Helmholtz second relation

Note 1.2.- Helmholtz s second relation gives the derivative of the ratio //,/T with the temperature ... 

However, in view of Helmholtz s second relation, we have ... 

This method is based on Helmholtz s second relation. Let us, respectively, write H and H"" for the partial molar enthalpies of the component i in the reference solution and in the solution at hand, for which the molar fraction of the component i is x. We have ... 

By expanding the Helmholtz free energy F at constant T in an arithmetic series in terms of ujk, we see that the linear terms vanish in view of the equilibrium condition (Euler relation for homogeneous functions of second order, F is given as... 

From E q. 11.1 Ob, it is seen that H is a characteristic function with natural variables S and P. Additional relations can be derived from the first and second laws when other experimental conditions are more easily controlled. For example, for a system of constant composition, the Helmholtz free energy, A (sometimes denoted as F), has natural variables T and V, and the Gibbs free energy, G, has natural variables T and P, as shown by E qs. 11.11b and 11.12b below ... 

The thermodynamic potentials, being a system s state functions of the corresponding (natural) parameters, arc of special importance in the system state description, their partial derivatives being the parameters of the system as well. The equalities between th( second mixed derivatives are a property of the state functions and lead to relation-ship.s between the system parameters (the Gibbs-Helmholtz equations). Hence, once any thermodynamic potential (usually, the Gibbs or the Helmholtz one) has been evaluated, by means of either simulation or experiment, this means the complete characterization of the thermodynamic properties of the system. 

We see that in order to express a in terms of g R), we must know the explicit dependence of g R) on the density. Thus, even if we have used the pressure equation in the integrand of (3.47), we need a second integration, over the density, to get the Helmholtz free energy per particle. The chemical potential follows from the relation... 

While the heat capacities at constant pressme are useful for calciflation of various thermodynamic quantities, related to Gibbs free energy for aqueous species, and chemical equilibria involving aqueous species, the isochoric heat capacity measurements are useftil in developing equations of state (EOS) because they yield valuable information about the second derivative of the pressure and of the Helmholtz energy with respect to temperature. Detailed comparisons of experimental Cv data with available EOS are needed to establish their accuracy. Cv experiments contain direct information on the curvature behavior of the p-T isochores, which are extremely important in the development of a reliable EOS. Furthermore, the temperature dependence of Cv at fixed density serves as a sensitive indicator of the phase transition boundary. Thus C measurements provide a tool for investigation of temperatures T and densities d on the coexistence ciuve, especially near the critical point. 

We can express the Helmholtz energy by relation [A2.40], The gas molecules are considered as indiscernible molecules, so the canonical partition function is given by relation [A2.36], Using Stirling s second approximation [A2.1], we see relation [A2.53] become ... 

The Clausis-Duhem inequality (22) has established a relation between the stress and the strain through the Helmholtz free energy y/. Considering a viscoelastic material and constant and uniform temperature, it can be assumed that the Helmholtz free energy will only be function of the total strain and an undetermined number of second order tensors internal state variables [q ] (a = l,.,n). So, Eq. (22)becomes ... 

With the help of the pair correlation function we can relate the first and the second binding Helmholtz energies, i.e.,... 

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