## 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 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

*a*(index notation), or by a roman bold-face letter, say

_{i}**a**(matrix notation). Similarly, square matrices are addressed by an italic letter with two subscripts, say

*A*or by

_{ij}**A**. The column

**a**is used as a shorthand for a collective of

*N*quantities, that is,

**a**=

**a**

_{1}

**a**

_{2}…

**a**

*. For example, for*

_{N}*N*molecules each with coordinates

**r**

*where*

_{i}**r**

*= (*

_{i}*x*,

_{i}*y*,

_{i}*z*), we denote the coordinates collectively by

_{i}**r**=

**r**

_{1}

**r**

_{2}…

**r**

*=*

_{N}*x*

_{1}

*y*

_{1}

*z*

_{1}

*x*

_{2}

*y*

_{2}

*z*

_{2}…

*x*or in a multidimensional integral we write ∫d

_{N}y_{N}z_{N}**r**where d

**r**= d

**r**

_{1}d

**r**

_{2}… d

**r**

*= d*

_{N}*x*

_{1}d

*y*

_{1}d

*z*

_{1}d

*x*

_{2}d

*y*

_{2}d

*z*

_{2}… d

*x*d

_{N}*y*d

_{N}*z*. If we denote the set

_{N}*b*by

_{i}**b**and the set

*a*by

_{i}**a**, we can therefore write

*c*= Σ

*=*

_{i}b_{i}a_{i}**b**

^{T}

**a**using the transpose

**b**

^{T}of

**b**. This allows us to write the derivatives of a function

*f*(

*a*) given by

_{i}*b*= ∂

_{i}*f*/∂

*a*collectively as

_{i}**b**= ∂

*f*/∂

**a**or of a set

*f*(

_{i}*a*) as

_{j}*B*= ∂

_{ij}*f*/∂

_{i}*a*(equivalently for

_{j}**f**we have

**B**= ∂

**f**/∂

**a**). Note, therefore, that we distinguish between a vector

**and its matrix representation**

*a***a**. The inner product

*c*of two vectors

**and**

*a***is**

*b**c*=

**·**

*a***(= Σ**

*b**) and written in matrix notation as*

_{i}b_{i}a_{i}*c*=

**a**

^{T}

**b**. For some further conventions on notation, we refer to

__Appendix B__.

Notes

__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.

References

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.