Energy thermodynamic internal

G Mass velocity, mass per unit area U Thermodynamic internal energy... 

In these equations pt is the mass density (g. cm.-3) of the fth chemical species, fc is the rate of production of the fth chemical species by chemical reaction (g. cm.-3 sec.-1), and Fi is the external body force per unit mass acting on the ith species. The velocity v is the local mass average velocity (that velocity measured by a Pitot tube), p is the over-all density of the fluid, and U is the local thermodynamic internal energy (per unit mass) of the mixture. The j, are the fluxes of the various chemical species in g. cm.-2 sec.-1 with respect to the local mass average velocity, v. It should be noted that 2j, = 0, 2/c,- = 0, and = p these relations are used in deriving the over-all equation of continuity [Eq. (4)] by adding up the individual equations of continuity given in Eq. (24). 

The thermal energy of a gas, which we have referred to before, can be equated with the thermodynamic internal energy, that is, E — Eq. First, let us calculate the expectation value... 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

It is important to mention briefly the issues of notation and units. A glossary of frequently used symbols is given in Appendix A. Note that the definition of thermodynamic work used here is the work done on a system (e.g., dw = -p dV). As a result, the first law of thermodynamics has the form dE= dq + dw. Note also that E is used for the thermodynamic internal energy and U for the potential energy instead of the lUPAC choices of... 

The First Law of Thermodynamics Internal Energy, Work, and Heat... 

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. 

Consider an ideal gas composed of diatomic molecules AB. In the limit of absolute zero temperature, all the AB molecules are in their ground states of electronic and nuclear motion, so DqN/ (where is the Avogadro constant and Dq is for the ground electronic state of AB) is the change in the thermodynamic interned energy U and enthalpy H for dissociation of 1 mole of ideal-gas diatomic molecules N/>,Dq = AUl = A//°o for AB(g) A(g) + B(g). 

The energy of the electron-transfer reaction, if sufficient, leads to the production of excited states. Since the excitation energy is primarily closest to a thermodynamic internal energy, the excited singlet state of energy ES (in electronvolts) can be produced as shown in Eq. 1.2... 

It is a measure of randomness of the molecules in the system, and is central to the quantitative description of the second law of thermodynamics. Internal energy, U, is the sum of the kinetic energy, potential energy, and vibrational energy of all the molecules in the system. 

Note 3.4 (On the constitutive relation and the Second Law of Thermodynamics). The Second Law of Thermodynamics essentially gives a relationship between the heat flux q that is externally supplied and the induced temperature field. Some scientists have stated that constitutive relations that depend on fields other than the temperature can also be derived by the Second Law however, as shown above, the Second Law does not consider fields other than the heat flux and temperature. All the constitutive relations can be derived from the First Law of Thermodynamics, internal energy and thermodynamic potentials induced by Legendre transformations (see Sect. 3.4). ... 

This result is not appropriate when the particles interact with a molecular field because the thermodynamic internal energy (not to be confused with the intramolecular energy) caused by molecular interactions is counted twice.This situation obtains because particles both generate and experience the molecular field. To correct this expression for the free energy we need to evaluate the contribution to the internal energy caused by the anisotropic interactions. The starting point is eqn (16) for the potential of mean torque for the nth conformer which can be rewritten in terms of the segmental interactions as... 

The calculated thermodynamic internal energy for the system obtained from Monte Carlo computer simulation based on the MCY-CI potential was -8.58 0.06 kcal/mol, compared with an observed value of -9.9 kcal/mol. The discrepancy of 13% is ascribed mainly to the assumption of pairwise additivity in the configurational potential. The calculated heat capacity, corrected for internal modes, is 17.9 cal/mol deg as compared with the experimental value at 25° of 18 cal/mol deg. 

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