Internal energy and heat

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Table 8. Comparison of internal energies and heat capacities for BNS and R potentials from Monte-Carlo simulation 72>...

Even if simple expressions for the directional field are not available, it is possible to draw certain conclusions about the other thermodynamic functions from (4.2). The orientational free energy Fi<>0 leads to the following additional contributions to the entropy, internal energy and heat capacity at constant volume... 

The determination of other parameters for the activation process at constant volume requires a knowledge of the rates at various tempera-tm%s, with the volume of the reaction mixture kept constant. This information is obtained by interpolation of the results at a number of different pressures on the assumption that the coefficients of expansion and compressibility of the reacting system are the same as in the pure solvent. Any resulting error is probably negligibly small when dilute solutions are studied. The internal energy and heat capacity of activation at constant volume are then given by... 

Properties of a system may either be independent of the amount of material present (intensive, like T and F) or proportional to the quantity of material present (extensive, like internal energy and heat capacity). 

We use the term thermal energy to designate energy in the form of internal energy and heat, and mechanical energy to designate mechanical and electrical work and... 

Thus the internal energy and heat capacity are simply related to the change in the partition function with temperature. For certain simple systems such as gases at low temperatures, the partition function can be estimated theoretically. For most systems of geological interest such as minerals and concentrated salt solutions, additional experimental information is required. This might take the form of spectroscopic data on electronic or molecular vibrational frequencies, or direct measurement of some of the non-ideal thermodynamic properties themselves. 

INTERNAL ENERGY AND HEAT CAPACITY OE MONATOMIC IDEAL GASES... 

Information from Section 28-5 can be used to calculate the internal energy and heat capacity of a monatomic gas because the complete temperature dependence of Z is accounted for by partition functions for translational motion and the electronic ground state. Molecules exhibit 3 degrees of freedom per atom. Hence, there are no internal degrees of freedom for a monatomic gas (i.e.. He, Ne, Ar, Kr, Xe) because all 3 degrees of freedom are consumed by translational motion in three different coordinate directions. The internal energy is calculated from equation (28-59) ... 

Contribution of Rotational Motion to the Internal Energy and Heat Capacity. 

Contribution of Vibrational Motion to the Internal Energy and Heat Capacity. As illustrated by equation (28-62), vibrational motion of diatomic and polyatomic nonlinear ideal gases contributes to the internal energy as follows ... 

In addition, tabulated quantities for the internal energies and heat capacities of the various species in the gas and liquid phases are necessary to complete the computations. Further, special attention is required to accommodate moving boundaries that account for piston and valve movements in engine applications. 

You should know what is meant by internal energy and heat. 

It should be emphasized that the above development is a lowest-order development in that only the first corrections to the local equilibrium values of the internal energy and heat capacity have been used, This is consistent with current practice in fluid dynamics and nonequilibiium thermodynamics. Further refinements can be developed if it is desired to make contact with some of the recent work in extended irreversible thermodynamics [36, 37]. 

We will now use the statistical definitions of the previous section to derive the expression for internal energy and heat capacity in the gas phase. We will begin with gases, since they form the simplest case. However, we will leave out the contribution from rotational levels since this contributim is of little interest in the remaining parts of the book. 

Within the scope of the present section we shall apply (13 54) to the internal energy and heat capacity only. The free energy and entropy on the Debye model may be worked out by similar methods. From (13 35) and (13 38) we have... 

All of the quantities in equation (2-30) (pressure, density, etc.) except internal energy and heat can be directly measured. If the latter items could be handled, equation (2-30) would then be extremely useful for many engineering applications. 

The method that is used is to consider an incompressible fluid (good approximation for most liquids and also for gases under certain conditions), and we can equate the internal energy and heat combination to a friction heating term ... 

It is noteworthy that a statistical mechanical calculation of absolute values of entropy or free energy is not required for determination of thermodynamic properties of matter. The functional dependence of the partition function on macroscopic properties, such as the total mass, volume, and temperature of the system, is sufficient to derive equations of state, internal energies, and heat capacities. For example, knowledge that the ideal gas partition function scales as is adequate to define and explain the ideal gas equation of state. 

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