## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 16. Some Special Topics: Phase Transitions

### 16.6. Final Remarks

We have come a long way: from a phenomenological description recognizing universality classes, via mean field theory, generalized homogenous functions, lattice gas and scaling to renormalization. These seven pillars (of wisdom?) constitute the modern basis of phase transition theory. The overall picture is complex, in particular the renormalization part, but is capable of catching most of the essentials of phase transformations quite well.

Notes

__1)__ In the past, the transitions were often labeled as first and second order, according to the discontinuity of their first- or second-order derivatives of the Gibbs energy. However, the second-order “class” appeared to be more complex than anticipated, and therefore these transitions are nowadays often labeled as continuous, due to the fact that in all cases a continuous transition from a one-phase state to a two-phase state occurs with a continuous change in order parameter (Δ*ρ* for fluids) over the transition. Although the label “first order” stuck, for consistency, we refer to this transition as discontinuous, the more so since the density behavior for fluids is discontinuous over the transition.

__2)__ Actually, the transition process is strongly influenced by gravity, and the density profile over the meniscus near the critical temperature can be described by an expression akin to the barometric formula (see __Problem 16.1__).

__3)__ A similar exercise for the Massieu functions, i.e. the entropy and its Legendre transforms, leads to concave functions of their extensive variables and convex functions of their intensive variables.

__4)__ Although the derivation of *κ _{T}* ≥ 0 is easier from

*F*than from

*S*,

*F*refers to isothermal conditions so that the thermal stability condition

*C*≥ 0 cannot be derived from

_{V}*F*.

__5)__ Note, though, that in practice doing reversible work along the curve bcd is impossible.

__6)__ Often, experimental data are given as in terms of log*P* instead of ln*P*.

__7)__ In the next paragraphs we use a labeling for the various critical exponents which is largely standardized. Unfortunately several labels, such as *β* and *α*, are also used for other properties.

__8)__ For example, in 3D *h*(*r*) can be approximately described by *h*(*r*) = (*a*/*r*) cos(*br *+ *c*) exp(−*r*/*ξ*).

__9)__ A square matrix **A** of order *m* operating on a column **u** yields a new column **u′**. If this operation yields a multiple of **u**, say λ**u**, the resulting equation (**A** − *λ***I**)**u** = 0 is an *eigenvalue equation* with *λ* the *eigenvalue*, **I** the unit matrix, and **u** the *eigenvector*. This equation can be solved by det(**A** − *λ***I**) = 0, yielding *m* eigenvalues and associated eigenvectors. Collecting the set of columns **u** in the matrix **U**, the complete set of equations can be written as **Λ = U**^{T}**AU**, where **Λ** is the diagonal matrix containing all eigenvalues. For a symmetric matrix **A**, **U** is an orthogonal matrix, i.e. **U**^{−1} = **U**^{T}. For a non-symmetric matrix **A** this is no longer the case, and we have to keep track of whether the eigenvector is positioned on the left or right side of the matrix **A**, i.e. whether we solve **Au** = λ**u** or **u**^{T}**A** = λ**u**^{T}.

References

1 (a) Antoine, C. (1888) *Compt. Rend.*, 107, 681 and 728; (b) Thomas, G.W. (1940) *Chem. Rev.*, 38, 1; (c) For data, see Reid R.C., Prausnitz J.M., and Poling B.E. (1988) *The Properties of Gases and Liquids*, 4th edn, McGraw-Hill.

2 Clusius, K. and Weigand, K. (1940) *Z. Phys. Chem.*, B46, 1.

3 Heller, P. (1967) *Rep. Progr. Phys.*, 30, 731. For an extensive review, see Sengers and Levelt-Sengers (1978).

4 (a) Fisher, M.E. (1964) *J. Math. Phys.*, 5, 944; (b) Fisher, M.E. (1967) *Rep. Prog. Phys.*, 30, 615.

5 See Ma (1976).

6 Barieau, R. (1966) *J. Chem. Phys.*, 45, 3175.

7 Widom, B. (1965) *J. Chem. Phys.*, 43, 3892 and 3898.

8 (a) Kadanoff, L.P. (1966) *Physics*, 2, 263; (b) Kadanoff, L.P. *et al.* (1967) *Rev. Mod. Phys.*. 39, 395.

9 Cooper, M.J. (1968) *Phys. Rev.*, 168, 183.

10 Wilson, K.G. (1971) *Phys. Rev.*, B4, 3174 and 3184.

11 See Gitterman and Halpern (2004).

12 This illustration was first given by Chowdhury, D. and Stauffer, D. (2000) *Principles of Equilibrium Statistical Mechanics*, Wiley-VCH Verlag GmbH, Weinheim.

13 Moelwyn-Hughes, E.A. (1961) *Physical Chemistry*, 2nd edn, Pergamon, Oxford.

14 Hocken, R. and Moldover, M.R. (1976) *Phys. Rev. Lett.*, 37, 29.

15 Ogita, N., Ueda, A., Matsubara, T., Matsuda, H., and Yonezawa, F. (1969) *J. Phys. Soc. Jpn*, 26 (Suppl.), 145.

16 Onsager, L. (1944) *Phys. Rev.*, 65, 117.

17 Pelissetto, A. and Vicari, E. (2002) *Phys. Rep.*, 368, 549.

Further Reading

Binney, J.J., Dowrich, N.J., Fisher, A.J., and Newman, M.E.J. (1992) *The Theory of Critical Phenomena*, Clarendon Press, Oxford.

Gitterman, M. and Halpern, V.H. (2004) *Phase Transitions*, World Scientific, Singapore.

Goldenfeld, N. (1993) *Lectures on Phase Transitions and the Renormalization Group*, Addison-Wesley.

Herbut, I. (2007) *A Modern Approach to Critical Phenomena*, Cambridge University Press, New York.

Ma, S.-K. (1976) *Modern Theory of Critical Phenomena*, Benjamin, Reading.

Sengers, J.V. and Levelt Sengers, J.M.H. (1978) Critical phenomena in classical fluids, in *Progress in Liquid Physics* (ed. C.A. Croxton), John Wiley & Sons, Ltd, Chichester, Ch. 4, pp. 103–174.

Stanley, H.E. (1971) *Introduction to Phase Transitions and Critical Phenomena*, Oxford University Press, New York.

Uzunov, D.I. (1993) *Introduction to the Theory of Critical Phenomena*, World Scientific, Singapore.