Calculating thermodynamic properties

We have now discussed the general first-principles methodology that will allow us to model gas-phase (isolated molecule) and solid-state systems. The remainder of the information presented in this chapter can be applied equally well to both types of simulation. 

Our basic optimized structure is the equilibrium structure, in which the atoms are (hypothetically) at rest and the structure therefore free from the effects of vibration. This is important, as this structure is impossible to obtain experimentally. However, if we wish to use our simulations to match experimental observations closely we must apply a series of corrections. 

The first correction we can apply is the harmonic zero-point energy (ZPE) correction, which is the energy difference between the vibrational ground state and the equilibrium structure. This is given by 

Therefore, if we know the harmonic vibrational frequencies for our model, we simply add up the 2 0) contribution from each vibrational mode, i, and multiply by Avogadro s number to report energies in units of kJ moP Thus we can see that those modes that correspond to a higher wavenumber (the bond stretches, particularly those that involve hydrogen, as a consequence of the reduced mass term given in the equation in Section 8.2.1) make the greatest contribution to the ZPE correction. 

Formally, the stability of a structure, with respect to other possible structures, is determined by the Gibbs free energy, G. This is written as 

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Calculating Thermodynamic Properties Using a Force Field... 

The shift makes the potential deviate from the true potential, and so any calculated thermodynamic properties will be changed. The true values can be retrieved but it is difficult to do so, and the shifted potential is thus rarely used in real simulations. Moreover, while it is relatively straightforward to implement for a homogeneous system under the influence of a simple potential such as the Lennard-jones potential, it is not easy for inhomogeneous systems containing rnany different types of atom. 

Unfortunately, this is not a feasible approach for calculating thermodynamic properties di to the large number of configurations that have extremely small (effectively zero) Boltzmar... 

One may wonder why it is important to distinguish between and keep track of these two energies and Dq, when it seems that one would do. Actually, both are important. The bond energy Dg dominates theoretical comparisons and the dissociation energy Dq, which is the ground state of the real molecule, is used in practical applications like calculating thermodynamic properties and reaction kinetics. 

The physical properties of the halogen fluorides are given in Table 1. Calculated thermodynamic properties can be found in Reference 24. 

This volume also contains four appendices. The appendices give the mathematical foundation for the thermodynamic derivations (Appendix 1), describe the ITS-90 temperature scale (Appendix 2), describe equations of state for gases (Appendix 3), and summarize the relationships and data needed for calculating thermodynamic properties from statistical mechanics (Appendix 4). We believe that they will prove useful to students and practicing scientists alike. 

Volumetric data are needed to calculate thermodynamic properties (enthalpy, entropy). They are also used for the metering of fluids, the sizing of vessels and in natural gas and oil reservoir calculations. 

Proceeding conceptually for a moment without these logistical difficulties, once we have determined the density of states we can calculate thermodynamic properties at any temperature of interest. The average potential energy is... 

TABLE III. Comparison of the Measured with the Calculated Thermodynamic Properties of Some Associating... 

In the previous discussion of laminar and turbulent flames, the effects of the physical and chemical parameters on flame speeds were considered and the trends were compared with the experimental measurements. It is of interest here to recall that it was not possible to calculate these flame speeds explicitly but, as stressed throughout this chapter, it is possible to calculate the detonation velocity accurately. Indeed, the accuracy of the theoretical calculations, as well as the ability to measure the detonation velocity precisely, has permitted some investigators to calculate thermodynamic properties (such as the bond strength of nitrogen and heat of sublimation of carbon) from experimental measurements of the detonation velocity. 

Self-consistent approaches in molecular modeling have to strike a balance of appropriate representation of the primary polymer chemistry, adequate treatment of molecular interactions, sufficient system size, and sufficient statistical sampling of structural configurations or elementary transport processes. They should account for nanoscale confinement and random network morphology and they should allow calculating thermodynamic properties and transport parameters. 

The constraint in Eq. (38) that enables the direct computation of Tg is obtained by the extension of the Lindemann criterion to the softening transformation in glass-forming liquids [42, 56, 129, 130], and the details of this relation are explained in Section VI. Within the schematic model for glass formation (with specified e, Es, and monomer structure), all calculated thermodynamic properties depend only on temperamre T, on pressure P, and on molar mass Mmoi (which is proportional to the number M of united atom groups in single chains). The present section summarizes the calculations for To, Tg, Ti, and Ta as functions of M for a constant pressure of P = 0.101325 MPa (1 atm). 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

Heats of formation assume resonance stabilizations 10.8 kcal mole-1 in ( CHaCN) 12.6 kcal mole-1 in (CH3CHCN) and in [(CH3)2CCN]. " Na = doubly bonded nitrogen in azo compounds. h This correction assumes that the barrier to rotation in the radical R is two-thirds the barrier in the corresponding alkane RH. See O Neal and Benson for further discussion of this point. AH° and to +2 cal mole-1 °K 1 for 5°. The following example shows how the table is used to calculate thermodynamic properties for the 2-butyl radical (12).41 H3C—ch—ch2—ch3 12 ... 

Table 4.7 Experimental (Lebedev et al. 2000) and calculated thermodynamic properties of in the crystalline state at T-298.15 K, J-K 1mol 1...

For the standard state of gaseous substances, we want to calculate thermodynamic properties in the 1.0-bar ideal gas state from measured values of properties at the measurement pressure, P. The calculation consists of the following three steps ... 

Six alternate methods for calculating thermodynamic properties are integrated into this program. 

Shimizu, K., and H. Murata Normal Vibrations and Calculated Thermodynamic Properties of Tetramethylsilane. J. molecular Spectroscopy 5, 44—51 (1960). 

In Section 2.1, we remarked that classical thermodynamics does not offer us a means of determining absolute values of thermodynamic state functions. Fortunately, first-principles (FP), or ab initio, methods based on the density-functional theory (DFT) provide a way of calculating thermodynamic properties at 0 K, where one can normally neglect zero-point vibrations. At finite temperatures, vibrational contributions must be added to the zero-kelvin DFT results. To understand how ab initio thermodynamics (not to be confused with the term computational thermochemistry used in Section 2.1) is possible, we first need to discuss the statistical mechanical interpretation of absolute internal energy, so that we can relate it to concepts from ab initio methods. 

The cubic equations of state have become the workhorse of the process industry, particularly in the case of natural gas. Most designs in the natural gas business are based on such equations. In chapter 2, the use of these equations for calculating thermodynamic properties was discussed. Here, their use for phase equilibrium calculations is presented. 

From equations (11)-(15) and from the definitions in the preceding section, all previously defined thermodynamic functions can be related to T, V, and Ni provided that Qi is a known function of T and V. The problem of calculating thermodynamic properties of gases therefore reduces to the problem of evaluating in terms of Tand K... 

The calculation of Af G° and Af H° of species from experimental data on apparent equilibrium constants and transformed enthalpies of reaction is described in R. A. Alberty, Thermodynamics of Biochemical Reactions, Wiley, Hoboken, NJ (2003) and a number of places in the literature. That is not discussed here because this package is oriented toward the derivation of mathematical functions to calculate thermodynamic properties at specified T, pH, and ionic strength. There are two types of biochemical reactants in the database ... 

Both MC and MD simulations have been used to calculate thermodynamic properties, most often the internal energy U, the virial pVYMgT, and the specific heat at constant volume Cy. Some of the rigid molecule models, e.g., the TIPS4 potential, were parameterized in part to give the correct molar volume at 300 K and zero pressure. As with the radial distribution functions, it is found that there is a reasonable variation between predicted values of these properties and that no one potential is clearly superior. 

A computer program for calculation of thermodynamic properties using the Peng-Robinson equation of state is available for a nominal fee (Carl L. Yaws, Box 10053, Lamar University, Beaumont, TX 77710, phone/FAX 409-880-8787). The computer program is executable and complete with data files. The program calculates thermodynamic properties at pressures and temperatures that are input by the user. Representative results are shown below ... 

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