Fundamental thermodynamic relations

From fundamental thermodynamic relations, the temperature and pressure dependence of Henry s constant can be shown (18,50,51) to be ... 

We start from the fundamental thermodynamic relation for an irreversible process (de Groot and Mazur, 1962),... 

This relation plays an important role in the derivation of the universal and fundamental thermodynamic relation... 

The fundamental thermodynamic relation is a function N > R (e, n) i->s(e, n), where s denotes entropy per unit volume. The function s(e,n) as well as all other functions introduced below are assumed to be sufficiently regular so that the operations made with them are well defined. We can see the fundamental thermodynamic relation s = s(e,n) geometrically (as Gibbs did, see Gibbs, 1984) as a two-dimensional manifold, called a Gibbs manifold, imbedded in the three-dimensional space with coordinates (e,n,s) by the mapping (e,n) —>(e, n,s(e,ri)). 

In view of the importance of Legendre transformations in equilibrium thermodynamics, we shall make, following Hermann (Hermann, 1984), an alternative formulation of the fundamental thermodynamic relation. We introduce a five-dimensional space (we shall use hereafter the symbol N to denote it) with coordinates (e,n,e ji, s) and present the fundamental thermodynamic relation as a two-dimensional manifold imbedded in the five-dimensional space N by the mapping... 

Question Where does the fundamental thermodynamic relation s = s(y) come from Given a physical system, what is the fundamental thermodynamic relation representing it in equilibrium thermodynamics ... 

Answer 1 The only way to find the fundamental thermodynamic relation s = s(y) inside the classical equilibrium thermodynamics is by making experimental measurements. Results of the measurements are usually presented in Thermodynamic Tables. 

The mesoscopic fundamental thermodynamic relation is now a function M —> R xi->h(x). In order to avoid possible confusion, we use in mesoscopic formulations the symbol h instead of s and call h(x) an eta-function instead of entropy function. In analogy with the geometrical representation of the fundamental thermodynamic relation introduced in Section 2.1, we present the mesoscopic fundamental thermodynamic relation as a manifold (denoted now by the symbol M), imbedded in the space (denoted now by the symbol M) with coordinates (x,x ji) as the image of the mapping... 

Now we want to pass from the fundamental thermodynamic relation h = h x) in M to the fundamental thermodynamic relation s = s(y) in N. First, we have to know how y is expressed in terms of x. Formally, we introduce... 

Let. .MA" V. o be the intersection of. t(A with the plane x 0. We note that its restriction to the plane x is a manifold of the states that we denote xeth(y ) and call equilibrium states. These are the states for which cp reaches its extremum if considered as a function of x. We shall denote the manifold of equilibrium states by the symbol Meth- Restriction of Mm x, 0 to the plane y is the manifold representing y expressed in terms of x (i.e. the function (4)), and its restriction to the plane (i/. yj is the Gibbs-Legendre manifold Af representing the dual form s s (y ) of the fundamental thermodynamic relation s = s(y) in N that is implied by the fundamental thermodynamic relation h = h(x) in M. This completes our presentation of the passage h(x) —> s(y). 

Answer 2 given above invites, of course, another question Where do the fundamental thermodynamic relation h = h x) and the relation y = y x) come from An attempt to answer this question makes us to climb more and more microscopic levels. The higher we stay on the ladder the more detailed physics enters our discussion of h = h(x) and y = y(x). Moreover, we also note that the higher we are on the ladder, the more of the physics enters into y = y(x) and less into h = h x). Indeed, on the most macroscopic level, i.e., on the level of classical equilibrium thermodynamics sketched in Section 2.1, we have s = s(y) and y y. All the physics enters the fundamental thermodynamic relation s s(t/), and the relation y = y is, of course, completely universal. On the other hand, on the most microscopic level on which states are characterized by positions and velocities of all ( 1023) microscopic particles (see more in Section 2.2.3) the fundamental thermodynamic relation h = h(x) is completely universal (it is the Gibbs entropy expressed in terms of the distribution function of all the particles) and all physics (i.e., all the interactions among particles) enters the relation y = y(x). 

In order to make the fundamental thermodynamic relation (23) explicit we have to solve Equation (22). We now proceed to do it. We begin by looking for a solution in the form... 

We note that if we insert this solution into (23) we obtain the well known van der Waals fundamental thermodynamic relation s(e, n) = const, n + Rn ln[( + an)3 2 — b)] provided wshort( 0 and wiong( >) are chosen as follows ... 

The system under consideration is at equilibrium. Let us suddenly switch on new interactions among the components. The components start to react chemically. They start to transform one into another. This new type of interactions brings the system initially out of equilibrium but eventually the system will reach a new equilibrium, called a chemical equilibrium. What is the fundamental thermodynamic relation of the system involving chemically reacting components at new chemical equilibrium ... 

Answer 3 The fundamental thermodynamic relation h = h(x) and the relation y = y x) in the space M emerge in an investigation of the time evolution of x representing the approach to equilibrium. 

We recall that both the manifold Meth and the fundamental thermodynamic relations y> are displayed on the manifold MN x,=0 (see the text following (6)). 

Everything that we have done so far in this example is completely standard. The next step in which we identify the slow fundamental thermodynamic relation in the state space M2 (i.e., we illustrate the point (IV) (see (120))) is new. Having found the slow manifold Msiow in an analysis of the time evolution in Mi, we now find it from a thermodynamic potential. We look for the thermodynamic potential ip(q,p, e, //. q, e, v) so that the manifold Msiow arises as a solution to... 

M w and its projection (vf.y 1"" becomes smaller (as small as possible) than for the slow manifold M w. Step 4) The slow fundamental thermodynamic relations associated with the slow manifolds M w and M w are identified. 

The four fundamental thermodynamic relations may now be cast in the form... 

The continuum mechanical analogues of the fundamental thermodynamic relations are written as ... 

We now do for reaction-equilibrium problems what we have done in 10.1 for phase-equilibrium problems we show how fundamental thermodynamic relations are used to develop computational strategies. We start by discussing the number of independent properties required to identify states in reacting systems ( 10.3.1) then we... 

Tabitha the Untutored claims the fugacity is the fugacity, no matter what, so it is legitimate to use FFF 1 for one component in a binary liquid mixture and use FFF 2 for the other component. Do you agree If so, justify. If not, what fundamental thermodynamic relations would be violated ... 

D. D. Perrin [537] was among the first researchers to examine the temperature dependence of pK in a systematic way, noting that for many dissociations of the form BH+ B -F H+ a temperature rise of 10°C can lead to a decrease of 0.1 to 0.3 pK units in the observed pK. From the fundamental thermodynamic relation -3AG/3T = AS, one can obtain the expression... 

If we assume that not more than one solute molecule can occupy a site at one time, the solute molecules will obey statistics of the Fermi-Dirac type an ensemble of solute particles obeying this principle is termed a spatial Fermi gas [54], The fundamental thermodynamic relations, such as the equilibrium distribution function, the free energy, the pressure dependence of the concentration of solute in contact with the ideal-gas phase, and the equation of state have been established in closed form for a spatial Fermi gas of identical as well as different particles [54]. Some of the results shall be review here. 

According to the fundamental thermodynamic relation [18], the change in the specific entropy of an ideal gas (gas species 1) in terms of the specific internal energy, the specific volume, the partial pressure and the equilibrium temperature is given by... 

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