Thermodynamic cycle integration

Considerable use has been made of the thermodynamic perturbation and thermodynamic integration methods in biochemical modelling, calculating the relative Gibbs energies of binding of inhibitors of biological macromolecules (e.g. proteins) with the aid of suitable thermodynamic cycles. Some applications to materials are described by Alfe et al. [11]. 

In the present chapter we have reviewed a numerically efficient and accurate equation of library state for high pressure fluids and solids. Thermodynamic cycle theories allow us to apply this model profitably to the reactions of energetic materials. The equation of state is based on HMSA integral equation theory, with a correction based on extensive Monte Carlo simulations. We have also shown that our equation of state can be used to accurately model the properties of molecular fluids and detonation products. The accuracy of the equation of state of polar fluids is significantly enhanced by using a multi-species or cluster representation of the fluid. 

When thermodynamic integration simulations and the thermodynamic cycle approach are used to evaluate free energy differences, the contribution of the kinetic energy usually cancels and therefore does not need to be calculated. Since Monte Carlo simulations generate ensembles of configurations stochastically, momenta are not available, and the contribution cannot be evaluated. 

Thermodynamic cycles have been generalized. The fundamental equation can be integrated and can be used to derive an expression characterizing the efficiency of a system for any kind of cyclic process by which work is produced [1]. 

This thermodynamic integration equation was first derived by Kirkwood it can also be obtained directly by integrating aF(X)/dX with respect to X from X = 0 to 1 where F is expressed in terms of Z (Eq. [3]). Equation [23] is similar to Eq. [18], where the specific heat, which is the derivative of the energy with respect to the temperature, is integrated. However, the latter derivative is generally smoother than that of Eq. [23] when creation and annihilation of particles is involved (see the section Thermodynamic Cycles ). 

From the magnetic entropy change it is easy to calculate the cooling capacity Q of the magnetic material, that is, the amount of heat that can be extracted from the cold end to the hot end of a refrigerator in one ideal thermodynamic cycle by integrating the following differential equation ... 

The general formalism for thermodynamic simulation methods follows from early work by Zwanzig [33], and provides a tool for the computation of thermodynamic properties, AA, AE and AS, as well as barriers for chemical processes occurring on long timescales [34]. These methods take on several guises in present implementations. The two approaches which we will describe are termed thermodynamic cycle perturbation theory (TP) [35] and thermodynamic integration (TI) [36]. Both of the methods are based on the definition of a hybrid Hamiltonian which represents some mixture of the initial state (1) and final state (2) of the system [37]. If /.represents the coordinate describing the pathway used to interconvert the two systems, then the hybrid Hamiltonian may be defined by [35, 37]. 

Example 8.9 shows two main features of thermodynamic cycles. First, because the internal energy is a state function, it sums to zero around the cycle You can check that AUi, -i- AUc + AUde + AU/ = Cv[(T2-T]) + (Ti T2) + (T4-Ti) + Ti - T4)] = 0. Second, w ork and heat are not state functions and do not sum to zero around the cycle. The engine performs work on each cycle. This is evident either from computing the sum Wh -1- Wc + Wde + Wf, or from the graph of the pressure versus volume in Figure 8.13. Because w ork is the integral of -pdV, the total work is the area inside the cycle. 

Therefore, the pK of group AH in water can be obtained from the computed value of AG , when the remaining free energy terms shown in the thermodynamic cycle are known. The AGg values can be obtained from ab initio methods or experiments, and the solvation free energy values can be calculated by Monte Carlo simulation methods combined with a reaction field approach, integral equation techniques, or continuum dielectric methods." However, because the proton solvation free energy, AG , (H+), is a quantity about which there exists considerable experimental uncertainty and for which there are no reliable calculations, frequently only the difference in the pfC s of two groups, AH and BH can be obtained" ... 

To summarize, the Carnot cycle or the Caratheodory principle leads to an integrating denominator that converts the inexact differential 8qrev into an exact differential. This integrating denominator can assume an infinite number of forms, one of which is the thermodynamic (Kelvin) temperature T that is equal to the ideal gas (absolute) temperature. The result is... 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

Many opportunities conversely are supported by reversible reactions of QM despite the noted complications. One example includes the synthesis and chiral resolution of binaphthol derivatives by two cycles of QM formation and alkylation.77 The reversibility of QM reaction may also be integrated in future design of self-assembling systems to provide covalent strength to the ultimate thermodynamic product. To date, QMs have already demonstrated great success in supporting the opposite process, spontaneous disassembly of dendrimers (Chapter 5). 

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