Absolute internal energy

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. 

Let denote the absolute internal energy of species K per unit mass of species K and let u denote the absolute internal energy per unit mass of mixture. Then... 

In order to show that the model of independent, coexistent continua represents correctly a real mixture of gases composed of different chemical species, we must compare the results obtained from this model with those of the kinetic theory of nonuniform gas mixtures (see Appendix D). Quantities such as the density p, the mass-weighted average velocity v j, and the body force fj have obviously analogous meanings in both the kinetic theory and the coexistent-continua model. On the other hand, the precise kinetic-theory meaning of terms such as the stress tensor, the absolute internal energy per unit mass and the heat-flux vector qf is not immediately apparent. In view of the known success of continuum theory for one-com-... 

The symbol U-(, v, t) will denote the total absolute internal energy of a molecule of type i traveling with velocity v and is given by... 

Here u is the intensive internal energy, is the number of molecules, Ujg is the molecular kinetic energy, and Up is the molecular potential energy the angle brackets indicate an average over all molecules. Note that in molecular theory we can write an equation that represents the absolute internal energy, but in thermodynamics we cannot. 

We cannot measure the absolute internal energy U or enthalpy H because the zero of energy is arbitrary. As a result, we are usually only interested in determining changes in these properties (At/ and A.H) during a process. However, it is possible to determine the absolute entropy of a substance. This is because of the third law of thermodynamics, which states that the entropy of a pure substance in its thermodynamically most stable form is zero at the absolute zero of temperature, independent of pressure. For the vast majority of substances, the thermodynamically most stable form at 0 K is a perfect crystal. An important exception is helium, which remains liquid, due to its large quantum zero-point motion, at 0 K for pressures below about 10 bar. 

The chemical potential, plays a vital role in both phase and chemical reaction equiUbria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, p approaches negative infinity when either P or x approaches 2ero. While these characteristics do not preclude the use of chemical potentials, the appHcation of equiUbrium criteria is faciUtated by the introduction of a new quantity to take the place of p but which does not exhibit its less desirable characteristics. 

Cp = specific heat e = specific internal energy h = enthalpy k =therm conductivity p = pressure, s = specific entropy t = temperature T = absolute temperature u = specific internal energy [L = viscosity V = specific volume f = subscript denoting saturated hquid g = subscript denoting saturated vapor... 

The work done by an expanding fluid is defined as the difference in internal energy between the fluid s initial and final states. Most thermodynamic tables and graphs do not presentbut only h, p, v, T (the absolute temperature), and s (the specific entropy). Therefore, u must be calculated with the following equation ... 

NOTE The commonly assumed zero enthalpy (internal energy) for air is approximately 14.7 pounds per square inch pressure absolute (psia), equivalent to 1.01 bar at 80 °F (27 °C). However, boiler pressure is commonly... 

In this expression U is the internal energy, T is the absolute temperature and S is the entropy. 

Thermodynamics deals with processes and reactions and is rarely concerned with the absolute values of the internal energy or enthalpy of a system, for example, only with the changes in these quantities. Hence the energy changes must be well defined. It is often convenient to choose a reference state as an arbitrary zero. Often the reference state of a condensed element/compound is chosen to be at a pressure of 1 bar and in the most stable polymorph of that element/compound at the... 

Absolutely everything possesses energy. We cannot see this energy directly, nor do we experience it except under certain conditions. It appears to be invisible because it is effectively locked within a species. We call the energy possessed by the object the internal energy , and give it the symbol U. 

A i //" j. In these cases, it is sometimes appropriate to make some sort of assumption regarding A i //",. For instance, when D and E in reaction 3.10 are radicals whose recombination is diffusion-controlled, it is expected that A 0. This hypothesis has, however, some subtleties that are important to mention. What is really assumed in these cases is that the internal energy of activation is zero at the absolute zero, that is, Atf/°j(0) = 0 [60]. Starting with equation 3.20, we can write the following series of equalities, where A= 1 — m, m being the molecularity of the forward reaction (rn = 1 and A= 0 in the case of reaction 3.10) ... 

The first factor is responsible for normal isotope effects, which arise because the bonds being affected by deuteriation are weakened in the transition state, but the absolute effect is greater on the bonds to deuterium rather than protium because the former have higher vibrational frequencies (typically by a factor of ca 1.37). This factor essentially reflects zero-point energy effects, so it becomes progressively more important at lower internal energies. 

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