Total versus Molar Properties

There is more than one difficulty introduced when physical properties of systems are used as variables in functions, as the reader will discover. However, the first difficulty encountered is relatively simple, although extremely important. That is that many physical properties such as the volume and various energy terms come in two forms—the total quantity in the system, and the quantity per mole or per gram of substance considered. For example, if we have 513 g of water in a beaker, its volume (V) is about 513 cm. We are, however, often concerned not with the total volume but the volume per mole (V, about 18 cm mol in this case), or the volume per gram (specific volume, about 1 cm g ). We use Roman capital letters for total properties, and the corresponding italic capitals for molar properties, and 

The reader is advised to notice whether Roman or Italic capitals are used in the equations to be presented, as it is an aid to clear thinking. In fact, many sections of this text will be quite confusing if the distinction is not made. Generally speaking, any functions or equations that are used in real applications will be in the molar form, while theoretical discussions are often more convenient using the total form. For example, much of the theoretical discussion in Chapters 4 to 6 inclusive will use total properties, but in Chapter 7 we start to discuss measurements and properties with numerical values, and of necessity we use molar properties. 

Of course, many equations look much the same with total and molar properties, because ratios are involved. That is, if (9U/c)S)v = T, then it is also true that (dU/dS)v = T or if dG/dP)T = V, then (dG/dP)T = V, so that the distinction may seem to be unimportant. However, sometimes it is important. For example, understanding the meaning of partial molar properties (Chapter 9) is not really possible without making this distinction clear, and many of the equations in Chapter 14 look quite different in the two forms. 

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For a binary mixture the characteristics of the partial molar properties can be illustrated graphically by plotting the state property m versus the mole fraction xi. The total differential of the partial molar property gives an equation of the form... 

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