Solid internal energy

Wsoi Stationary-phase and sorbent solid internal energy, J/kg... 

Here, we present in suminary form the equations that describe solid strain, solid stress, and solid internal energy of the porous medium. First, the internal energy statement ... 

And, the total solid internal energy associated with pressure and stresses is ... 

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. 

Thermodynamic paths are necessary to evaluate the enthalpy (or internal energy) of the fluid phase and the internal energy of the stationary phase. For gas-phase processes at low and modest pressures, the enthalpy departure function for pressure changes can be ignored and a reference state for each pure component chosen to be ideal gas at temperature and a reference state for the stationarv phase (adsorbent plus adsorbate) chosen to be adsorbate-free solid at. Thus, for the gas phase we have... 

For shock waves in solids, the shock pressure P is typically much greater than the initial pressure Pq, which is normally ambient atmospheric conditions, so that Pq is usually neglected. Eq can also be taken to be zero, sinee internal energy is a thermodynamie state funetion and ean be refereneed to any initial state. Removing Eq and Pq from the jump conditions results in their eommon form... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Clearly, Aid is equal to the heat transferred in a constant pressure process. Often, because biochemical reactions normally occur in liquids or solids rather than in gases, volume changes are small and enthalpy and internal energy are often essentially equal. 

Heat is one of the many forms of energy and mainly arises from chemical sources. The heat of a body is its thermal or internal energy, and a change in this energy may show as a change of temperature or a change between the solid, liquid and gaseous states. 

Solids and liquids also have internal energy. In the case of solids, translational motion is usually very limited and rotational motion is only present in special circumstances the common form of internal energy is usually vibrational. In liquids, all three forms of energy are usually present, although in some instances, some forms of motion may be restricted. 

Statistical thermodynamics provides the relationships that we need in order to bridge this gap between the macro and the micro. Our most important application will involve the calculation of the thermodynamic properties of the ideal gas, but we will also apply the techniques to solids. The procedure will involve calculating U — Uo, the internal energy above zero Kelvin, from the energy of the individual molecules. Enthalpy differences and heat capacities are then easily calculated from the internal energy. Boltzmann s equation... 

In contrast, the internal temperature of a material does not change as the material undergoes a change of state. (Thus, its internal energy does not change at that point). Therefore, for a chcuige of state between solid and liquid, we would have ... 

The vast majority of the reactions carried out in industrial scale batch reactors involve reactants in condensed phases. Since the specific volumes of both liquids and solids are very small, the difference between internal energy and enthalpy for these materials is usually negligible. Thus one often sees the statement that for batch reactions taking place at constant volume ... 

This shows that for solids and liquids the enthalpy depends upon both temperature and pressure. This is in contrast to the internal energy, which depends upon temperature only. Note that for solids and liquids cp = cv. 

In lattice statics simulations all vibrational effects are neglected2 and the internal energy of the solid U is simply equal to < >, and the entropy is zero. Such minimizations give the crystal structure and internal energy (often referred to as the lattice energy) of the low-temperature phase. In the static limit at 0 K and zero pressure3 the crystal structure is thus determined by the equation... 

Following from Equation (3.3), we say that internal energy is a state function. A more formal definition of state function is, A thermodynamic property (such as internal energy) that depends only on the present state of the system, and is independent of its previous history . In other words, a state function depends only on those variables that define the current state of the system, such as how much material is present, whether it is a solid, liquid or gas, etc. 

Figure 5.14 The dimensionless internal energy versus volume fraction, indicating empirically defined zones of liquid-like and solid-like behaviour...

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