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Realistic Thermodynamic Curves Describing a Second-Order Phase Transition Paul W. R. Bessonette and Mary Anne White* Department of Chemistry, Dalhousie University, Halifax, NS B3H 4J3, Canada
Importance of Phase Transitions Phase transitions are commonly recognized in the context of links between the solid, liquid, and gaseous phases of a substance. The melting of ice cubes and the boiling of liquid water serve as examples of phase transitions that we observe in everyday life. The full scope of phase transitions, however, extends far beyond this. Many substances exhibit polymorphism—that is, phase transformations between two or more solid phases upon changing the temperature and/or pressure. For example, ice is known to have at least 11 different solid phases (1, 2), most of which are observed upon application of high pressure. An unusual phase transition occurs in liquid helium whereby the “high-temperature” liquid phase transforms to a second liquid phase (3) at a temperature of 2.2 K. When materials undergo phase transitions they can change their optical, thermal, electronic, magnetic, and/or mechanical properties, and therefore delineation of phase transitions and their driving forces can be very important in understanding properties of materials. Despite the number and variety of phase transitions that are observed, they are all thermodynamically bound by the common feature that the Gibbs energy function (G ) must be continuous upon passing from one equilibrium phase to another. In this paper we consider the temperature dependence of G for a variety of types of phase transitions and correct a diagrammatic error in the current selection of introductory physical chemistry textbooks. Ehrenfest Classification of Phase Transitions In 1933, Paul Ehrenfest (4) put forth a classification of phase transitions in matter based upon the nature of the continuity in the Gibbs energy function for a substance at a phase transition in the following manner.
First-Order Transitions A first-order phase transition is one for which the first derivatives of G given by S = ᎑ ∂G ∂T
(1) P
(2) T
are discontinuous through the transition (S is the entropy, V *Corresponding author. Email:
[email protected].
220
C p = T ∂S ∂T
= ᎑T P
∂2G ∂T
(3)
2 P
and
α = 1 ∂V V ∂T
= 1 ∂ ∂G P V ∂T ∂P
(4) T P
at the transition. A common example of a first-order phase transition is melting (solid → liquid).
Second-Order Transitions A second-order phase transition as defined by Ehrenfest is one for which the first derivatives of the Gibbs energy function (S and V ) are continuous through the transition, but the second derivatives of G are now discontinuous. Thus, there can be no discontinuities in the entropy or volume (∆trS = ∆trV = 0), but the heat capacity and thermal expansion as defined by eqs 3 and 4, respectively, show finite jump discontinuities at the temperature of the phase transition. Perhaps the best example of an ideal second-order phase transition is the normal-to-superconducting phase transition at zero applied magnetic field that occurs in certain materials at low temperature. An illustrative example of this is shown by low-temperature heat capacity measurements of tin (5). Higher-Order Transitions Third- and higher-order phase transitions can be defined in a completely analogous fashion based upon discontinuities in the higher-order derivatives of the Gibbs energy function. The Ehrenfest classification has been further discussed and elaborated by Pippard (6 ). Thermodynamic Curves
and
V = ∂G ∂P
is the molar volume). Thus, a first-order transition shows an abrupt, finite, nonzero change in entropy (∆ trS ≠ 0) and volume (∆ trV ≠ 0) at the transition temperature, Ttr. As a consequence there must also be a discontinuity in H (i.e., latent heat; ∆ trH = Ttr × ∆ trS ≠ 0). The discontinuous behavior of S and V at the transition temperature leads to infinite values of both the heat capacity at constant pressure (Cp ) and the thermal expansion (α) defined by
First-Order Transitions A thermodynamically consistent set of theoretical curves describing the temperature-dependent behavior of G, S, V, H, and Cp for a first-order phase transition is well established, and presented diagrammatically in many introductory physical chemistry textbooks (see, for example, references [7–9]). Such
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In the Classroom
a set of curves is shown in Figure 1.
Second-Order Transitions: A Diagrammatic Error Can Lead to Physically Unrealistic Results The analogous thermodynamic curves for second-order phase transitions are not so apparent. A survey of all the introductory physical chemistry textbooks available at present (10) showed that, although many discuss second-order phase transitions, only two (7, 8) show diagrams of G(T ) for a second-order phase transition, and, in both cases G(T ) is drawn in the same incorrect manner, leading to a physically impossible situation, as we now show. In both reference 7 and reference 8, the G vs T curve shows an inflection point at the transition temperature (Ttr), as shown in Figure 2a. While an inflection point in G(T ) meets the criterion of continuity of the first derivatives of G(T ), the second derivative of an inflection point in a curve is zero (thus Cp = 0 at Ttr) by definition, and this implies that (dG/dT ) must have a local extremum at Ttr (i.e., minimum or maximum in S ). Specifically, using the definitions of S and Cp given in eqs 1 and 3, respectively, the G(T ) function of Figure 2a results in S(T ) and Cp(T ) as shown in Figures 2b and 2c, respectively; that is, an inflection in G(T ) (Fig. 2a) leads to a minimum in the entropy curve at Ttr, and negative heat capacity below Ttr—both unrealistic situations! Realistic Depiction of G(T) for Ehrenfest Second-Order Transitions To resolve this problem, we approached G(T ) for a second-order transition from a physically realistic picture of S(T ), as shown in Figure 3a. Note that S(T ) is continuous at Ttr, in accord with eq 1 and the definition of a secondorder transition, and S increases with increasing temperature (this follows from eq 3 with Cp(T ) > 0 at all temperatures). From S(T ), G(T ) can be determined by rearranging eq 1: T
G – G0 = ᎑
SdT
(5)
0
and G(T ) is shown in Figure 3b. From eq 3, Cp(T ) has been derived and this is shown in Figure 3c. Note that in Figure 3b the change in the Gibbs energy curve at the temperature of the transition is very subtle, and can really only be seen easily in its derivatives, S(T ) and V(T ) (see Fig. 3d for the latter). The Cp(T ) curve shown (Fig. 3c) is very similar to that observed experimentally in tin (5). Being related to the second partial derivative of the Gibbs energy with respect to temperature, the heat capacity shows a finitejump discontinuity at Ttr consistent with the definition of a second-order phase transition. As a further step, the enthalpy curve, H, can be calculated by integrating the heat capacity curve T
H – H0 =
0
C pd T
(6)
with the result shown in Figure 3e. Figures 3d and 3e show ∆trH = 0 and ∆trV = 0, consistent with the Ehrenfest definition of a second-order phase transformation. We are not the first to resolve this problem. Correct G(T ) curves are published in Alberty’s introductory physical chemistry textbook (11), but this textbook is no longer in print and its successor (9) does not include G(T ) diagrams
Figure 1. Depictions of various thermodynamic parameters as functions of temperature in the region of a first-order phase transition. (a) Gibbs energy function, G ; (b) entropy, S; (c) molar volume, V ; (d) enthalpy, H ; (e) heat capacity, Cp .
Figure 2. An unrealistic depiction of thermodynamic parameters as functions of temperature in the region of an Ehrenfest second-order phase transition. Inflection in Gibbs energy function (a) as shown in two introductory physical chemistry textbooks (7, 8) leads to a minimum in the entropy (b) and negative heat capacity (c).
for second-order phase transitions. A correct diagram is also given in Rock’s introductory thermodynamics textbook (12).
Other Second-Order Phase Transitions The Ehrenfest classification of second-order phase transitions is based on mean field theory. The heat capacity, thermal expansion coefficient, and isothermal compressibility are always finite but have discontinuities at the phase transition temperature. (See Figure 3c for Cp(T ).) Most experimentally observed second-order phase transitions do not fit the Ehrenfest category because mean field theory fails to describe these fluctuation-dominated phase transitions. There is still no finite jump in the second derivatives of G, but the heat capacity, thermal expansion coefficient, and isothermal compressibility now all approach infinity at the phase transition temperature (critical point). Corresponding thermodynamic curves are shown in Figure 4. Examples of critical second-order transitions include Ising transitions in uniaxial magnets, the λ-point in helium, and Heisenberg transitions in magnets.
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In the Classroom
H(T ), and V(T ) are summarized in Figures 1, 3, and 4 for first-order, Ehrenfest second-order, and critical second-order transitions, respectively. The derivation of the set of curves for an Ehrenfest second-order transition is also a useful exercise to demonstrate the intricate relationships among thermodynamic quantities. In general, these sets of thermodynamic curves summarize experimental evidence for phase transitions and their driving force (lowest G phase is most stable). Acknowledgments We are grateful to R. J. C. Brown for useful discussions. This work is supported by the Natural Sciences and Engineering Research Council of Canada (scholarship to PWRB, grants to MAW) and the Killam Trust (scholarship to PWRB and Research Professorship in Materials Science to MAW). Literature Cited
Figure 3. Realistic depictions of various thermodynamic parameters as functions of temperature in the region of an Ehrenfest second-order phase transition. (a) entropy, S; (b) Gibbs energy function, G ; (c) heat capacity, Cp ; (d) molar volume, V; (e) enthalpy, H .
Figure 4. Depictions of various thermodynamic parameters as functions of temperature in the region of a critical (fluctuationdominated) second-order phase transition. (a) entropy, S ; (b) Gibbs energy function, G ; (c) heat capacity, C p; (d) molar volume, V; (e) enthalpy, H .
1. Matsuo, T.; Tajima, Y.; Suga, H. J. Phys. Chem. Solids 1986, 47, 165–173. 2. Klug, D. D.; Handa, Y. P.; Tse, J. S.; Whalley, E. J. Chem. Phys. 1989, 90, 2390–2392. 3. Atkins, K. R. Liquid Helium; Cambridge University Press: New York, 1959; Chapter 2 and references cited therein. 4. Ehrenfest, P. Commun. Kamerlingh Onnes Lab., Leiden Suppl. 1933, 75b. 5. Keesom, W. H.; Van Laer, P. H. Physica 1938, 5, 193–201. 6. Pippard, A. B. Elements of Classical Thermodynamics; Cambridge University Press: New York, 1957, p 136. 7. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 2nd ed.; Houghton Mifflin: Boston, 1995. 8. Atkins, P. W. Physical Chemistry, 5th ed.; Freeman: New York, 1994. 9. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 2nd ed.; Wiley: New York, 1997. 10. The Book Buyers Guide, a supplement to Journal of Chemical Education, September 1997, lists 15 books that could be classified as introductory physical chemistry textbooks. Of these, one (Ladd, M. Introduction to Physical Chemistry, 3rd ed.; Cambridge University Press) was not yet published when this paper was written. 11. Alberty, R. A. Physical Chemistry, 7th ed.; Wiley: New York, 1987. 12. Rock, P. A. Chemical Thermodynamics; University Science Books: Mill Valley, CA, 1983.
Conclusions The S, G, Cp , V, and H curves that we have presented here are both physically realistic and thermodynamically consistent for a second-order phase transition, according to the Ehrenfest definition. This contrasts with the diagrammatic representations available in the present range of introductory physical chemistry textbooks, showing a Gibbs energy function with an inflection point at the transition temperature (7, 8); we show this to be physically unrealistic. The proof of the latter would make a useful exercise in a course in which phase transitions are discussed. Furthermore, the subtle shape change in a feasible G(T ) curve at a second-order transition temperature shows that calculations of G(T ) alone (e.g., from molecular dynamics) do not readily identify a second-order phase transition; derivatives of G (i.e., S, V, and Cp) make the transition much more apparent. From an experimental perspective, the order of a phase transformation can be delineated by measurements of H(T ) or V(T ) where the presence of a discontinuity indicates that the phase transformation is first order. Thermodynamic curves for G(T ), S(T ), C p (T ), 222
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