Gases, critical constants

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

Since most synthetic and natural gas systems will contain some amount (however small) of heavy undefined components, we have been searching for improved methods of predicting critical properties and an equation of state which does not use critical constants (or quasi critical constants) to determine the parameters for the equation. Development of improved critical property prediction methods appears to be a waste of time. Wilson and Cunningham (6) have presented an equation—the Parameters From Group Contributions (PFGC) equation of state which satisfies our needs. As the name implies, the parameters in this equation of state are estimated by group contribution rather... 

The macroscopic properties of the three states of matter can be modeled as ensembles of molecules, and their interactions are described by intermolecular potentials or force fields. These theories lead to the understanding of properties such as the thermodynamic and transport properties, vapor pressure, and critical constants. The ideal gas is characterized by a group of molecules that are hard spheres far apart, and they exert forces on each other only during brief periods of collisions. The real gases experience intermolecular forces, such as the van der Waals forces, so that molecules exert forces on each other even when they are not in collision. The liquids and solids are characterized by molecules that are constantly in contact and exerting forces on each other. 

Let us examine the value of Z under different conditions. The first term is always greater than one, which represents the repulsion term making the volume greater than the ideal gas volume and the second term reduces the value of Z, which represents the attraction term. At a fixed value of T above the critical temperature, compression will cause V to decrease so that Z will drop below one, and further compression will cause V to decrease even more so that Z will rise above one. When the temperature is at or below the critical temperature, compression will eventually cause the gas to condense into a liquid at or above the critical pressure Pc. The relations between the critical constants and the values of van der Waals a and b are... 

The principle of corresponding states provides a convenient and rough means for detg the properties of a dense gas or a liquid. The only info required is the value of two of the critical constants for the substance under consideration. As the critical volume s very difficult to measure, even approximately, it is more convenient to use the expression in terms of pf,Tt (Eq 4.1-3) rather than in terms of Vr,Tr (Eq 4.1-2). The values of the critical constants may in turn be estimated from more readily available data, such as bp, mp, etc. For example, if bp of a substance is T, its critical temp Tc, is approximately 3/2 T j and if mp is Tm its Tc is approx... 

Table 2.4 displays critical constants Tc, Pc, Vc and critical compressibility factor Zc for a number of common gases. (Accurate determination of the critical point is experimentally challenging, and quoted values are generally uncertain in the final decimal.) One can see from the table that many common gases (including N2, 02, and CH4) are actually supercritical fluids ( permanent gases ) under ambient temperature conditions, incapable of liquefaction by any applied pressure whatsoever. (Aspects of cryogenic gas-liquefaction techniques are discussed in Section 3.6.3.)... 

EXAMPLE 3-8 Calculate the pseudocritical temperature and pseudocritical pressure of the gas given in Example 3-5. Use the critical constants given in Appendix A. 

The Van der Waals equation is one of those happy approximations which somehow make reasonable predictions well outside the region in which their assumptions are valid. It can even be used to predict the critical constants of a gas. A Van der Waals gas should have a critical pressure and temperature given respectively by... 

Critical Constants of a Gas The most characteristic property of gases is that their molecules lie far apart from one another and are in continuous rapid motion. Each molecule, therefore, leads almost an independent existence. This is particularly so when temperature is high and pressure is low. 

Thus, knowing the critical constants of a gas, it is possible to calculate the van der Waals constants and vice versa. 

Example A certain gas has the following values of its critical constants Pc = 45.6 atm, Vm c = 0.987 dm3 moH and Tc = 190.6 K. Calculate the van der Waals constants of this gas. Also, estimate the radius of the gas molecules assuming that they are spherical. Solution ... 

How well or poorly the ideal gas equation of state fits PVT data for a species often depends on the values of the system temperature and pressure relative to two physical properties of the species—the critical temperature (Tc) and critical pressure (Pc)- Values of these critical constants can be looked up in Table B.l and in most standard chemical reference handbooks. Let us first consider their physical significance and then see how they are used in nonideal gas calculations. 

If the gas is either hydrogen or helium, determine adjusted critical constants from the empirical formulas... 

To perform PVT calculations for nonideal gas mixtures, you may use Kay s rule. Determine pseudocritical constants (temperature and pressure) by weighting the critical constants for each mixture component by the mole fraction of that component in the mixture then calculate the reduced temperature and pressure and the compressibility factor as before. 

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). 

White, Friedman, and Johnston (343) have measured the critical constants for normal hydrogen and have found 33.244 K. and 12.797 atmospheres. Woolley, Scott, and Brickwedde have presented data on the dissociation energy and the thermodynamic properties for the ideal diatomic gas, including contributions from nuclear spin. We have omitted the spin entropy in compiling our tables. Thermodynamic properties for the ideal monatomic gas have been computed at the National Bureau of Standards (395). Note that the reference state represents 2 gram atomic weights for this element. 

The terms vapor and gas are used very loosely. A gas that exists below its critical temperature is usually called a vapor because it can condense. If you continually compress a pure gas at constant temperature, provided that the temperature is below the critical temperature, some pressure is eventually reached at which the gas starts to condense into a liquid. Further compression does not increase the pressure but merely increases the fraction of gas that condenses. A reversal of the procedure just described will cause the liquid to be transformed into the gaseous state again, (i.e., vaporize). From now on, the word vapor will be reserved to describe a gas below its critical point in a process in which the phase change is of primary interest, while the word gas or noncondensable gas will be used to describe a gas above the critical point or a gas in a process in which it cannot condense. 

If the values of a and 6 which make the van der Waals equation represent the P-V relationship of a gas at a particular temperature are inserted in equation (5.14), the critical pressure, volume and temperature obtained are only in moderate agreement with the experimental results. This is not unexpected for, as already pointed out, a and 6 are not strictly constant, if the van der Waals equation is assumed to hold. If the P-V data from which a and 6 are derived are obtained at a temperature that is some distance from the critical, they will clearly not prove satisfactory for the evaluation of critical constants. In actual practice the procedure adopted is to calculate a and 6 from the observed critical data by means of the equations (5.14). A number of the results obtained in this manner are gi ren in Table I if the pressure P of the gas is in atm. and the molar volume F in liter mole S a will be in liter atm. mole since o/F must have the same dimensions as P, and 6 will be in liter mole , since this has the dimensions of F. 

The change of heat content for a given pressure change can thus be evaluated from he critical constants. If heat content data are to be corrected for departure from deal behavior, P2 is set equal to zero P2 is then the heat content of the gas at this pressure when it behaves ideally. Since the heat content of an ideal gas is independent of the pressure, P2 gives the required corrected value. 

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