Thermodynamic integration routes

This is precisely expression (3.24) we obtained from the thermodynamic integration route using the approximation (3.22). Using the weU-known result that for an arbitrary probability distribution the following inequality holds... 

Thermodynamic integration provides an alternative approach to calculation of free energies, which is always useful for verifying the results of other methods. In some cases, it provides the only workable route to calculation of free energies, even if artificial integration paths must be devised. ... 

Jefferson Tester We find ourselves at MIT with the exact same dilemma in dealing with a physical chemistry textbook like Castellan, and an undergraduate favorite, Smith and Van Ness. The fourth edition of Smith and Van Ness does include a very brief treatment of statistical mechanics. Although it s a start, a more effective route we have found is to integrate statistical and molecular concepts from a physical chemistry text into our introductory chemical engineering thermodynamics course. In this situation, we have a two-semester sequence that uses both Castellan and Smith and Van Ness as textbooks. 

Studies of the thermodynamic properties obtained from solutions of the SSOZ equation have been much more extensive but the success has been mixed. The first such calculations were those of Lowden and Chandler who obtained the pressure of hard diatomic fluids from the RISM (SSOZ-PY) equation. They used two routes to the equation of state a compressibility equation of state in which they integrated the bulk modulus calculated from the site-site correlation functions via... 

The most accurate route to the thermodynamic properties from the SSOZ equation seems to be the energy equation." The integral from which the internal energy is obtained (see Eq. (2.3.1)) seems to be relatively insensitive to errors in the predicted site-site correlation functions. It might on this basis be reasonably assumed that calculations of the Helmholtz free energy via integration of the Gibbs Helmholtz equation... 

There are many different routes for calculating phase equilibria that are covered in detail in other chapters of this volume thermodynamic scaling Monte Carlo (chapter by Valleau), Gibbs-Duhem integration along coexistence lines (chapter by Kofke), and pseudo-ensemble methods (chapter by de Pablo and Escobedo). Thus these methods are not discussed here. 

While linking structure and thermodynamics based on the virial expression is not straightforward, this link can in fact be established using an alternative desciiptiOTi based on Kirkwood-Buff (KB) theory [76], Whereas the virial route requires information on the effective potential, the KB description does not make any assumption on the nature of the potentials, is exact, and its central quantities can be interpreted in terms of local solution structure. To this end, we consider the derivatives of the salt activity with respect to the density at constant pressure p and temperature T. For the systems shown in Fig. 5 these derivatives show the same order as the osmotic coefficients/salt activities for the different ions [70]. Hence, the microscopic mechanism explaining the order among the derivatives of the salt activity for the different ions also explains the Hofmeister series for the activities obtained by integration of the derivatives. Based on this, the relation between... 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

In Chapter 9 w e will use the Euler relationship to establish the Maxwell relations between the thermodynamic quantities. Here we derive the Euler relationship. Figure 5.12 shows four points at the vertices of a rectangle in the xy plane. Using a Taylor series expansion, Equation (4.22), compute the change in a function Af through tw o different routes. First integrate from point A to point B to point C. Then integrate from point A to point D to point C. Compare the results to find the Euler reciprocal relationship. For Af = f(x + Ax,y + Ay) -f(x,y), the hrst terms of the Taylor series are... 

Having determined the structure of the polymer liquid, it is in principle possible to compute most thermodynamic properties of interest. Whereas the structure or radial distribution functions at liquid density are primarily controlled by the repulsive part of the intersite potentials, thermodynamic quantities will also be sensitive to the attractive potentials. In the case of a one-component melt, thermodynamic quantities of interest include the pressure P, isothermal compressibility k, and the internal or cohesive energy U. Since in general one theoretically knows g(r) only approximately, the thermodynamic properties derived from structure will be approximate. Moreover, integral equation theory leads to thermodynamically inconsistent results in the sense that the predictions depend on the particular thermodynamic route used to relate the thermodynamic quantity to the structure. ... 

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