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 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_i (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 A_ij or by A. The column a is used as a shorthand for a collective of N quantities, that is, a = a₁a₂ … a_N. For example, for Nmolecules each with coordinates r_i where r_i = (x_i,y_i,z_i), we denote the coordinates collectively by r = r₁r₂ … r_N = x₁y₁z₁x₂y₂z₂ … x_Ny_Nz_N or in a multidimensional integral we write ∫dr where dr = dr₁dr₂ … dr_N = dx₁dy₁dz₁dx₂dy₂dz₂… dx_Ndy_Ndz_N. If we denote the set b_i by b and the set a_i by a, we can therefore write c = Σ_ib_ia_i = b^Ta using the transpose b^T of b. This allows us to write the derivatives of a function f(a_i) given by b_i = ∂f/∂a_i collectively as b = ∂f/∂aor of a set f_i(a_j) as B_ij = ∂f_i/∂a_j (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 (= Σ_ib_ia_i) and written in matrix notation as c = a^Tb. 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.