Property relations fundamental

Equation 6.6 expresses the basic relation connecting the Gibbs energy to the temperature and pressure in any closed system  

This equation may be applied to a single-phase fluid in a closed system wherein no chemical reactions occur. For such a system the composition is necessarily constant, and therefore  

The subscript n indicates that the numbers of moles of all chemical species are held constant. 

Consider now the more general case of a single-phase, open system that can interchange matter with its surroundings. The total Gibbs energy nG is still a function of T and P. Since material may be taken from or added to the system, nG is now also a function of the numbers of moles of the chemical species present. Thus, 

The summation is over all species present, and subscript nj indicates that all mole numbers except the ith are held constant. The derivative in the final term is important enough to be given its own symbol and name. Thus, by definition the chemical potential of species i in the mixture 

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This fundamental property relation is the basis for development of aU. other equations relating the properties of PTTsystems. 

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

Equations 54 and 58 through 60 are equivalent forms of the fundamental property relation apphcable to changes between equihbtium states in any homogeneous fluid system, either open or closed. Equation 58 shows that ff is a function of 5" and P. Similarly, Pi is a function of T and C, and G is a function of T and P The choice of which equation to use in a particular apphcation is dictated by convenience. Elowever, the Gibbs energy, G, is of particular importance because of its unique functional relation to T, P, and the the variables of primary interest in chemical technology. Thus, by equation 60,... 

An alternative form of the fundamental property relation given by equation 60 is provided by the mathematical identity of equation 166 ... 

Fundamental Property Relation. The fundamental property relation, which embodies the first and second laws of thermodynamics, can be expressed as a semiempifical equation containing physical parameters and one or more constants of integration. AH of these may be adjusted to fit experimental data. The Clausius-Clapeyron equation is an example of this type of relation (1—3). 

Funda.menta.1 PropertyRela.tion. For homogeneous, single-phase systems the fundamental property relation (3), is a combination of the first and second laws of thermodynamics that may be written as... 

Equation (4-8) is the fundamental property relation for singlephase PVT systems, from which all other equations connecting properties of such systems are derived. The quantity is called the chemical potential of ecies i, and it plays a vital role in the thermodynamics of phase ana chemical equilibria. 

Equations (4-8) and (4-14) through (4-16) are equivalent forms of the fundamental property relation. Each expresses a property nU, nH,... 

Equation (4-66) is a useful alternative to the fundamental property relation given by Eq. (4-16). AU terms in this equation have the units of moles moreover, the enthalpy rather than the entropy appears on the right-hand side. 

For convenience, the three other fundamental property relations, Eos. (4-16), (4-80), and (4-82), expressing the Gibbs energy and refated properties as functions of T, P, and the are collected nere ... 

Finally, a Gibbs/Diihem equation is associated with each fundamental property relation ... 

Our initial purpose in this chapter is to develop from the first and second laws the fundamental property relations which underlie the mathematical structure of thermodynamics. From these, we derive equations which allow calculation of enthalpy and entropy values from PVT and heat-capacity data. We then discuss the diagrams and tables by which both measured and calculated property values are presented for convenient use. Finally, we develop generalized correlations which allow estimates of property values to be made in the absence of complete experimental information. 

These fundamental property relations are general equations for a homogeneous fluid of constant composition. 

This is a fundamental property relation for residual properties applicable to constant-composition fluids. From it we get immediately that ... 

Liquids are usually moved by pumps, generally rotating equipment The same equations apply to adiabatic pumps as to adiabatic compressors. Thus, Eqs. (7.25) through (7.27) and (7.29) are valid. However, application of Eq. (7.26) for the calculation of = —Aff requires values of the enthalpy of compressed liquids, and these are seldom available. The fundamental property relation, Eq. (6.8), provides an alternative. For an isentropic process,... 

Equation (10.2) is the fundamental property relation for single-phase fluid syst of constant or variable mass and constant or variable composition. It is foundation equation upon which the structure of solution thermodynamics built. It is applied initially in the following section, and will appear again subsequent chapters. 

We first develop an alternative form of Eq. (10.2), just as was done in 6.2, where the fundamental property relation was restricted to phases of con composition. We make use of the same mathematical identity ... 

Just as the fundamental property relation of Eq. (13.12) provides complete property information from a canonical equation of state expressing G/RT as a function of T, P, and composition, so the fundamental residual-property relation, Eq. (13.13) or (13.14), provides complete residual-property information from a PVT equation of state, from PVT data, or from generalized PVT correlations. However, for complete property information, one needs in addition to PVT data the ideal-gas-state ieat capacities of the species that comprise the system. 

The fundamental property relations for homogeneous fluids of constant composition given by Eqs. (6.7) through (6.10) show that each of the thermodynamic properties U, H, A, and G is functionally related to a special pair of variables. In particular, Eq. (6.10),... 

Equation (7.10) applies to the steady-state flow of fluid through a control volume to which there is but one entrance and one exit. In addition, we have the fundamental property relation of Eq. (6.8) ... 

In Chap. 6 we treated the thermodynamic properties of constant-composition fluids. However, many applications of chemical-engineering thermodynamics are to systems wherein multicomponent mixtures of gases or liquids undergo composition changes as the result of mixing or separation processes, the transfer of species from one phase to another, or chemical reaction. The properties of such systems depend on composition as well as on temperature and pressure. Our first task in this chapter is therefore to develop a fundamental property relation for homogeneous fluid mixtures of variable composition. We then derive equations applicable to mixtures of ideal gases and ideal solutions. Finally, we treat in detail a particularly simple description of multicomponent vapor/liquid equilibrium known as Raoult s law. 

Earlier we stated that the fundamental property relation for a closed, equilibrium system has the form U(S,V), When we allow the system to be open we must include the effects of varying mass and composition on the interna] energy. It turns out that the extended property relation for an open, equilibrium system is U(S,VrN-,N2,., ., Nn) where Nt is the mole number of component i. We can differentiate this function and compare it to Eq. (A.3) to obtain... 

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