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Research: Science and Education

The Gibbs Phase Rule Revisited: Interrela Interrelattionships ionships betw betw een Component Components s and Phases J osep oseph h S. S. Alpe Alper Department Department of Chem C hemistry, Univers Universiity of Massachuset M assachusetts ts– –Boston, Boston Boston,, MA MA 02125 021 25;; j.alp j.alpe [email protected] .edu

The Gibbs phase rule appears to be extremely simple; indeed, that is the source of its elegance: F = = C – – P + 2, where C is the number of components in a chemical system, P is is the number of phases, and F is is the number of degrees of freedom. Students of physical chemistry usually have little difficulty counting the number of phases in the systems they encounter. A phase is defined as a state of matter that is uniform throughout, both in chemical composition and in physical properties. I n dete determini rmini ng P , students need only realize that there can be at most one gas phase, but that there can be more than one solid phase (e.g., a mixture of zinc turnings and iron filings) and more than one liquid phase (e.g., a mixture of toluene and wat water). r) . H oweve owever, r, counti count ing components components, although alt hough a seemingly trivial task, presents unexpected difficulties, especially in multiphase systems involving chemical reactions. In this paper, I discuss the problem of counting components in multiphase systems involving chemical reactions. No new principles are required to solve this problem (1, 2 ). ). Its solution simply requires careful attention to the nature of the constraints that provide relationships among the species constituting the system. As a consequence of this more detailed examination, we find that the number of components depends on the phase structure of the system. In some situations, the number of components can be affected by the temperature and pressure of the system. In other situations, a system containing arbitrary amounts of two different chemical compounds should be regarded as a one-component system. Counting Components C , the number of components, is defined as the minimum number of independent chemical species necessary to define the composition of all of of the phases present in the system (3 ). ). The number number of independent species peciesisequal to t o the the number number of constituents (the total number of chemical species used to characterize the system) minus the number of constraints on the concentration of these constituents ( 4 ). The T he cons constr tra ai nts nt s include both equations involving equilibrium constants, which provide relationships among the chemical species at equilibrium, and equations describing stoichiometric relationships among the species irrespective of whether the system is in equilibrium. I n applying applying these principl pri nciple es, it i t is i mportant to re r emembe memberr tha t hatt the phase rule describes the intensive state of the system; the relative amounts of the various phases cannot be determined (5 ). In other words, the phase rule says nothing about the ). total number of moles in each phase. The number and even the choice of constituents in a system is to some extent arbitrary, depending on the problem at hand or on personal personal choice choice.. H owever, ver, irr i rre especti pecti ve of the t he choice of constituents, the number of components is an invari varia ant. nt . For example mpl e, i n a proble probl em involvi i nvolving ng the t he phas phase equiqui librium of a system containing liquid water, ice, and water vapor, vapor, the t here re is jus just one cons constti tuent uent,, H 2O, and thus one component. However, in a problem involving the dissociation of

water, there are three constituents, namely, H 2O, H 3O+, and OH . There is still only one component because there are two constraining relationships among these three constituents:

+ K w = [H 3O ][OH ]

and [H 3O+] = [OH ]

(1)

In determining the number of components from the constituents, the textbooks focus on determining the number of independent chemical reactions without regard to phase. In fact, an article appearing in this Journal discussing the Brinkley method for determining the number of components notes that this method is best suited for one-phase systems such as high-temperature mixtures in the gaseous phase or for complicated aqueous solutions of electrolytes (6 ). The Brinkley method uses linear algebra to analyze the matrix whose (i ,j )th entry is the number of atoms of element E j in the chemical species Si . The authors do apply the Brinkley method to a two-phase system, namely a solution of electrolytes i nvolving nvolvi ng an an i nsolu nsolubl ble e species pecies(AlCl (Al Cl3 dissolved in water with the formation of Al(OH) 3(s)) (s)).. H owe owever, ver, the t hey y do not not dwell on the effect of the second phase in determining the number of components. The complications in determining the number of components in multiphase systems can be clearly seen in a comparison between two seemingly similar equilibria and

NH 4Cl(s Cl( s) = NH N H 3(g) + H Cl(g)

(2)

CaCO3(s) = CaO(s) + CO 2(g)

(3)

Let us first treat the case in which, for each equilibrium, we introduce nt roduce arbitr rbi tra ary amounts of of the three hree cons constti tue tuents nt s i nto nt o a closed container. In the first equation, there are three consti tue tuents nt s, but only only two two compone component nts s. A constr constra ai nt on the t he conconstituents is provided by the equilibrium constant expression, which relates the partial pressures of the two gaseous products: K 1 = p NH 3 p HCl

(4)

where K 1 is the equilibrium constant and p X denotes the partial pressure of the species X. T he same reasoning oni ng appli appli es to the t he second equat equati on. The T he equilibrium expression is K 2 = p (5) CO2 and agai gai n, C = = 2. Now consider the case in which, for each equilibrium, we introduce the reactant only into the container. For the first equilibrium, there is an additional stoichiometric relationship, n (NH (NH 3) = n (HCl), (HCl), where the notation n (X) means the number (X) of moles of X. We conclude that the number of components is reduced to one. This reasoning is correct. C = 1, P = 2 (on (one e gaseous and one solid phase), and so, and the number of degrees of freedom is F = = 1. This degree of freedom can be taken to be the temperature. Specifying the temperature determines K 1, 1, which together with the stoichiometric relationship determines p NH 3 and p HCl.

JChe JChemEd.c d.chem.wis .wisc.edu .edu • Vol. 76 N o. 11 N ovem ovember 1999 199 9 • Jour Journal of Che Chemical ical Educ ducation

1567

Research: Science and Education

If we apply the same reasoning to the second equilibrium, we note that n (CaO) = n (CO2) and again conclude that C = 1. However, this reasoning is false. If we apply the phase rule to the decomposition of CaCO 3, we have C = 1, P = 3 (two solid phases and one gaseous phase), and thus F = 0. This result cannot be correct. There is certainly more than one temperature at which CO 2(g) exists in equilibrium with CaCO3(s) and CaO(s). Clearly, the source of the problem lies in the additional solid phase in the second equil ibrium. After the equilibrium has been established, we can add arbitrary amounts of CaO without affecting the equilibrium. Thus C = 2. Note that in the equilibrium expression K 2, CaO(s) does not appear. Since it is in a phase by itself, the composition (concentration) of the solid CaO phase is independent of the amount of that phase. For the usual choice of standard states, this concentration can be set equal to 1. Thus the existence of the stoichiometric relati onship n (CaO) = n (CO2), relating the number of moles of the two species, is irrelevant in determining the number of components. In the NH 4Cl example, on the other hand, addition of either HCl or NH 3 drives the equilibrium to the left. Both HCl and NH 3 appear in the equilibri um expression K 1, and the stoichiometric relationship between the two species does affect the number of components. The general principle becomesclear from these examples. The only stoichiometric relationships that reduce the number of components are those in which every species in the relationship appears in an equation involving an equilibrium constant. Stoichiometric relationships involving at least one species in a solid phase or at least one pure liquid species in a phase by itself are irrelevant in determining the number of components. This approach to the determination of the number of components clarifies the example mentioned above involving an aqueous solution of AlCl3. Let us take as constituents the species H2O, Al3+, Cl , H +, OH , and Al(OH)3(s). There are two relevant equilibrium constants: K w and K sp, the solubility product constant for Al(OH)3. There is only one stoichiometric constraint involving species all of which appear in the equilibrium equations involving K w and K sp. This constraint is the charge balance equation

3[Al 3+] + [H +] = [Cl ] + [OH ]

(6)

which comprises all the charged species in solution. Thus the number of components is 3 (6 constituents, 2 equilibrium constants, 1 stoichiometric constraint). These components can be chosen to be H 2O, AlCl 3, and Al(OH)3. This result can also be obtained by noting that the component Al(OH)3 is required to specify the composition of the solid phase. Suppose now, contrary to fact, that Al(OH)3 were soluble and appeared in the aqueous phase. I n this case, the equation involving K sp would be replaced by K = [Al3+][OH ]/[Al(OH)3]

(7)

The species Al(OH)3 now appears in an equation involving an equilibrium constant, so that a second stoichiometric constraint becomes relevant. This constraint is the mass balance equation 3[Al(OH)3] + 3[Al 3+] = [Cl ] (8)

reflecting the fact that no matter what aqueous species form, the ratio of the total (analytical) concentration of Al to that of Cl must remain 1:3. Thus, assuming Al(OH) 3 is soluble, there would be just two components, H 2O and AlCl 3. C a s

a Function of Temp era ture a nd Pressure

It is obvious that the number of phases present in a system depends on the temperature T and pressure p . H owever, it is not often appreciated that the number of components is not an invariant of a system, but that it too is a function of p and T . As a fi rst example, let us return to the decomposition of NH 4Cl. At a pressure of 1 atm and between temperatures 191 and 239 K, NH 3 exists in the liquid phase while HCl remains a gas. Now only HCl appears in the equilibrium equation because NH 3 is a pure liquid in a phase by itself. Consequently, there are no longer any stoichiometric constraints so that the number of components is increased from 1 to 2. It seems strange that changing the temperature (and by extension, the pressure) of a system can result in a change in the number of components as well as a change in the number of phases. Another situation in which the number of components can vary depending upon the temperature and pressure occurs under conditions of metastable equilibrium. An arbitrary mixture of hydrogen, oxygen, and water vapor at equilibrium has two components because of the equilibrium constant relating the three species. In this case P = 1 and there are three degrees of freedom: for example, p , T , and the mole fraction of one of the three constituents. A mixture of these three gases might also be prepared by introducing arbitrary amounts of each of the gases into a container being careful to avoid an explosion. The rate of reaction is so slow at normal temperatures and pressures that we can consider the three unreacted gases to be in equilibrium. In Richard Feynman’s words, a system is in equilibri um “ if all the ‘fast’ things have happened and all the ‘slow’ things not” (7 ). Under these conditions, C = 3 (there is no equilibrium constant expression) and there are four degrees of freedom: p , T , and the mole fractions of two of the three gases. Distinguishing Com pone nts from Phases

In most cases, that i s, exercises in physical chemistry texts, there is never any question about whether two species represent two components or should be considered to be two phases of the same component. Liquid water and ice are two phases of the same component. Similarly, C(graphite) and C(diamond) are also regarded as two phases of the same component, namely carbon. But are things always so simple? Under ordinary conditions, C(gr) and C(d) are not in chemical equilibrium; C(gr) is the more stable form. However, as we know from ordinary experience, the two forms of carbon are in metastable equilibrium and can exist independently of each other. For all practical purposes, the graphite– diamond system can be regarded as being in equilibrium and subject to the phase rule, just as is the case for the hydrogen– oxygen–water vapor system discussed above. Consequently, under these conditions for the graphite–carbon system, C = 2. It is clear that P = 2. Applying the phase rule, we find F = 2. The two degrees of freedom are p and T . (Each of the two solid phases consists of only one species, so the compositions of both are determined.)

1568 Journal of Chemical Education • Vol. 76 No. 11 November 1999 • JChemEd.chem.wisc.edu

Research: Science and Education

If however, C(gr) and C(d) are at true chemical equilibrium, the chemical potentials of the two species become equal. As a result of this constraint, C = 1. Since P remains equal to two, there is just one degree of freedom, p or T . From the constraint on the chemical potentials, we can derive the Clausius equation dp /dT = ∆ trsS / ∆ trsV

(9)

where ∆ tr sS and ∆ trsV are the changes in entropy and volume across the phase boundary, which provides the functional relationship between p and T . In view of this discussion, consider a hypothetical isomerization reaction, A = B, that has reached equilibrium. (We assume that the initial amount of each species is arbitrary.) I f one or both of the two i somers are gases, C = 1 because there are two constituent species, one equilibrium expression, but no stoichiometric relation between A and B. In the case of one gaseous constituent P = 2, and so F = 1. If both constituents are gases, P = 1 and F = 2, the mole fraction of A in the gas phase providing the additional degree of freedom. If both A and B are solids, the system is totally analogous to the graphite–diamond example. If the rate of interconversion of the two solids is very slow, A and B are in metastable equilibrium, and we can take C = 2. There are two different compounds A and B with no equilibrium constant expression relating them. However, there may be some regions of the p –T plane in which p and T take on values for which the isomerization proceeds and the reaction reaches equilibrium. For these values of p and T , A and B are in chemical equilibrium. However, unlike a typical chemical reaction involving gases or solutions, there is no equilibrium constant for this reaction relating the concentrations of A and B. Each of the chemical species A and B is in a phase by itself so that the mole fraction (or concentration) of each species is unity. Even at equilibrium, the ratio of the number of moles of A and B is arbitrary. Although there is no equilibrium constant, we can analyze this equilibrium. Instead of treating the system consisting of A(s) and B(s) as a two-component system, we treat it as a one-component system with two phases that are in equilibrium. The reduction in the number of components from 2 to 1 results from the fact that at equilibrium, the chemical potentials of the two species are equal, just as is the case in the graphite–diamond example. Since C = 1 and P = 2, the phase rule tells us that F = 1. Consequently, there must be a relationship between p and T . This relationship is given by a Clausius equation which is derived by equating the chemical potentials of A and B. Starting from d µA( p ,T ) = dµ B( p ,T ) we obtain the Clausius-like relation dp /dT = ∆ S / ∆ V

(10)

where ∆ S and ∆ V are the changes in entropy and volume upon isomerization. But is it legitimate to call a system containing A(s) and B(s) a one-component system?In the derivation of the usual Clausius equation, as in the determination of the number of constraints in the derivation of the phase rule itself, we equate chemical potentials of the same substance in two different phases. I have purposely introduced the new word “substance” to leave open the question of how the relative similarity between graphite and diamond compares with that between the two isomers A and B. Graphite and diamond differ in the arrangement of their carbon atoms; in graphite the carbon atoms lie in sheets, in diamond they form a tetrahedral structure. Supposing for specificity that the two isomers are hydrocarbons, then the isomers differ only in the arrangement of their carbon and hydrogen atoms. Thermodynamics is totally independent of the details or even the existence of the molecular structure. Consequently, if chemical equilibrium can be established between any two isomers (they need not be hydrocarbons), these two species are thermodynamically just as similar to each other as are graphite and diamond. If the latter pair can be considered to be a one-component system, so can the former. Thisisomerization example shows that there are surprising subtleties associated with the application of the Gibbs phase rule. For certain chemical equilibria, such as one involving the solid phases of two isomers, a seemingly two-component system should be regarded asa one-component system. More generally, in determining the number of components, a thorough understanding of the phase structure of the system is critical. C is not independent of P . Acknowledgments

I would like to thank R. I. Gelb for his critical reading of the manuscript and for several helpful discussions. I would also like to thank the reviewers for several valuable suggestions. Literature Cited 1. Denbigh, K. The Principles of Chemical Equili bri um , 3rd ed.; Cambridge University Press: London, 1971. 2. Findlay, A.; Campbell, A. N.; Smith, N . O. T he Phase Rul e and I ts Appli cati ons , 9th ed.; Dover: New York, 1951. 3. Atkins, P. Physi cal Chemistr y , 6th ed.; Freeman: New York, 1998; pp 192–193. 4. Noggle, J. H. Physi cal Chemistr y , 3rd ed.; H arper Colli ns: New York, 1996; pp 332–333. 5. Albert y, R. A.; Silbey, R. J. Physi cal Chemi str y , 2nd ed.; Wiley: New York, 1997; pp 141–143. 6. Zhao, M.; Wang, Z.; Xiao, L. J. Chem. Educ . 1992, 69 , 539–542. 7. Feynman, R. Stati sti cal M echanics: A Set of Lectu res ; Benjamin: Reading, MA, 1972; p 1.

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The Gibbs Phase Rule Revisited: Interrela Interrelattionships ionships betw betw een Component Components s and Phases J osep oseph h S. S. Alpe Alper Department Department of Chem C hemistry, Univers Universiity of Massachuset M assachusetts ts– –Boston, Boston Boston,, MA MA 02125 021 25;; j.alp j.alpe [email protected] .edu

The Gibbs phase rule appears to be extremely simple; indeed, that is the source of its elegance: F = = C – – P + 2, where C is the number of components in a chemical system, P is is the number of phases, and F is is the number of degrees of freedom. Students of physical chemistry usually have little difficulty counting the number of phases in the systems they encounter. A phase is defined as a state of matter that is uniform throughout, both in chemical composition and in physical properties. I n dete determini rmini ng P , students need only realize that there can be at most one gas phase, but that there can be more than one solid phase (e.g., a mixture of zinc turnings and iron filings) and more than one liquid phase (e.g., a mixture of toluene and wat water). r) . H oweve owever, r, counti count ing components components, although alt hough a seemingly trivial task, presents unexpected difficulties, especially in multiphase systems involving chemical reactions. In this paper, I discuss the problem of counting components in multiphase systems involving chemical reactions. No new principles are required to solve this problem (1, 2 ). ). Its solution simply requires careful attention to the nature of the constraints that provide relationships among the species constituting the system. As a consequence of this more detailed examination, we find that the number of components depends on the phase structure of the system. In some situations, the number of components can be affected by the temperature and pressure of the system. In other situations, a system containing arbitrary amounts of two different chemical compounds should be regarded as a one-component system. Counting Components C , the number of components, is defined as the minimum number of independent chemical species necessary to define the composition of all of of the phases present in the system (3 ). ). The number number of independent species peciesisequal to t o the the number number of constituents (the total number of chemical species used to characterize the system) minus the number of constraints on the concentration of these constituents ( 4 ). The T he cons constr tra ai nts nt s include both equations involving equilibrium constants, which provide relationships among the chemical species at equilibrium, and equations describing stoichiometric relationships among the species irrespective of whether the system is in equilibrium. I n applying applying these principl pri nciple es, it i t is i mportant to re r emembe memberr tha t hatt the phase rule describes the intensive state of the system; the relative amounts of the various phases cannot be determined (5 ). In other words, the phase rule says nothing about the ). total number of moles in each phase. The number and even the choice of constituents in a system is to some extent arbitrary, depending on the problem at hand or on personal personal choice choice.. H owever, ver, irr i rre especti pecti ve of the t he choice of constituents, the number of components is an invari varia ant. nt . For example mpl e, i n a proble probl em involvi i nvolving ng the t he phas phase equiqui librium of a system containing liquid water, ice, and water vapor, vapor, the t here re is jus just one cons constti tuent uent,, H 2O, and thus one component. However, in a problem involving the dissociation of

water, there are three constituents, namely, H 2O, H 3O+, and OH . There is still only one component because there are two constraining relationships among these three constituents:

+ K w = [H 3O ][OH ]

and [H 3O+] = [OH ]

(1)

In determining the number of components from the constituents, the textbooks focus on determining the number of independent chemical reactions without regard to phase. In fact, an article appearing in this Journal discussing the Brinkley method for determining the number of components notes that this method is best suited for one-phase systems such as high-temperature mixtures in the gaseous phase or for complicated aqueous solutions of electrolytes (6 ). The Brinkley method uses linear algebra to analyze the matrix whose (i ,j )th entry is the number of atoms of element E j in the chemical species Si . The authors do apply the Brinkley method to a two-phase system, namely a solution of electrolytes i nvolving nvolvi ng an an i nsolu nsolubl ble e species pecies(AlCl (Al Cl3 dissolved in water with the formation of Al(OH) 3(s)) (s)).. H owe owever, ver, the t hey y do not not dwell on the effect of the second phase in determining the number of components. The complications in determining the number of components in multiphase systems can be clearly seen in a comparison between two seemingly similar equilibria and

NH 4Cl(s Cl( s) = NH N H 3(g) + H Cl(g)

(2)

CaCO3(s) = CaO(s) + CO 2(g)

(3)

Let us first treat the case in which, for each equilibrium, we introduce nt roduce arbitr rbi tra ary amounts of of the three hree cons constti tue tuents nt s i nto nt o a closed container. In the first equation, there are three consti tue tuents nt s, but only only two two compone component nts s. A constr constra ai nt on the t he conconstituents is provided by the equilibrium constant expression, which relates the partial pressures of the two gaseous products: K 1 = p NH 3 p HCl

(4)

where K 1 is the equilibrium constant and p X denotes the partial pressure of the species X. T he same reasoning oni ng appli appli es to the t he second equat equati on. The T he equilibrium expression is K 2 = p (5) CO2 and agai gai n, C = = 2. Now consider the case in which, for each equilibrium, we introduce the reactant only into the container. For the first equilibrium, there is an additional stoichiometric relationship, n (NH (NH 3) = n (HCl), (HCl), where the notation n (X) means the number (X) of moles of X. We conclude that the number of components is reduced to one. This reasoning is correct. C = 1, P = 2 (on (one e gaseous and one solid phase), and so, and the number of degrees of freedom is F = = 1. This degree of freedom can be taken to be the temperature. Specifying the temperature determines K 1, 1, which together with the stoichiometric relationship determines p NH 3 and p HCl.

JChe JChemEd.c d.chem.wis .wisc.edu .edu • Vol. 76 N o. 11 N ovem ovember 1999 199 9 • Jour Journal of Che Chemical ical Educ ducation

1567

Research: Science and Education

If we apply the same reasoning to the second equilibrium, we note that n (CaO) = n (CO2) and again conclude that C = 1. However, this reasoning is false. If we apply the phase rule to the decomposition of CaCO 3, we have C = 1, P = 3 (two solid phases and one gaseous phase), and thus F = 0. This result cannot be correct. There is certainly more than one temperature at which CO 2(g) exists in equilibrium with CaCO3(s) and CaO(s). Clearly, the source of the problem lies in the additional solid phase in the second equil ibrium. After the equilibrium has been established, we can add arbitrary amounts of CaO without affecting the equilibrium. Thus C = 2. Note that in the equilibrium expression K 2, CaO(s) does not appear. Since it is in a phase by itself, the composition (concentration) of the solid CaO phase is independent of the amount of that phase. For the usual choice of standard states, this concentration can be set equal to 1. Thus the existence of the stoichiometric relati onship n (CaO) = n (CO2), relating the number of moles of the two species, is irrelevant in determining the number of components. In the NH 4Cl example, on the other hand, addition of either HCl or NH 3 drives the equilibrium to the left. Both HCl and NH 3 appear in the equilibri um expression K 1, and the stoichiometric relationship between the two species does affect the number of components. The general principle becomesclear from these examples. The only stoichiometric relationships that reduce the number of components are those in which every species in the relationship appears in an equation involving an equilibrium constant. Stoichiometric relationships involving at least one species in a solid phase or at least one pure liquid species in a phase by itself are irrelevant in determining the number of components. This approach to the determination of the number of components clarifies the example mentioned above involving an aqueous solution of AlCl3. Let us take as constituents the species H2O, Al3+, Cl , H +, OH , and Al(OH)3(s). There are two relevant equilibrium constants: K w and K sp, the solubility product constant for Al(OH)3. There is only one stoichiometric constraint involving species all of which appear in the equilibrium equations involving K w and K sp. This constraint is the charge balance equation

3[Al 3+] + [H +] = [Cl ] + [OH ]

(6)

which comprises all the charged species in solution. Thus the number of components is 3 (6 constituents, 2 equilibrium constants, 1 stoichiometric constraint). These components can be chosen to be H 2O, AlCl 3, and Al(OH)3. This result can also be obtained by noting that the component Al(OH)3 is required to specify the composition of the solid phase. Suppose now, contrary to fact, that Al(OH)3 were soluble and appeared in the aqueous phase. I n this case, the equation involving K sp would be replaced by K = [Al3+][OH ]/[Al(OH)3]

(7)

The species Al(OH)3 now appears in an equation involving an equilibrium constant, so that a second stoichiometric constraint becomes relevant. This constraint is the mass balance equation 3[Al(OH)3] + 3[Al 3+] = [Cl ] (8)

reflecting the fact that no matter what aqueous species form, the ratio of the total (analytical) concentration of Al to that of Cl must remain 1:3. Thus, assuming Al(OH) 3 is soluble, there would be just two components, H 2O and AlCl 3. C a s

a Function of Temp era ture a nd Pressure

It is obvious that the number of phases present in a system depends on the temperature T and pressure p . H owever, it is not often appreciated that the number of components is not an invariant of a system, but that it too is a function of p and T . As a fi rst example, let us return to the decomposition of NH 4Cl. At a pressure of 1 atm and between temperatures 191 and 239 K, NH 3 exists in the liquid phase while HCl remains a gas. Now only HCl appears in the equilibrium equation because NH 3 is a pure liquid in a phase by itself. Consequently, there are no longer any stoichiometric constraints so that the number of components is increased from 1 to 2. It seems strange that changing the temperature (and by extension, the pressure) of a system can result in a change in the number of components as well as a change in the number of phases. Another situation in which the number of components can vary depending upon the temperature and pressure occurs under conditions of metastable equilibrium. An arbitrary mixture of hydrogen, oxygen, and water vapor at equilibrium has two components because of the equilibrium constant relating the three species. In this case P = 1 and there are three degrees of freedom: for example, p , T , and the mole fraction of one of the three constituents. A mixture of these three gases might also be prepared by introducing arbitrary amounts of each of the gases into a container being careful to avoid an explosion. The rate of reaction is so slow at normal temperatures and pressures that we can consider the three unreacted gases to be in equilibrium. In Richard Feynman’s words, a system is in equilibri um “ if all the ‘fast’ things have happened and all the ‘slow’ things not” (7 ). Under these conditions, C = 3 (there is no equilibrium constant expression) and there are four degrees of freedom: p , T , and the mole fractions of two of the three gases. Distinguishing Com pone nts from Phases

In most cases, that i s, exercises in physical chemistry texts, there is never any question about whether two species represent two components or should be considered to be two phases of the same component. Liquid water and ice are two phases of the same component. Similarly, C(graphite) and C(diamond) are also regarded as two phases of the same component, namely carbon. But are things always so simple? Under ordinary conditions, C(gr) and C(d) are not in chemical equilibrium; C(gr) is the more stable form. However, as we know from ordinary experience, the two forms of carbon are in metastable equilibrium and can exist independently of each other. For all practical purposes, the graphite– diamond system can be regarded as being in equilibrium and subject to the phase rule, just as is the case for the hydrogen– oxygen–water vapor system discussed above. Consequently, under these conditions for the graphite–carbon system, C = 2. It is clear that P = 2. Applying the phase rule, we find F = 2. The two degrees of freedom are p and T . (Each of the two solid phases consists of only one species, so the compositions of both are determined.)

1568 Journal of Chemical Education • Vol. 76 No. 11 November 1999 • JChemEd.chem.wisc.edu

Research: Science and Education

If however, C(gr) and C(d) are at true chemical equilibrium, the chemical potentials of the two species become equal. As a result of this constraint, C = 1. Since P remains equal to two, there is just one degree of freedom, p or T . From the constraint on the chemical potentials, we can derive the Clausius equation dp /dT = ∆ trsS / ∆ trsV

(9)

where ∆ tr sS and ∆ trsV are the changes in entropy and volume across the phase boundary, which provides the functional relationship between p and T . In view of this discussion, consider a hypothetical isomerization reaction, A = B, that has reached equilibrium. (We assume that the initial amount of each species is arbitrary.) I f one or both of the two i somers are gases, C = 1 because there are two constituent species, one equilibrium expression, but no stoichiometric relation between A and B. In the case of one gaseous constituent P = 2, and so F = 1. If both constituents are gases, P = 1 and F = 2, the mole fraction of A in the gas phase providing the additional degree of freedom. If both A and B are solids, the system is totally analogous to the graphite–diamond example. If the rate of interconversion of the two solids is very slow, A and B are in metastable equilibrium, and we can take C = 2. There are two different compounds A and B with no equilibrium constant expression relating them. However, there may be some regions of the p –T plane in which p and T take on values for which the isomerization proceeds and the reaction reaches equilibrium. For these values of p and T , A and B are in chemical equilibrium. However, unlike a typical chemical reaction involving gases or solutions, there is no equilibrium constant for this reaction relating the concentrations of A and B. Each of the chemical species A and B is in a phase by itself so that the mole fraction (or concentration) of each species is unity. Even at equilibrium, the ratio of the number of moles of A and B is arbitrary. Although there is no equilibrium constant, we can analyze this equilibrium. Instead of treating the system consisting of A(s) and B(s) as a two-component system, we treat it as a one-component system with two phases that are in equilibrium. The reduction in the number of components from 2 to 1 results from the fact that at equilibrium, the chemical potentials of the two species are equal, just as is the case in the graphite–diamond example. Since C = 1 and P = 2, the phase rule tells us that F = 1. Consequently, there must be a relationship between p and T . This relationship is given by a Clausius equation which is derived by equating the chemical potentials of A and B. Starting from d µA( p ,T ) = dµ B( p ,T ) we obtain the Clausius-like relation dp /dT = ∆ S / ∆ V

(10)

where ∆ S and ∆ V are the changes in entropy and volume upon isomerization. But is it legitimate to call a system containing A(s) and B(s) a one-component system?In the derivation of the usual Clausius equation, as in the determination of the number of constraints in the derivation of the phase rule itself, we equate chemical potentials of the same substance in two different phases. I have purposely introduced the new word “substance” to leave open the question of how the relative similarity between graphite and diamond compares with that between the two isomers A and B. Graphite and diamond differ in the arrangement of their carbon atoms; in graphite the carbon atoms lie in sheets, in diamond they form a tetrahedral structure. Supposing for specificity that the two isomers are hydrocarbons, then the isomers differ only in the arrangement of their carbon and hydrogen atoms. Thermodynamics is totally independent of the details or even the existence of the molecular structure. Consequently, if chemical equilibrium can be established between any two isomers (they need not be hydrocarbons), these two species are thermodynamically just as similar to each other as are graphite and diamond. If the latter pair can be considered to be a one-component system, so can the former. Thisisomerization example shows that there are surprising subtleties associated with the application of the Gibbs phase rule. For certain chemical equilibria, such as one involving the solid phases of two isomers, a seemingly two-component system should be regarded asa one-component system. More generally, in determining the number of components, a thorough understanding of the phase structure of the system is critical. C is not independent of P . Acknowledgments

I would like to thank R. I. Gelb for his critical reading of the manuscript and for several helpful discussions. I would also like to thank the reviewers for several valuable suggestions. Literature Cited 1. Denbigh, K. The Principles of Chemical Equili bri um , 3rd ed.; Cambridge University Press: London, 1971. 2. Findlay, A.; Campbell, A. N.; Smith, N . O. T he Phase Rul e and I ts Appli cati ons , 9th ed.; Dover: New York, 1951. 3. Atkins, P. Physi cal Chemistr y , 6th ed.; Freeman: New York, 1998; pp 192–193. 4. Noggle, J. H. Physi cal Chemistr y , 3rd ed.; H arper Colli ns: New York, 1996; pp 332–333. 5. Albert y, R. A.; Silbey, R. J. Physi cal Chemi str y , 2nd ed.; Wiley: New York, 1997; pp 141–143. 6. Zhao, M.; Wang, Z.; Xiao, L. J. Chem. Educ . 1992, 69 , 539–542. 7. Feynman, R. Stati sti cal M echanics: A Set of Lectu res ; Benjamin: Reading, MA, 1972; p 1.

JChemEd.chem.wisc.edu • Vol. 76 No. 11 November 1999 • Journal of Chemical Education

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