Differential of the internal energy

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 ... 

As we saw in Section 1.3, the prototypical lamella representing the confined fluid from a purely thermodynamic perspective may be deformed in a number of ways. For example, the most general expression for the exact differential of the internal energy in Eq. shows that in the context of... 

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 ... 

This equation is of fundamental importance. Its thermodynamic interpretation is that it gives the change in internal energy as a result of a small change of state under reversible conditions. Its mathematical interpretation is that it expresses the differential of the internal energy, using entropy and volume as the independent variables. Internal energy is a state function, therefore, the above is an exact differential. As... 

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. 

To ealeulate the pressure, we need to have the differential of the internal energy in variables P and T, an expression whieh is of the form ... 

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) ... 

The differentials of Q and W in (3.2) are given as 5Q and 5W, respectively, while the differential of the internal energy U is given as dU. This special notation has been used to illustrate the fundamental difference between state variables such as t/, and energy quantities such as Q and W. 

The mathematical difference between dU and 5Q, or between dU and 5W is as follows The differential of the internal energy dU is an exact (or total) differential 5Q and 5W are not exact differentials (see Mathematical appendix,... 

For any reversible process where only volume work occurs, it will apply that ad that 6Q = SQrev = TdS. Under these circumstances the differential of the internal energy U can thus be expressed on the following form... 

The fundamental equations (f) denote exact differentials of the thermodynamic state functions U, H, G, and A (see Mathematical appendix, subchapter A6 on Exact differential) for example, the differential of the internal energy U S,V) means that... 

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