Internal energy functions

The stress relation obtained from an expansion of the internal energy function to fourth order in the finite strain t] takes the following form [79D01] ... 

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

The thermal internal energy function calculated at 298.15 K [E — 0] is also listed in Table 8.1. The translational and rotational contributions are found using Eqs. 8.80 and 8.82, respectively. The vibrational contributions (Eq. 8.84) are much less, as expected. Mode 2 makes a significant contribution to the total internal energy at this temperature. Vibrational modes 5 and 6 also make smaller, but nonnegligible, contributions. The electronic contribution was calculated directly from Eq. 8.76. Through application of Eq. 8.118, the total enthalpy is [H — Ho] - 11146.71 J/mole. 

Far more useful are statements that establish the mathematical nature of the internal energy function. Any of the following statements can be considered fully equivalent to the first law, once U is defined in terms of heat and work ... 

We can also derive certain provocative consequences of the first law that make no direct reference to the internal energy function. Consider, for example, a cyclic process that involves passing from A to B by path 1 (with heat qx and work w ), then returning from B to A by alternative path 2 (with heat q2 and work w2) ... 

More generally, we can recognize that the natural arguments of the internal energy function U are a fundamental set of extensive properties Xlf X2,..., Xi9... (whose number remains to be established),... 

Equation (6.27) merely says that if the independent extensive arguments of U are multiplied by A [cf. (6.25b-d)], then U itself must be multiplied by the same factor [cf. (6.25a)]. [Mathematically, the property (6.27) identifies the internal energy function (6.26) as a homogeneous function of first order, and the consequence to be derived is merely a special case of what is called Euler s theorem for homogeneous functions in your college algebra textbook.]... 

In order to better understand the physical nature of the chemical potential jxt of a chemical substance, let us first review the major mathematical features of the Gibbsian thermodynamics formalism. The starting point is the Gibbs fundamental equation for the internal energy function... 

For chemical purposes, the internal energy (8.71) must include chemical work terms, one for each of the c independent chemical components participating in active equilibria. Together with the usual extensities XL for heat (S) and pres sure-volume work (V), the arguments of the internal energy function must be extended to include c additional chemical extensities (such as the mole numbers n2,. .., nc, with conjugate chemical potentials ... 

U(x) is the internal energy function of K. (To make sure that U is in fact defined for ail states, one assumes that some adiabatic transition always exists between any pair of given states,) The energy of a compound standard system is the sum of the energies of its constituent standard systems. Further, U iuusi be a muiiulonie function of t, and it is convenient to choose the scale of t such that dU/dt > 0. 

O Donoghue, S. I., and Nilges, M. (1997). Tertiary structure prediction using mean-force potentials and internal energy functions Successful prediction for coiled-coil geometries. Fold. Des. 2, 47-52. 

At constant T and P (easy to experimentally control), the condition for equilibrium is AG = 0. The Gibbs energy function (Eq. 2.11) is a generally more useful function than the internal energy function (Eq. 2.6) that requires constant S and V at equilibrium (AU = 0). 

Each site is assigned the same energy (-zJqhQ2) (where z is the number of nearest neighbors) so the internal energy functional is... 

The First Law of Thermodynamics guarantees the existence of a function of state E, termed the internal energy (function) which may be correlated with any state of any system for any process whatsoever, between a given initial state (1) and final state (2), the difference AE E2 - Ei depends solely on the coordinates x and x2 characterizing these states and is independent of the path connecting them. 

Prove that for the internal energy function the following Gibbs-Duhem relation must hold Ed(l/T) + Vd(P/T)... 

In this Section the internal energy function has been introduced in the form E - E(T,V), whereas in Section 1.18 it has been formulated as E - E(S,V). Considering that thermodynamic functions of state should be useful in deriving various intensive and extensive variables, are the two formulations equivalent If not, which one is more fundamental In a similar vein discuss the relation between H... 

This expression shows that volume and entropy serve as thermodynamic variables, or as control variables, for the internal energy function of the system E = E(S, V). Nevertheless, the intensive variables are those of the surroundings, and are therefore well defined, even when the processes in the system proper are far removed from equilibrium. For the present we exclude other types of work that are treated in Chapter 5. We defer the generalization of the present treatment to the case of open systems to Section 1.20. 

Lang, G., Heusler, K.E. Can the internal energy function of sohd interfaces be of a nonho-mogeneous nature J. Electroanal. Chem. 472(2), 168-173 (1999)... 

A further phenomenological theory, which uses the concept of strain-energy functions, deals with more general kinds of stress than uniaxial stress. When a rubber is strained work is done on it. The strain-energy function, U, is defined as the work done on unit volume of material. It is unfortunate that the symbol U is conventionally used for the strain-energy function and it will be important in a later section to distinguish it from the thermodynamic internal-energy function, for which the same symbol is also conventionally used, but which is not the same quantity. 

The internal energy function U, however, is peculiar to thermodynamics. It represents the kinetic and potential energies of the molecules, atoms, and subatomic particles that make up the system on a microscopic scale. There is no known way to determine absolute values of U. Fortunately, only changes of AU are needed and these can be derived experimentally. When the state of the system is fixed, the internal energy U is fixed. If Eq. 5.8 is used in Eq. 5.7, the first law of thermodynamics can be written as ... 

In canonical thermostatics a heterogeneous system is characterized by a set of intensive and extensive quantities. However, the state can be unambiguously defined even by a proper subset of the corresponding quantities. The characteristic functions Ux of a system can be derived from its internal energy function (7 by a Legendre transformation to satisfy the expression... 

Theorem 1.4.1. If the rate functions/i,. ..,/u and the internal energy function U satisfy the conditions of Postulate 1.2.1 then each invariant manifold Amq) contains at least one equilibrium point. 

Let us consider an ideal gas mixture of N species, subject to R independent reaction with ratesThe internal energy function is... 

The mathematical assumptions behind the theory of fading memory have been recently reviewed by Drapaca et al. [150]. Definition (88) leads to the so-called Strong Principle of Fading Memory [108] (SPFM) 1, which defines the class of admissible internal energy functionals ... 

The universality of the aforementioned internal energy function F[n] allows us to define the ground-state wave function go to an N-particle system that delivers the minimum of F r and reproduces Uq. Kohn and Sham (KS) [101] made this more practical by nsing noninteractive electrons that are known as Kohn-Sham electrons. They pnt forward a mathematical assumption [99,101-103] ... 

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