The Compressibility Factor Equation of State

The compressibility factor of a gaseous species is defined as the ratio 

If the gas behaves ideally, z = 1. The extent to which z differs from 1 is a measure of the extent to which the gas is behaving nonideally. 

Equation 5.4-1 may be rearranged to form the compressibility factor equation of state, 

An alternative to using a nonideal gas equation of state like those described in Section 5,3 is to determine z and substitute it into Equation 5.4-2a, 5.4-2b, or 5.4-2c. The next section describes an application of this method to a system for which tabulated z values are available. The following sections outline methods for estimating z in the absence of tabulated data. 

Perry s Chemical Engineers Handbook (see footnote 2), pp. 2-140 through 2-150, gives values of z(T, P) for air, argon, COj, CO, H2, CH4, N2, O2, steam, and a limited number of other compounds. Once z is known, it can be substituted in the compressibility factor equation of state, which may in turn be solved for whichever variable is unknown. 

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Use the compressibility charts to determine the compressibility factor, and then solve for the unknown variable from the compressibility-factor equation of state (Equation 5.4-2),... 

The compressibility-factor equation of state used in conjunction with the generalized compressibility chart is not generally as accurate as a multiple-constant equation of state for PVT calculations under highly nonideal conditions. Furthermore, it lacks precision and cannot readily be adapted to computer calculations. Its advantages include relative computational simplicity and (as will be shown) adaptability to multicomponent gas mixtures. 

The basis of the generalized compressibility charts is the law of corresponding states, an empirical rule stating that the compressibility factor of a species at a given temperature and pressure depends primarily on the reduced temperature and reduced pressure, T, - T Tc and Pr = P Pc- Once you have determined these quantities, you may use the charts to determine z and then substitute the value in the compressibility-factor equation of state and solve for whichever variable is unknown. 

A 10-liter cylinder containing oxygen at 175 atm absolute is used to supply O2 to an oxygen tent. The cylinder can be used until its absolute pressure drops to 1.1 atm. Assuming a constant temperature of 27°C. calculate the gram-moles of O2 that can be obtained from the cylinder, using the compressibility-factor equation of state when appropriate. 

The ideal gas laws will be used frequently throughout this text, primarily for reasons of conceptual and algebraic simplicity. The assumption of ideal gas behavior permits more attention to be focused on reaction engineering concepts, at the expense of actual gas behavior. The ideal gas equation, PV = nRT is just one of many equations of state. If the ideal gas equation is not valid, any other (valid) equation of state could be used to express a concentration Ca- For example, using the compressibility factor equation of state,... 

Since non-ideal gases do not obey the ideal gas law (i.e., PV = nRT), corrections for nonideality must be made using an equation of state such as the Van der Waals or Redlich-Kwong equations. This process involves complex analytical expressions. Another method for a nonideal gas situation is the use of the compressibility factor Z, where Z equals PV/nRT. Of the analytical methods available for calculation of Z, the most compact one is obtained from the Redlich-Kwong equation of state. The working equations are listed below ... 

Other volume explicit equations of state are sometimes necessary, such as the compressibility factor equation, V = zRT/P, or the truncated virial equation,... 

While not immediately apparent, the PFGC-MES equation of state for the compressibility factor, z, cubic in behavior. Because of the complexity of the formulation an iterative solution procedure is used to determine the appropriate root for the... 

Calculate the compressibility factor for the mixture. In a manner similar to that used in the previous problem, an expression for the fugacity coefficient in vapor mixtures can be derived from any equation of state applicable to such mixtures. If the Redlich-Kwong equation of state is used, the expression is... 

However, this equation can not be used for solids. For such cases Cowan and Fickett suggest an equation of the following form with the factors p V), a(l/) and b(lO as polynomial functions of the compression of the material rj relative to the crystal density of the solid in the standard state, which is known as the Cowan-Fickett equation of state for solids ... 

Explain in your own words and without the use of jargon (a) the three ways of obtaining values of physical properties (b) why some fluids are referred to as incompressible (c) the liquid volume additivity assumption and the species for which it is most likely to be valid (d) the term equation of state (e) what it means to assume ideal gas behavior (f) what it means to say that the specific volume of an ideal gas at standard temperature and pressure is 22.4 L/mol (g) the meaning of partial pressure (h) why volume fraction and mole fraction for ideal gases are identical (i) what the compressibility factor, z, represents, and what its value indicates about the validity of the ideal gas equation of state (j) why certain equations of state are referred to as cubic and (k) the physical meaning of critical temperature and pressure (explain them in terms of what happens when a vapor either below or above its critical temperature is compressed). 

Given any three of the quantities P, V (or F), n (or n), and T for an ideal gas, (a) calculate the fourth one either directly from the ideal gas equation of state or by conversion from standard conditions (b) calculate the density of the gas and (c) test the assumption of ideality either by using a rule of thumb about the specific volume or by estimating a compressibility factor and seeing how much it differs from 1. 

The graphs are based on the Peng-Robinson equation of state (1) as improved by Stryjek and Vera (2, 3). The equations for thermodynamic properties using the Peng-Robinson equation of state are given in the appendix for volume, compressibility factor, fugacity coefficient, residual enthalpy, and residual entropy. Critical constants and ideal gas heat capacities for use in the equations are from the data compilations of DIPPR (8) and Yaws (28, 29, 30). 

In this equation V is total volume, and Z = PV/RT is the compressibility factor computed from an equation of state, and V is the molar volume of the mixture. Most equations of state used in engineering are pressure explicit, that is, they arc in a form in which the pressure is explicit and the volume dependence is more complicated. One such example is the virial equation... 

Find the compressibility factor of ammonia gas at conditions from 50 to 250 atm and 4(X) K using the Redlich-Kwong equation of state in Excel. (Hint Before beginning your spreadsheet, think about how you can organize it so that you can copy formulas from cell to cell easily.)... 

It follows that atoms or molecules interacting with the same pair potential s( )(rya), but with different s and cj, have the same thermodynamic properties, derived from A INkT, at the same scaled temperature T and scaled density p. They obey the same scaled equation of state, with identical coexistence curves in scaled variables below the critical point, and have the same scaled vapour pressures and second virial coefficients as a function of the scaled temperature. The critical compressibility factor P JRT is the same for all substances obeying this law and provides a test of the hypothesis. Table A2.3.3 lists the critical parameters and the compressibility factors of rare gases and other simple substances. 

FIGURE 8.14 Compressibility factor FEj/RF versus P/T at T= 493 K for SAN and EVOH copolymers the solid curve was calculated with the S-S equation of state for SAN copolymers. It follows the linear regression curve through aU data points. 

In the van der Waals model (4.5.54), the first term makes the compressibility factor larger than the ideal-gas value to account for repulsive forces among the molecules. The second term makes Z smaller, to account for attractive forces. So the two terms compete in their effects on Z one term or the other may dominate, depending on state condition (T and p). In the low-density limit, the van der Waals equation collapses to the ideal-gas law, while in the high temperature limit it approximates the hard-sphere equation of state. Formally, the parameters a and b depend on state condition as well as the kind of molecules, but in practice values for a and b are usually assumed to be constant for a particular substance (see 4.5.10). 

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