Total differential of the internal energy

For a closed system undergoing processes in which the only kind of work is expansion work, the first law becomes dU = dq + dw = dq - pb dV. Since it will often be useful to make a distinction between expansion work and other kinds of work, this book will sometimes write the first law in the form 

Consider a closed system of one chemical component (e.g., a pure substance) in a single homogeneous phase. The only kind of work is expansion work, with V as the work variable. This kind of system has two independent variables (Sec. 2.4.3). During a reversible process in this system, the heat is = T dS, the work is dm = —pdV, and an infinitesimal internal energy change is given by 

In the conditions of validity shown next to this equation, C=1 means there is one component (C is the number of components) and P = l means there is one phase (P is the number of phases). 

The appearance of the intensive variables T and p in Eq. 5.2.2 implies, of course, that the temperature and pressure are uniform throughout the system during the process. If they were not uniform, the phase would not be homogeneous and there would be more than two independent variables. The temperature and pressure are strictly uniform only if the process is reversible it is not necessary to include reversible as one of the conditions of validity. 

A real process approaches a reversible process in the limit of infinite slowness. For all practical purposes, therefore, we may apply Eq. 5.2.2 to a process obeying the conditions of validity and taking place so slowly that the temperature and pressure remain essentially uniform—that is, for a process in which the system stays very close to thermal and mechanical equilibrium. 

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To this end, we consider the thermodynamic functions of a homogeneously stressed solid, e.g., [L.D. Landau, E.M. Lifshitz (1989) W. W. Mullins, R. Sekerka (1985)]. In contrast to the unstressed solid, the internal energy of which is U(S, K ,), the internal energy of a stressed solid is given as U(S, VuJk,nj). For the total differential of the internal energy one has1... 

The total differential of the internal energy U is then given by Eq. 2.4 ... 

The total differential of the internal energy U of a system can be written as a function of independent state variables such as the temperature, volume and composition of the system as shown in Eq. 5.1 ... 

There are several examples where we can relate thermodynamically measurable parameters to the molecular properties. The link between the internal pressure and the van der Waals constants, a and b is a good example. The internal pressure of a fluid is defined from pure thermodynamics if we consider the total differential of the internal energy of a fluid as a function of its entropy and volume, U = f(S,V)... 

The resemblance between Eq. (101) and the total differential of the internal energy with respect to the number of moles of a chemical species, n, and the volume of the container, V, at absolute zero ... 

CHAPTER 5 THEIiMODYNAMIC POTENTIALS 5.2 Total Differential of the internal energy... 

CHAPTER 5 THERMODYNAMIC POTENTIALS 5.2 TOTAL Differential of the internal Energy... 

For the moment we shall confine our attention to closed systems with one component in one phase. The total differential of the internal energy in such a system is given by Eq. 5.2.2 dt/ = T dS — pdV. The independent variables in this equation, S and V, are called the natural variables of U. 

Incorporating such a departure in a total differential of the internal energy of the surface phase, and using Euler s theorem on homogeneous functions, the following relation can be shown to be valid for the surface phase (Guggenheim, 1967) ... 

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