Liquid internal energy

A reactive species in liquid solution is subject to pemianent random collisions with solvent molecules that lead to statistical fluctuations of position, momentum and internal energy of the solute. The situation can be described by a reaction coordinate X coupled to a huge number of solvent bath modes. If there is a reaction... 

The integrated terms are simply the specific heat of the unit mass of adsorbent and its associated adsorbate. The specific heat at constant volume has been used for the adsorbate since, theoretically, there is no expansion of the adsorbate volume and the heat required to raise the temperature is the change in internal energy. In practice there will be some expansion and a pessimistically high estimate could use the specific heat at constant pressure The specific heat of the adsorbed phase is in any case difficult to estimate and it is common to approximate it to that of saturated liquid adsorbate at the same temperature. 

To use a thermodynamic graph, locate the fluid s initial state on the graph. (For a saturated fluid, this point lies either on the saturated liquid or on the saturated vapor curve, at a pressure py) Read the enthalpy hy volume v, and entropy from the graph. If thermodynamic tables are used, interpolate these values from the tables. Calculate the specific internal energy in the initial state , with Eq. (6.3.23). 

When thermodynamic tables are used, read the enthalpy hf, volume Vj, and entropy Sf of the saturated liquid at ambient pressure, po, interpolating if necessary. In the same way, read these values (hg, Vg, Sg) for the saturated vapor state at ambient pressure. Then use the following equation to calculate the specific internal energy... 

The vessel is assumed to be filled with saturated liquid and vapor. The specific internal energy of the saturated liquid can be computed by substituting the appropriate thermodynamic data of Table 9.3 in Eq. (6.3.23) ... 

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. 

This means, of course, that an energy equation is necessary for the description of gas-liquid flows, along with the usual equations of movement and continuity. Transformation of the internal energy of dissolved gas into medium movement energy is what causes the observed pressure drop at the die entrance, e.g. the apparent decline in the amount of energy required to transport the gas-containing melt. 

Steam tables indicate an arbitrary zero internal energy and entropy for water in its liquid state, at the triple point of water. 

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. 

From laboratory experiments, the internal energy of liquid water at room temperature is —8.1 kcal/mol. From our simulations, the two-body liquid yields an internal energy of —6.8 kcal/mol. The three-body liquid improves to —7.7 kcal/mol and the four-body liquid brings it to —8.95 kcal/mol. The quantum cor ... 

L, or G refer to either soUd, liquid or gas. Note that this internal energy differs from the heat of transformation, H. (This concept is perhaps one of the more difficult ones for one to grasp. However, the internal energy is that energy the substance has at a given temperature). Thus, the relation between Hs,l,g and Cp is ... 

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. 

Chemical potential, p, is another name for total internal energy. Convenient units for it are energy per mole. In terms of the work done on (PdV), and the entropy (S) of a gaseous or liquid substance, it may be written in differential form (Callen, 1960) ... 

Equations similar to 12.3-10 to -15 may be written in terms of internal energy, U, with Cv, the heat capacity at constant volume, replacing CP. For liquid-phase reactions, the difference between the two treatments is small. Since most single-phase reactions carried out in a BR involve liquids, we continue to write the energy balance in terms of H, but, if required, it can be written in terms of U. In the latter case, it is usually necessary to calculate AU from AH and Cv from CP, since AH and CP are the quantities listed in a database. Furthermore, regardless of which treatment is used, it may be necessary to take into account the dependence of AH (or AU) and CP (or C,) on T ... 

The simplest way to treat an interface is to consider it as a phase with a very small but finite thickness in contact with two homogeneous phases (see Fig. 16.1). The thickness must be so large that it comprises the region where the concentrations of the species differ from their bulk values. It turns out that it does not matter, if a somewhat larger thickness is chosen. For simplicity we assume that the surfaces of the interface are flat. Equation (16.1) is for a bulk phase and does not contain the contribution of the surfaces to the internal energy. To apply it to an interface we must add an extra term. In the case of a liquid-liquid interface (such as that between mercury and an aqueous solution), this is given by 7 cL4, where 7 is the interfacial tension - an easily measurable quantity - and A the surface area. The fundamental equation (16.1) then takes on the form ... 

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. 

Although we have looked already at boiling and condensation, until now we have always assumed that no work was done. We now see how invalid this assumption was. A heater located within the distillation apparatus, such as an isomantle, supplies heat energy q to molecules of the liquid. Heating the flask increases the internal energy U of the liquids sufficiently for it to vaporize and thence become a gas. 

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