tt and at the bottom as Tb. We note further that on the assumption that A and B molecules are very similar, the temperature range in which ordering in the mixed phase would take place is far below the temperature range of kinetic arrest. As a result there would be no detectable irreversibility (for example, hysteresis) in traversing this latter range in the course of calorimetric measurements. Let the chemical potentials of the substances in the pure phases be μoA and μ oB respectively. Then the chemical potentials in the mixed phase will be given by m
o
m
m
o
m
μA = μA + RT ln XA and μ B = μ B + RT ln X B : The Xs denote the mole fractions; the superscript m denoting the mixed phase. The terms RT ln X arise, of course, from the expression for the entropy of random mixing, S = k ln [(NA + NB)! / NA! NB!]. We will assume ideal entropies of mixing throughout, although the conclusions will apply qualitatively to non-ideal solutions also. 3. Solubilities under different assumptions Let us imagine a solvent C in which A and B are of limited solubility. This is equivalent to finding a solvent in which the standard chemical potentials μCA and μ CB are greater than those of the pure phases. Under this assumption the solubilities of the pure substances in the solvent C (expressed by the mole fractions of the saturated solutions XAs and XBs) would be given by o
C
s
o
C
s
μA = μA + RT ln XA and μB = μ B + RT ln XB or s
XA = exp
h i o C s μA – μA = RT and similarly for XB :
ð1Þ
If however the mixed phase is brought into solution equilibrium with the solvent C, the solubilities would be given by o
m
C
s
μA + RT ln XA = μA + RT ln XA which can be rearranged to s
m
XA = XA = exp
h i o C μA –μA = RT
ð2Þ
and a similar equation for XBs. As by hypothesis the exponent on the right hand side of the above equation is negative, the solubilities in the solvent are limited. If the entropy of mixing in the mixed phase were to disappear on kinetic arrest, the solubilities of A and B from the mixed crystal in the solvent would become equal to the solubilities of the pure substances as given in Eq. (1). Let us denote the solubility of A above the temperature of kinetic arrest by XAs (Tt) and below by XAs (Tb). While the right hand sides of Eqs. (1) and (2) are clearly functions of temperature, both implicitly through the μ’s and explicitly, the change in solubilities on kinetic arrest that would be produced by a vanishing of the mixing entropy can be assumed to be much greater than that produced by the temperature dependence of the right hand sides over the small
s
m
s
XA ðTt Þ = XA = XA ðTb Þand similarly for B: This can be rearranged to: s
s
m
XA ðTb Þ = XA ðTt Þ = 1 = XA : Let us consider the mixed phase of equal mole fraction of each component, so the predicted increase in solubilities of each is a doubling. In our earlier paper we showed that such an increase in solubility permits the construction of a cyclical process violating the second law. In the case we are discussing here, the pure phases of A and B would be combined to form the mixed phase at Tt, cooled to Tb, then dissolved in the solvent to form solutions with concentrations higher than the saturated concentrations at Tt. Then with the use of semipermeable membranes the concentrations are reduced to the levels corresponding to the saturated solutions at Tt, performing osmotic work. Following this, the solutions are heated to Tt, and the pure phases are reconstituted from the saturated solutions, a process involving no additional work as ΔG is zero for this step. The osmotic work performed at Tb exceeds the work allowed by the Carnot efficiency of a cycle with upper and lower temperatures as close together as Tt and Tb. 4. A counter-intuitive consequence of a vanishing entropy As we noted earlier, if the entropy of mixing vanished on kinetic arrest, the solubilities of the substances A and B from the mixed phase would become equal to the solubilities of the respective pure phases. Specifically this means that the chemical potential, solubility, and for that matter vapor pressure, of A from the mixed phase below Tb would be independent of the concentration, so that even at infinite dilution equal to that of pure A. The possibility of chemical potentials in various infinitely dilute solutions being equal to the chemical potentials of the respective pure solutes is certainly counter-intuitive. Whether those who argue for a vanishing of configurational entropy on kinetic arrest can provide a plausible rationalization of this remains to be seen. In the above discussion, we have assumed an ideal solution for simplicity. In non-ideal solutions the entropy will be a different function of concentration, but the counter-intuitive consequences of its vanishing would be essentially the same. 5. The question of glass solubility We acknowledged in our earlier paper [12] that the concept of the solubility of a glass is problematic. Solubility implies an equilibrium between a substance in pure form and a substance in a dissolved state. If there were such an equilibrium, not only would glass be dissolving, but glass would be precipitating on the surface of the pure substance at the same time. Under these conditions, the structure of the surface layer, in contact with the solvent, would be far from kinetically arrested, so those who argue that the kinetically arrested state should have zero entropy of mixing can reasonably maintain that the experiment described is not capable of resolving the controversy. This raises the question: is there any conceivable experiment that would demonstrate that the entropy of mixing has vanished? We proposed an alternative experiment in our earlier paper. Suppose we do the experiment not under equilibrium conditions, which would require that the glass be dissolved in a solution which is saturated at the concentration XAs (Tb), but rather dissolving the glass in a solution of concentration XAs (Tt). Under these conditions the glass would not be in equilibrium with the solution, and its rate of dissolution will exceed the rate of redeposition from the solution. At least initially the redeposition of A would be so much slower than
M. Goldstein / Journal of Non-Crystalline Solids 357 (2011) 463–465
dissolution that the entropy of the surface layer (assuming it had vanished) would not have increased enough to lower the solubility to XAs (Tt). One might expect that the concentration would first increase above XAs (Tt), and then gradually fall as the precipitation reaction rate increases with the increasing concentration. Even if the concentration in solution never reaches XAs (Tb), any increase in concentration above XAs (Tt) would provide an opportunity for performing enough osmotic work to violate the second law. 6. Possible experimental tests Simple experimental tests of these conclusions appear possible. Such tests of course need not be performed only on the ideal solutions used for illustrative purposes in the preceding discussion. Any glassy solution for which either the vapor pressure or solubility of at least one component can be measured is suitable. An alternative appropriate to ionic glasses or disordered metal alloys is measurement of the electromotive force above and below the kinetic freezing temperature [15–17]. An interesting possible system is a two component one of polymer and low molecular weight solvent. It is a common observation that the vapor pressure of a solvent from such a system below the glass temperature decreases as the solvent evaporates, but the experiment called for by the above analysis is somewhat different, and to our knowledge has not been tried. It would require preparing a series of solutions of different concentrations of the low molecular weight solvent above the Tgs of the various solutions, cooling all of them to a single temperature below the Tg of any of the series, and then measuring the initial vapor pressures from each member of the series, before enough evaporation has occurred to change the concentrations appreciably. 7. Summary We have shown that the vanishing of the entropy of mixing on kinetic arrest leads to the counter-intuitive result that the chemical
465
potential of a solvent in an infinitely dilute solution that has undergone such arrest can equal or the chemical potential of the pure solvent. We have suggested possible experiments to test this conclusion. In addition, we have applied the reasoning of our earlier paper to show that the claim that the entropy vanishes would lead to a dilemma; either it violates the second law of thermodynamics, or else it cannot be demonstrated by any conceivable experiment. One can argue that statistical mechanics requires us to believe in the vanishing of entropy anyway, even if no experiment could demonstrate it. That option would however require us to introduce drastic modifications to the second law, a price not everyone is willing to pay. We acknowledge that we have not offered a statistical mechanical justification for our results, but believe that a valid one will not involve a contradiction between statistical and calorimetric entropies.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
F. Simon, Physica IV (1937) 1089. D. Kivelson, H. Reiss, J. Phys. Chem. B 103 (1999) 8337. H. Reiss, J. Non-Cryst. Solids 355 (2009) 617. P.K. Gupta, J.C. Mauro, J. Chem. Phys. 126 (2007) 224504. R.J. Speedy, J. Phys. Chem. B 103 (1999) 4060. R. Fowler, E. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, UK, 1949, pp. 202–212. E.D. Eastman, R.D. Miller, J. Chem. Phys. 1 (1933) 444. J. Jäckle, Philos. Mag. B44 (1981) 533. A.B. Bestul, S.S. Chang, J. Chem. Phys. 43 (1965) 4532. M. Goldstein, J. Chem. Phys. 64 (1976) 4767. S.A. Langer, J.P. Sethna, E.R. Grannan, Phys. Rev. B 41 (1990) 2261. M. Goldstein, J. Chem. Phys. 128 (2008) 154510. P.K. Gupta, J.C. Mauro, J. Chem. Phys. 129 (2008) 067101. M. Goldstein, J. Chem. Phys. 129 (2008) 067102. I. Gutzow, J. Non-Cryst. Solids 45 (1981) 301. G.P. Johari, J. Chem. Phys. 113 (2000) 751. G.P. Johari, J. Chem. Phys. 132 (2010) 124509.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
On the reality of the residual entropy of glasses and disordered crystals: The entropy of mixing Martin Goldstein 299 Riverside Drive, No. 7A, New York, NY 10025, United States
a r t i c l e
i n f o
Available online 26 August 2010 Keywords: Residual entropy; Thermodynamics; Glasses; Disordered crystals
a b s t r a c t We have previously shown that the assumption that the configurational entropy of a supercooled liquid vanishes at Tg leads to a non-trivial violation of the second law. Here we consider the example of the entropy of mixing. We use as a model system two similar chemical substances which form an ideal solution in a mixed phase. We apply the reasoning of our earlier paper to show that this vanishing would lead to a dilemma; either it violates the second law of thermodynamics, or else it cannot be demonstrated by any conceivable experiment. We show further that the vanishing of the entropy of mixing on kinetic arrest leads to the counter-intuitive result that the chemical potential of each component in an infinitely dilute kinetically arrested (or glassy) solution can equal or the chemical potential of the pure component. The most parsimonious conclusion from these results is that residual entropies are real. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The idea that when certain degrees of freedom become kinetically arrested the entropy associated with them vanishes has been debated for the better part of a century, beginning when the quantum mechanical basis of the third law of thermodynamics was recognized [1]. Those on the affirmative side can be divided into two camps; one asserting that the “residual entropy” determined by calorimetric measurements is entirely an artifact of performing such measurements on a system out of equilibrium [2–4], and the other that the residual entropy has the physical meaning usually attributed to it even though in reality it is not the true entropy at 0 K [5]. The experimental evidence provided by usually excellent agreement between apparent residual entropies and statistical calculations for those disordered crystals for which such a calculation is possible [6] would appear to count strongly against the first view. Those who believe in the reality of residual entropy argue that the vanishing of entropy on kinetic arrest, claimed by its proponents to be a clear requirement of the statistical mechanical definition of entropy, would imply a contradiction between statistical mechanics and classical thermodynamics, and they prefer to retain the latter in unmodified form [7,8]. In particular, they argue that while the existence of residual entropy is based on calorimetric measurements performed in part on systems out of equilibrium, classical thermodynamics permits such measurements, or at least provides limits on the error in entropy deduced from them. Direct calculation of the entropy errors has shown that they are small
E-mail address: [email protected]. 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.05.089
compared to the residual entropy itself [8–11]. Langer et al. have proposed a statistical ensemble in which each system in the ensemble is subjected to an identical thermal history on cooling into the glassy state, resulting in a non-zero residual entropy [11]. In an earlier paper, we reviewed attempts to answer the question experimentally, and concluded that by and large they supported the reality of residual entropy [12]. Further, we have argued that a spontaneous vanishing of entropy on cooling a substance through a kinetic transition between ergodic and non-ergodic conditions would not only represent a violation of the second law of thermodynamics, but that this violation is not a trivial one correctible by a minor reformulation of that law, but instead leads directly to the possibility of construction of a perpetual motion machine of the second kind [12–14]. In particular we showed that if the configurational entropy of a glass vanished at the glass transition temperature Tg, the solubility of the glass in a solvent would be greatly increased. This would permit a cyclic process in which osmotic work in diluting this solution to the concentration corresponding to the solubility of the supercooled liquid or crystal phase could be performed in gross violation of the predicted Carnot efficiency of the cycle. In this paper we apply this reasoning to the case of the entropy of mixing. We have several reasons for considering this case: first, because it has been used as an example to argue for the vanishing of residual entropy [3]; second because in many systems it is an entropy easy to calculate and visualize, unlike the configurational entropy of a supercooled liquid; third, because in many such systems there is no hysteresis or other indications of irreversibility on going through the temperature of kinetic arrest; and finally, such a vanishing of the entropy of mixing has striking and quite counter-intuitive consequences.
464
M. Goldstein / Journal of Non-Crystalline Solids 357 (2011) 463–465
temperature range between Tt and Tb. This provides the approximate equations:
2. Description of the model We consider two chemically similar substances A and B, which are capable of forming an ideal solution in a mixed phase. The molecules in both the pure phases and the mixed phase will be assumed to be capable of diffusion, with an exponential temperature dependence reflecting the existence of a potential barrier to diffusive motion. It follows that there will be some narrow temperature range in which diffusive motion undergoes kinetic arrest, at least on any conceivable experimental time scale. Let us denote a temperature at the top of this range as Tt and at the bottom as Tb. We note further that on the assumption that A and B molecules are very similar, the temperature range in which ordering in the mixed phase would take place is far below the temperature range of kinetic arrest. As a result there would be no detectable irreversibility (for example, hysteresis) in traversing this latter range in the course of calorimetric measurements. Let the chemical potentials of the substances in the pure phases be μoA and μ oB respectively. Then the chemical potentials in the mixed phase will be given by m
o
m
m
o
m
μA = μA + RT ln XA and μ B = μ B + RT ln X B : The Xs denote the mole fractions; the superscript m denoting the mixed phase. The terms RT ln X arise, of course, from the expression for the entropy of random mixing, S = k ln [(NA + NB)! / NA! NB!]. We will assume ideal entropies of mixing throughout, although the conclusions will apply qualitatively to non-ideal solutions also. 3. Solubilities under different assumptions Let us imagine a solvent C in which A and B are of limited solubility. This is equivalent to finding a solvent in which the standard chemical potentials μCA and μ CB are greater than those of the pure phases. Under this assumption the solubilities of the pure substances in the solvent C (expressed by the mole fractions of the saturated solutions XAs and XBs) would be given by o
C
s
o
C
s
μA = μA + RT ln XA and μB = μ B + RT ln XB or s
XA = exp
h i o C s μA – μA = RT and similarly for XB :
ð1Þ
If however the mixed phase is brought into solution equilibrium with the solvent C, the solubilities would be given by o
m
C
s
μA + RT ln XA = μA + RT ln XA which can be rearranged to s
m
XA = XA = exp
h i o C μA –μA = RT
ð2Þ
and a similar equation for XBs. As by hypothesis the exponent on the right hand side of the above equation is negative, the solubilities in the solvent are limited. If the entropy of mixing in the mixed phase were to disappear on kinetic arrest, the solubilities of A and B from the mixed crystal in the solvent would become equal to the solubilities of the pure substances as given in Eq. (1). Let us denote the solubility of A above the temperature of kinetic arrest by XAs (Tt) and below by XAs (Tb). While the right hand sides of Eqs. (1) and (2) are clearly functions of temperature, both implicitly through the μ’s and explicitly, the change in solubilities on kinetic arrest that would be produced by a vanishing of the mixing entropy can be assumed to be much greater than that produced by the temperature dependence of the right hand sides over the small
s
m
s
XA ðTt Þ = XA = XA ðTb Þand similarly for B: This can be rearranged to: s
s
m
XA ðTb Þ = XA ðTt Þ = 1 = XA : Let us consider the mixed phase of equal mole fraction of each component, so the predicted increase in solubilities of each is a doubling. In our earlier paper we showed that such an increase in solubility permits the construction of a cyclical process violating the second law. In the case we are discussing here, the pure phases of A and B would be combined to form the mixed phase at Tt, cooled to Tb, then dissolved in the solvent to form solutions with concentrations higher than the saturated concentrations at Tt. Then with the use of semipermeable membranes the concentrations are reduced to the levels corresponding to the saturated solutions at Tt, performing osmotic work. Following this, the solutions are heated to Tt, and the pure phases are reconstituted from the saturated solutions, a process involving no additional work as ΔG is zero for this step. The osmotic work performed at Tb exceeds the work allowed by the Carnot efficiency of a cycle with upper and lower temperatures as close together as Tt and Tb. 4. A counter-intuitive consequence of a vanishing entropy As we noted earlier, if the entropy of mixing vanished on kinetic arrest, the solubilities of the substances A and B from the mixed phase would become equal to the solubilities of the respective pure phases. Specifically this means that the chemical potential, solubility, and for that matter vapor pressure, of A from the mixed phase below Tb would be independent of the concentration, so that even at infinite dilution equal to that of pure A. The possibility of chemical potentials in various infinitely dilute solutions being equal to the chemical potentials of the respective pure solutes is certainly counter-intuitive. Whether those who argue for a vanishing of configurational entropy on kinetic arrest can provide a plausible rationalization of this remains to be seen. In the above discussion, we have assumed an ideal solution for simplicity. In non-ideal solutions the entropy will be a different function of concentration, but the counter-intuitive consequences of its vanishing would be essentially the same. 5. The question of glass solubility We acknowledged in our earlier paper [12] that the concept of the solubility of a glass is problematic. Solubility implies an equilibrium between a substance in pure form and a substance in a dissolved state. If there were such an equilibrium, not only would glass be dissolving, but glass would be precipitating on the surface of the pure substance at the same time. Under these conditions, the structure of the surface layer, in contact with the solvent, would be far from kinetically arrested, so those who argue that the kinetically arrested state should have zero entropy of mixing can reasonably maintain that the experiment described is not capable of resolving the controversy. This raises the question: is there any conceivable experiment that would demonstrate that the entropy of mixing has vanished? We proposed an alternative experiment in our earlier paper. Suppose we do the experiment not under equilibrium conditions, which would require that the glass be dissolved in a solution which is saturated at the concentration XAs (Tb), but rather dissolving the glass in a solution of concentration XAs (Tt). Under these conditions the glass would not be in equilibrium with the solution, and its rate of dissolution will exceed the rate of redeposition from the solution. At least initially the redeposition of A would be so much slower than
M. Goldstein / Journal of Non-Crystalline Solids 357 (2011) 463–465
dissolution that the entropy of the surface layer (assuming it had vanished) would not have increased enough to lower the solubility to XAs (Tt). One might expect that the concentration would first increase above XAs (Tt), and then gradually fall as the precipitation reaction rate increases with the increasing concentration. Even if the concentration in solution never reaches XAs (Tb), any increase in concentration above XAs (Tt) would provide an opportunity for performing enough osmotic work to violate the second law. 6. Possible experimental tests Simple experimental tests of these conclusions appear possible. Such tests of course need not be performed only on the ideal solutions used for illustrative purposes in the preceding discussion. Any glassy solution for which either the vapor pressure or solubility of at least one component can be measured is suitable. An alternative appropriate to ionic glasses or disordered metal alloys is measurement of the electromotive force above and below the kinetic freezing temperature [15–17]. An interesting possible system is a two component one of polymer and low molecular weight solvent. It is a common observation that the vapor pressure of a solvent from such a system below the glass temperature decreases as the solvent evaporates, but the experiment called for by the above analysis is somewhat different, and to our knowledge has not been tried. It would require preparing a series of solutions of different concentrations of the low molecular weight solvent above the Tgs of the various solutions, cooling all of them to a single temperature below the Tg of any of the series, and then measuring the initial vapor pressures from each member of the series, before enough evaporation has occurred to change the concentrations appreciably. 7. Summary We have shown that the vanishing of the entropy of mixing on kinetic arrest leads to the counter-intuitive result that the chemical
465
potential of a solvent in an infinitely dilute solution that has undergone such arrest can equal or the chemical potential of the pure solvent. We have suggested possible experiments to test this conclusion. In addition, we have applied the reasoning of our earlier paper to show that the claim that the entropy vanishes would lead to a dilemma; either it violates the second law of thermodynamics, or else it cannot be demonstrated by any conceivable experiment. One can argue that statistical mechanics requires us to believe in the vanishing of entropy anyway, even if no experiment could demonstrate it. That option would however require us to introduce drastic modifications to the second law, a price not everyone is willing to pay. We acknowledge that we have not offered a statistical mechanical justification for our results, but believe that a valid one will not involve a contradiction between statistical and calorimetric entropies.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
F. Simon, Physica IV (1937) 1089. D. Kivelson, H. Reiss, J. Phys. Chem. B 103 (1999) 8337. H. Reiss, J. Non-Cryst. Solids 355 (2009) 617. P.K. Gupta, J.C. Mauro, J. Chem. Phys. 126 (2007) 224504. R.J. Speedy, J. Phys. Chem. B 103 (1999) 4060. R. Fowler, E. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, UK, 1949, pp. 202–212. E.D. Eastman, R.D. Miller, J. Chem. Phys. 1 (1933) 444. J. Jäckle, Philos. Mag. B44 (1981) 533. A.B. Bestul, S.S. Chang, J. Chem. Phys. 43 (1965) 4532. M. Goldstein, J. Chem. Phys. 64 (1976) 4767. S.A. Langer, J.P. Sethna, E.R. Grannan, Phys. Rev. B 41 (1990) 2261. M. Goldstein, J. Chem. Phys. 128 (2008) 154510. P.K. Gupta, J.C. Mauro, J. Chem. Phys. 129 (2008) 067101. M. Goldstein, J. Chem. Phys. 129 (2008) 067102. I. Gutzow, J. Non-Cryst. Solids 45 (1981) 301. G.P. Johari, J. Chem. Phys. 113 (2000) 751. G.P. Johari, J. Chem. Phys. 132 (2010) 124509.