Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
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 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 = a1a2 … aN. For example, for Nmolecules each with coordinates ri where ri = (xi,yi,zi), we denote the coordinates collectively by r = r1r2 … rN = x1y1z1x2y2z2 … xNyNzN 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.
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