Thermodynamic relations

Thermod5mamic relations are derived to determine changes in properties such as enthalpy, entropy, and Gibbs function based on the known basic properties. The following thermodynamic relations are derived from the first law and by using relations among work, enthalpy, entropy, and Gibbs function  

Reliable values of thermodynamic functions of H bonds are derived from the equilibrium constant, K, and its variation with temperature. The experimental techniques vary only in their approach to finding the concentration or pressure values needed to determine K, The basic relations are 

In equation it is common to use concentration or pressure and to adjust the experimental conditions such that these quantities are neeuly equal to activity and fugacity then no appreciable error is involved. It is important to remember that the units of K influence both AS and AF, and that values of K (or AS and AF) are not directly comparable unless the units are the same (25, p. 269). 

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If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

The fugacity coefficient can be found from the equation of state using the thermodynamic relation (Beattie, 1949) ... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

Mathematical Consistency Requirements. Theoretical equations provide a method by which a data set s internal consistency can be tested or missing data can be derived from known values of related properties. The abiUty of data to fit a proven model may also provide insight into whether that data behaves correctiy and follows expected trends. For example, poor fit of vapor pressure versus temperature data to a generally accepted correlating equation could indicate systematic data error or bias. A simple sermlogarithmic form, (eg, the Antoine equation, eq. 8), has been shown to apply to most organic Hquids, so substantial deviation from this model might indicate a problem. Many other simple thermodynamics relations can provide useful data tests (1—5,18,21). 

From fundamental thermodynamic relations, the temperature and pressure dependence of Henry s constant can be shown (18,50,51) to be ... 

Known as the Clapeyron equation, this is an exacl thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation... 

The ionic and elecU oii or positive hole chemical potentials are related tluough thermodynamic relations such as... 

Certain thermodynamic relations exist between the state variables. In general for a binary alloy we choose p, T and Xg (the at% of component B) as the independent variables - though presently we shall drop p. The volume 1/ and the composition Xa (= 1 - Xg) are then determined they are the dependent variables. Of course, the weight percentages Wa and Wg can be used instead. 

From Boyles Law, it is known that the pressure is directly proportional to the temperature, therefore, it was shown that the kinetic energy of the molecules related directly to the temperature of the gas. A simple thermodynamic relation holds for this ... 

Air treatment, thermodynamic Relating to the various thermodynamic changes that occur in the specific volume, enthalpy, and wet and dry bulb temperatures of treated air. 

The experiments result in an explicit measure of the change in the shock-wave compressibility which occurs at 2.5 GPa. For the small compressions involved (2% at 2.5 GPa), the shock-wave compression is adiabatic to a very close approximation. Thus, the isothermal compressibility Akj- can be computed from the thermodynamic relation between adiabatic and isothermal compressibilities. Furthermore, from the pressure and temperature of the transition, the coefficient dO/dP can be computed. The evaluation of both Akj-and dO/dP allow the change in thermal expansion and specific heat to be computed from Eq. (5.8) and (5.9), and a complete description of the properties of the transition is then obtained. 

A typical example of an ideal polarizable interface is the mercury-solution interface [1,2]. From an experimental point of view it is characterized by its electrocapillary curve describing the variation of the interfacial tension 7 with the potential drop across the interface, 0. Using the thermodynamic relation due to Lippmann, we get the charge of the wall a (-a is the charge on the solution side) from the derivative of the electrocapillary curve ... 

The ability of N to exist in its compounds in at least 10 different oxidation states from —3 to +5 poses certain thermodynamic and mechanistic problems that invite systematic treatment. Thus, in several compounds N exists in more than one oxidation state, e.g. [N- "H4] + [N "02] , [N-" H4] + [N 03] , [N-"2H5] + [N 03]-, [N- "H4] + [N-3 3]-, etc. Furthermore, we have seen (p. 423) that, under appropriate conditions, NH3 can be oxidized by O2 to yield N2, NO or NO2, whereas oxidation by OCl yields N2H4 (p. 427). Likewise, using appropriate reagents, N2H4 can be oxidized either to N2 or to HN3 (in which the average oxidation number of N is — ). The thermodynamic relations between these various hydrido and 0x0 species containing N can be elegantly codified by means of their... 

Summary of Thermodynamic Relations (Basis Unit mass of Fluid)... 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

We now consider the thermodynamic relations for a system which is normally defined. A system having n independent variables is said to have n degrees of freedom. 

At constant temperature, the activity coefficient depends on both pressure and composition. One of the important goals of thermodynamic analysis is to consider separately the effect of each independent variable on the liquid-phase fugacity it is therefore desirable to define and use constant-pressure activity coefficients which at constant temperature are independent of pressure and depend only on composition. The definition of such activity coefficients follows directly from either of the exact thermodynamic relations... 

Thermodynamic Functions of the Condensed Phases. Tabulated thermodynamic functions for the condensed phases of plutonium dioxide and a detailed description of their calculation are given elsewhere (21). The AG (Pu02 c) is represented by the equation given in Table I. The AGf values were calculated using standard thermodynamic relations and the data given below. 

It is necessary to be able to calculate the energy and momentum of a fluid at various positions in a flow system. It will be seen that energy occurs in a number of forms and that some of these are influenced by the motion of the fluid. In the first part of this chapter the thermodynamic properties of fluids will be discussed. It will then be seen how the thermodynamic relations are modified if the fluid is in motion. In later chapters, the effects of frictional forces will be considered, and the principal methods of measuring flow will be described. 

The vapor pressure of a liquid depends on how readily the molecules in the liquid can escape from the forces that hold them together. More energy to overcome these attractions is available at higher temperatures than at low, and so we can expect the vapor pressure of a liquid to rise with increasing temperature. Table 8.3 shows the temperature dependence of the vapor pressure of water and Fig. 8.3 shows how the vapor pressures of several liquids rise as the temperature increases. We can use the thermodynamic relations introduced in Chapter 7 to find an expression for the temperature dependence of vapor pressure and trace it to the role of intermolecular forces. 

RT = 1.35kcal/mol at room temperature. This is an important thermodynamic relation, as it relates microscopic physical theories (which serve as the basis for computational models) to experimentally measurable quantities. 

By determining the formula for the apparent activation energy, and recalling the thermodynamic relations (Eq. 48) for the equilibrium constant, we can obtain a relationship between the apparent activation energy and the coverages ... 

In contradiction to the usual treatment, relation (5) is not general (see below). For this reason, as well as the lack of an exact thermodynamic relation with the changes in Gibbs energy, the term compensation voltage is also recommended instead of emf, which is commonly used for... 

Expressions for the entropy and heat of mixing which are appropriate in the light of the above revision may be obtained from AFm through use of the standard thermodynamic relations... 

Parameter occurring in thermodynamic relations for dilute polymer solutions (Chaps. XII and XIV). 

Since optical measurements of monolayers at the water-oil interface are rather difficult to carry out, a configuration was suggested where a monolayer at the water-air interface was in contact with an oil lens which was partly wetting the monolayer [23]. The thermodynamic relation between this monolayer and that residing at the water-oil interface was discussed. This configuration was utilized in the X-ray diffraction experiments [24] where the structural changes of dipalmitoyl phosphatidylcholine (DPPC) and DPPE were followed. 

The three quantities of interest — the free energy difference, AA, the difference in potential energies, AIJq i, and the entropy difference, AS o >i, are connected through a basic thermodynamic relation... 

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