General Thermodynamic Relations

As Einstein noted (see the introduction to Chapter 1), it is remarkable that the two laws of thermodynamics are simple to state but they relate so many different quantities and have a wide range of applicability. Thermodynamics gives us many general relations between state variables which are valid for any system in equilibrium. In this section we shall present a few important general relations. We will apply them to particular systems in later Chapters. As we shall see in Chapters 15-17, some of these relations can also be extended to nonequilibrium systems that are locally in equilibrium. 

One of the important general relations is the Gibbs-Duhem equation. This relation shows that the intensive variables T, p and Pjt independent. It 

This relation follows from the assumption that entropy is an extensive function of U, V and Nk and from the Euler theorem. The differential of (5.2.2) is 

Equation (5.2.4) is called the Gibbs-Duhem equation. It shows that changes in the intensive variables T, p and cannot all be independent. We shall see in Chapter 7 that the Gibbs-Duhem equation can be used to understand the equilibrium between phases, and the variation of boiling point with pressure as described by the Clausius-Clapeyron equation. 

At constant temperature and pressure, from (5.2.4) it follows that Yhk k d ik)pT — Since the change in the chemical potential d ik)pj = (0p /SiV,)d iV,-, we can write this expression as  

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The Josephson current, being an equilibrium supercurrent between two superconductors, can be calculated from the general thermodynamical relation... 

All the general thermodynamic relations can be applied with minor symbolic modifications to the partial molar quantities ... 

The denominator on the right side of Eq. (4) is the heat capacity at constant pressure Cp. The numerator is zero for an ideal gas [see Eq. (1)]. Accordingly, for an ideal gas the Joule-Thomson coefficient is zero, and there should be no temperature difference across the porous plug. Eor a real gas, the Joule-Thomson coefficient is a measure of the quantity [which can be related thermodynamically to the quantity involved in the Joule experiment, Using the general thermodynamic relation ... 

Brunauer and co-workers [21,22] used the following general thermodynamic relation [23] to obtain the pore size distribution (referred to as the "modelless" method)... 

The corresponding reference curve for can be found by numerical differentiation through the general thermodynamical relation q = (3 x/91nT)g — fegT. 

Using the general thermodynamical relations expressing the enthalpy H and the volume Fas a function of the free enthalpy G, H = d([iG)/d[i and V — dG/dP, one can deduce the following informations from the correlation time. At constant pressure, the slope of lnxc as a function of /i = 1 //cBT is the activation enthalpy. Similarly, at constant temperature, a study as a function of pressure gives the activation volume as A Fa — knTdln(xc)/dP. A more complete discussion may be found in ref. 88. 

In the first part of this paper general thermodynamic relations are presented for the calculation of thermodynamic properties from functions representing P-F-T and specific heat, velocity of sound, or Joule-Thomson data. In the latter part of the paper the equations for thermodynamic properties are developed in terms of zero-pressure (i.e., ideal gas) specific heats and are applied to a particular equation of state. 

Section 1. The standard reference for general thermodynamic relations is... 

Equilibrium thermodynamics too has its extremum principles. In this chapter we will see that the approach to equilibrium under different conditions is such that a thermodynamic potential is extremized. Following this, in preparation for the applications of thermodynamics in the subsequent chapters, we will derive general thermodynamic relations. 

In multicomponent systems, thermodynamic functions such as volume V, Gibbs free energy G, and many other thermodynamic functions that can be expressed as functions of p, Tand Nk are extensive functions of Nk. This extensivity gives us general thermodynamic relations, some of which we will discuss in this section. Consider the volume of a system as a function of p, T and Nk V = V p,T,Nk). At constant p and T, if all the mole numbers were increased by a factor X, the volume V would also increase by the same factor. This is the property of extensivity we have already discussed several times. In mathematical terms, we have... 

The formalism and general thermodynamic relations that we have seen in the previous chapters have a wide applicability. In this chapter we will see how thermodynamic quantities can be calculated for gases, liquids and solids. We will also study some basic features of equilibrium between different phases. 

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