Internal energy differential relationships

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

The partial derivative (dU/dT)p is not Cy, but if it could be expanded into some relationship with (dU/dT)y, we would have succeeded in introducing Cy into Equation (4.58). The necessary relationship can be derived by considering the internal energy U as sl function of T and V and setting up the total differential ... 

We may note that the energy conservation principle (or, equivalently, the first law of thermodynamics) has not improved the balance between the number of unknown, independent variables and differential relationships between them. Indeed, we have obtained a single independent scalar equation, either (2 47 ) or (2-51), but have introduced several new unknowns in the process, the three components of q and either the specific internal energy e or enthalpy h. A relationship between e or h and the thermodynamic state variables, say, pressure p and temperature 9, can be obtained provided that equilibrium thermodynamics is assumed to be applicable to a fluid element that moves with a velocity u. In particular, a differential change in 9 orp leads to a differential change in h for an equilibrium system ... 

Here is a useful leisure time exercise for a very attentive reader. The purpose is to understand the connection between virial expansion (8.7) and the well known van der Waals equation of state (i.e., the relationship between volume, pressure, and temperature) for an ordinary imperfect gas. You may have studied van der Waals equation in general physics and/or general chemistry class, it reads p- -a/V )(V — b) = NksT. Say, the volume is V, and the number of molecules in the gas is N. Then n = N/V. You can work out the pressure by differentiation p = — (9F/9V), where free energy F is defined by formula (7.19), F = U — TS = t/ + / , the internal energy U is given by (8.7), and... 

For entropy to be of any practical value, we must be able to relate it to quantities that can be measured experimentally. Here is how we develop this relationship. Since entropy is a state function, we can express it as a mathematical function of two intensive properties. We choose internal energy U, and volume V, and write S = S(U, V). This unusual choice is perfectly permissible.iThe differential of entropy in terms of these independent variables is... 

The Nomenclature Committee of the International Confederation for Thermal Analysis (ICTA) has defined DSC as a technique in which the difference in energy inputs into a substance and a reference material is measured as a function of temperature whilst the substance and reference material are subjected to a controlled temperature program. Two modes, power compensation DSC and heat flux DSC, can be distinguished depending on the method of measurement used1 . The relationship of these techniques to classical differential thermal analysis (DTA) is discussed by MacKenzie2). 

---