Virial remainder

The highest power of the series terms chosen to define each isotherm reflected the extent of the nonideality. For the ethylene isotherms, a cubic series was used for temperatures from 0° to 25°C and a quadratic series was used for temperatures from 75° to 175°C. At 50°C, a quadratic series as well as a cubic series were used. For the helium isotherms, a quadratic series was used with the virial coefficient of the quadratic term treated as a constant obtained from published values rather than as a parameter. The other parameters were evaluated more accurately with the quadratic coefficient treated as a constant rather than as a parameter since the contribution of this term was so small for our range of pressures. The term functioned only as a virial remainder. In the helium data analyses, the parameters were common to all of the data. For the ethylene data analyses, only the virial coefficient parameters were common to all of the data an initial density parameter was required for each sequence... 

For this experiment, the treatment of N as a constant results in the best approximation of the series coefficients as virial coefficients. The most accurate as a true virial coefficient is, of course, the second (linear) coefficient since the experiment was optimized for the determination of the linear behavior and not for the nonlinear behavior of the sample gas. However, the role of the nonlinear terms is more than that of virial remainders terms to accommodate only the analytical fit to the data. The nonlinear coefficients are characteristic of the true virial coefficients to the extent suggested to us by the variations of their values and by the definitiveness of their temperature behavior. Our listed values of the third virial coefficient agree within 2% with those determined by Douslin and Harrison (2). In Table II, we present what we consider to be our best values for the virial coefficients for N constant. Where several groups of... 

Although we have omitted an identifying subscript in the preceding equations, their application so far has been to the development of generated correlations for pure gases only. In the remainder of this section we show how the virial equation may be generalized to allow calculation of fugacity coefficients < , of species in gas mixtures. 

The theory of McMillan and Mayer is exact, but only useful in dilute solutions. It delivers thermodynamic functions as a power series in the solute concentrations and it is quite difficult to compute, or even to interpret the coefficients higher than the second virial coefficient, Bj. About 6 years after the McMillan-Mayer theory was developed a new solution theory appeared, not subject to this difficulty, that of Kirkwood and Buff (Kirkwood and Buff 1951), of course this new theory had computational problems of its own. KB (Kirkwood-Buff) theory is also known as fiuctuation theory for reasons that will become obvious below. It is the basis for the rest of this volume and therefore will occupy the remainder of this chapter. 

There is no known PVTx equation of state that is suitable for calculation of fiigacity coefficients for alt mixtures at all possible conditions of interest. The choice of an equation of state for an engineering calculation is therefore often made on an ad hoc basis. Guidelines are available, but they reflect the inevitable compromise between simplicity and accuracy. We treat in the remainder of this section three popular classes of equations of state commonly employed for practical calculations the virial equations, u for gases at low to moderate densities the cubic equations of state (exemplified by the R lich-Kwong equations), used for dense gases and liquids and equations inspired by the so-called chemical theories, used for associating vapors and vapor mixtures. 

Potentials, variables, and laws comprise the majority of the thermodynamic infrastructure. The remainder is obtained via equations of state ranging from the rigorous to empirical. The equations vary in their names, history, and familiarity. This section will concentrate on the empirical category for its ease of application. The rigorous category includes infinite series or virial-type equations. These describe systems precisely, although at the cost of weighing an infinite number of terms. The empirical variety affords simplicity via economy. 

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