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

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

1. Introduction

1.4. Notation

Within these notes we use at many occasions thermodynamics, and for that topic it is essential to agree on some conventions. For summations over particles, molecules, and so on, a lowercase Latin index, say i or j, is used, while for a summation over chemical components a lowercase Greek index, say α or β, is used. Furthermore, a superscript * is used for a pure compound, for example, the partial volume c1-math-5001 of component α, and a superscript ° for a reference state, for example, the pressure P°, conventionally taken as 1 bar.

With respect to mathematical notation, scalars are addressed via an italic letter, say a, and vectors by an italic bold-face letter, say a. Column matrices are labeled by, say ai (index notation), or by a roman bold-face letter, say a(matrix notation). Similarly, square matrices are addressed by an italic letter with two subscripts, say Aij or by A. The column a is used as a shorthand for a collective of N quantities, that is, a = a1a2aN. For example, for Nmolecules each with coordinates ri where ri = (xi,yi,zi), we denote the coordinates collectively by r = r1r2rN = x1y1z1x2y2z2xNyNzN or in a multidimensional integral we write ∫dr where dr = dr1dr2 … drN = dx1dy1dz1dx2dy2dz2… dxNdyNdzN. If we denote the set bi by b and the set ai by a, we can therefore write c = Σibiai = bTa using the transpose bT of b. This allows us to write the derivatives of a function f(ai) given by bi = ∂f/∂ai collectively as b = ∂f/∂aor of a set fi(aj) as Bij = ∂fi/∂aj (equivalently for f we have B = ∂f/∂a). Note, therefore, that we distinguish between a vector a and its matrix representation a. The inner product c of two vectors a and b is c = a·b (= Σibiai) and written in matrix notation as c = aTb. For some further conventions on notation, we refer to Appendix B.


1) Although we denote for convenience the basic entities as molecules, the term is also supposed to include atoms and ions, whenever appropriate.

2) We “forget” for convenience phase transformations.

3) Liquids and gases together are often indicated as fluids.

4) The pair correlation function and its properties will be discussed more extensively in Chapter 6.

5) The probability is scaled in such a way that its average value is unity.


1 (a) Witten, T.A. (2004) Structured Fluids, Oxford University Press, Oxford; (b) Jones, R.A.L. (2002) Soft Condensed Matter, Oxford University Press, Oxford.

2 See, e.g., van Emmerik, E.P. (1991) J.J. van Laar (1860–1938), A mathematical chemist. Thesis, Delft University of Technology.

Further Reading

Barrat, J.-L. and Hansen, J.-P. (2005) Basic Concepts for Simple and Complex Liquids, Cambridge University Press, Cambridge.

Barker, J.A. (1963) Lattice Theories of the Liquid State, Pergamon, London.

Barton, A.F.M. (1974) The Dynamic Liquid State, Longman, New York.

Beck, T.L., Paulatis, M.E., and Pratt, L.R. (2006) The Potential Distribution Theorem and Models of Molecular Solutions, Cambridge University Press, Cambridge.

Ben-Naim, A. (1974) Water and Aqueous Solutions: Introduction to a Molecular Theory, Plenum, London.

Ben-Naim, A. (2006) Molecular Theory of Solutions, Oxford University Press, Oxford.

Croxton, C.A. (1974) Liquid State Physics: A Statistical Mechanical Introduction, Cambridge University Press, Cambridge.

Debenedetti, P.G. (1996) Metastable Liquids: Concept and Principles, Princeton University Press, Princeton.

Egelstaff, P.A. (1994) An Introduction to the Liquid State, 2nd edn, Clarendon, Oxford.

Eyring, H. and Jhon, M.S. (1969) Significant Liquid Structures, John Wiley & Sons, Ltd, London.

Fawcett, W.R. (2004) Liquids, Solutions and Interfaces, Oxford University Press, Oxford.

Fisher, I.Z. (1964) Statistical Theory of Liquids, University of Chicago Press, Chicago.

Frisch, H.L. and Salsburg, Z.W. (1968) Simple Dense Fluids, Academic, New York.

Frenkel, J. (1946) Kinetic Theory of Liquids, Oxford University Press, Oxford (see also Dover, 1953).

Guggenheim, E.A. (1952) Mixtures, Oxford, Clarendon.

Hansen, J.-P. and McDonald, I.R. (2006) Theory of Simple Liquids, 3rd edn, Academic, London (1st edn 1976, 2nd edn 1986).

Henderson, D. (ed.) (1971) Physical Chemistry, and Advanced Treatise, vols. VIIIa and VIIIb, Academic, New York.

Hildebrand, J.H. and Scott, R.L. (1950) Solubility of Non-Electrolytes, 3rd edn, Reinhold (1st edn 1924, 2nd edn 1936).

Hildebrand, J.H. and Scott, R.L. (1962) Regular Solutions, Prentice-Hall, Englewood Cliffs, NJ.

Hildebrand, J.H., Prausnitz, J.M., and Scott, R.L. (1970) Regular and Related Solutions, Van Nostrand-Reinhold, New York.

Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B. (1954) Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York.

Kalikmanov, V.I. (2001) Statistical Physics of Fluids, Springer, Berlin.

Kohler, F. (1972) The Liquid State, Verlag Chemie, Weinheim.

Kruus, P. (1977) Liquids and Solutions, Marcel Dekker, New York.

Larson, R.G. (1999) The Structure and Rheology of Complex Fluids, Oxford University Press, New York.

Lucas, K. (2007) Molecular Models of Fluids, Cambridge University Press, Cambridge.

Lee, L.L. (1988) Molecular Thermodynamics of Nonideal Fluids, Butterworths, Boston.

March, N.H. and Tosi, M.P. (2002) Introduction to the Liquid State Physics, World Scientific, Singapore.

March, N.H. and Tosi, M.P. (1976) Dynamics of Atoms in Liquids, McMillan, London (see also Dover, 1991).

Marcus, Y. (1977) Introduction to Liquid State Chemistry, John Wiley & Sons, Ltd, London.

Murrell, J.N. and Jenkins, A.D. (1994) Properties of Liquids and Solutions, 2nd edn, John Wiley & Sons, Ltd, Chichester.

Prigogine, I. (1957) The Molecular Theory of Solutions, North-Holland, Amsterdam.

Pryde, J.A. (1966) The Liquid State, Hutchinson University Library, London.

Rice, S.A. and Gray, P. (1965) The Statistical Mechanics of Simple Liquids, Interscience, New York.

Rowlinson, J.S. and Swinton, F.L. (1982) Liquids and Liquid Mixtures, 3rd edn, Butterworth, London.

Temperley, H.N.V., Rowlinson, J.S., and Rushbrooke, G.S. (1968) Physics of Simple Liquids, North-Holland, Amsterdam.

Temperley, H.N.V. and Trevena, D.H. (1978) Liquids and Their Properties, Ellis Horwood, Chichester.

Ubbelohde, A.R. (1978) The Molten State of Matter, John Wiley & Sons, Ltd, Chichester.

Wallace, D.C. (2002) Statistical Physics of Crystals and Liquids: A Guide to Highly Accurate Equations of State, World Scientific, Singapore.

Watts, R.O. and McGee, I.J. (1976) Liquid State Chemical Physics, John Wiley & Sons, Inc., New York.