Mean compressibility factor

A gaseous mixture at 25°C (298 K) and 120 atm (12,162 kPa) contains 3% helium, 40% argon, and 57% ethylene on a mole basis. Compute the volume of the mixture per mole using the following (a) ideal-gas law, (b) compressibility factor based on pseudoreduced conditions (Kay s method), (c) mean compressibility factor and Dalton s law, (d) van der Waal s equation and Dalton s law, and (e) van der Waal s equation based on averaged constants. 

Calculate the volume using the mean compressibility factor and Dalton s law. Dalton s law... 

Mean compressibility factor. Another approach toward the treatment of gaseous mixtures is to say that pV = z riRT where Zm can be called the mean compressibility factor. With such a relationship the only problem is how to evaluate the mean compressibility fector satisfactorily. One obvious technique that might occur to you is to make Zm a mole average as follows ... 

Kay s method is known as a two-parameter rule since only pc and Tc for each component are involved in the calculation of z. If a third parameter such as the Pitzer acentric fector, or Vc is included in the determination of the mean compressibility factor, then we would have a three-parameter rule. All the pseudocritical methods do not provide equal accuracy in predicting p V-T properties, but most suffice for engineering work. Stewart et al. reviewed 21 different methods of determining the pseudoreduced parameters by three-parameter rules (see Table 3.6). Although Kay s method was not the most accurate, it was easy to use and not consid-... 

This permeation relationship can be used in the separation calculations for liquid mixtures by assuming that P, = Dj/zRT in the gas-phase format and all units are consistent, with the permeability incorporating the averaged or mean compressibility factor for the reject and permeate. Using the mean compressibility factor partially offsets the effect of pressure difference on the flux relationship for component /. In other words, the permeability coefficient itself can be perceived as dependent on the initial pressure P and final pressure Py and changes with the particular operating conditions. 

Utilizing the mean compressibility factor, the diffusivity becomes... 

Retention is usually measured in units of time for convenience. Voliime units are more exact. Table 1.1, after suitable corrections have been applied (26). Under average chromatographic conditions liquids can be considered incompressible, but not so for gases, and in gas chromatography elution volumes are corrected to a mean column pressure by multiplying them by the gas compressibility factor, j, equation (1.2)... 

Hazen and Finger (1979) extended equation 1.110 to mean polyhedral compressibility (mean compressibility of a given coordination polyhedron within a crystal structure), suggesting that it is related to the charge of ions in the polyhedron through an ionicity factor, analogous to what we have already seen for thermal expansion—i.e.. 

If a chromatogram contains a peak for a compound that is not retained on the stationary phase, it is possible to calculate the mean linear velocity 0 of the carrier gas in the column. It is also possible to determine u0 at the outlet of the instrument at atmospheric pressure P0 by putting a flow meter at the end of the column. The ratio between these two velocities is equal to the compression factor J. which is related to the pressure differential between the inlet and outlet, P/Pq (P is the pressure at the head of the column) ... 

A more systematic way of approaching deviations from ideal gas behavior is by means of the compression factor, Z, defined as the ratio of the volume of the gas to that of an ideal gas at the same temperature and pressure ... 

The deviation of real gases from ideal gases is best explained by means of compressibility factor (Z). It is defined as,... 

From a data collection with 349 experimental values for the critical compression factor (Reid et al., 1987) obtained with organic and inorganic compounds and elements, a mean value of Zc = 0.2655 is obtained with a standard deviation of a = 0.0346. 

The value of the compressibility factor Z for hydrogen at high pressures and lotv temperatures in Figure 1.1 shotvs that, at ambient temperature, a value of 1.2 is reached at 300 bar, and at lotv temperatures even earlier. This means that a calculation of the hydrogen mass in a container from a measurement of temperature and pressure using the ideal gas equation vrill result in a mass 20% greater than in reality. 

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). 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

The conclusion just reached forms the bams of the generalized, or reduced, compresmbility curves (Fig. 4). From actual experiments on a numbw of gases, the mean observed compressibility factors at various temperatures and pressures have been derived, and the values of k are plotted against the corresponding reduced pressures, with the reduced temperature as parameter. From these curves it is posmble to derive, with a fair degree of accuracy, the value of either the pressure, volume or temperature of any gas, if the other two variables are given. The determination of the volume can be achieved directly from Fig. 4, but the evaluation of either pressure or temperature is not quite as simple.f... 

Show that at moderate and low pressures (and moderate temperatures) f/P of a pure gas is approximately equal to its compressibility factor (x). Verify this for T equal to 2 and values of of 1.20 or more by means of Figs. 4 and 18. 

Although the meaning of the term hardness or softness does not necessarily identify the physical properties, using a thermodynamic approach this chemical hardness can be shown to be related to the compressibility factor, which in turn predicts mechanical hardness of minerals [153]. The definition of softness suggests that there would be correlation between softness and polarizability (a). Some workers [154]... 

For compression gases, we may normally take the dimensionless compressibility factor, Z as unity or at least constant in the pressure range we are studying, and consider it no further. The dependence of specific volume on pressure and temperature means that we may eliminate this variable from our chosen parameter set. Further, it has been found convenient to use the grouping R T instead of temperature alone. Thus the parameter set becomes ... 

For the case of gas-liquid chromatography the relation v = M /p is used again, but u5vr° is expressed as = Z RT/p, where is the compressibility factor (mean value) of carrier gas. According to relations 47, 48 and 50 the following equations expressing G (GLC) and /k(GLC) are obtained ... 

The derivation is possible by substituting the pressure P by means of the compressibility factor z ... 

These forms of a generalized equation of state only require the critical temperature and the critical pressure as substance-specific parameters. Therefore, these correlations are an example for the so-called tsvo-parameter corresponding-states principle, which means that the compressibility factor and thus the related thermodynamic properties for all substances should be equal at the same values of their reduced properties. As an example, the reduced vapor pressure as a function of the reduced temperature should have the same value for all substances, provided that the regarded equation of state can reproduce the PvT behavior of the substance on the basis of the critical data. In reality, the two-parameter corresponding-states principle is only well-suited to reflect the properties of simple, almost spherical, nonpolar molecules (noble gases as Ar, Kr, Xe). For all other molecules, the correlations based on the two-parameter corresponding-states principle reveal considerable deviations. To overcome these limitations, a third parameter was introduced, which is characteristic for a particular substance. The most popular third parameter is the so-called acentric factor, which was introduced by Pitzer ... 

The largest value of the three solutions (1) corresponds to a vapor compressibility factor, and the lowest value (2) corresponds to a liquid one, whereas the middle value (3) has no physical meaning. Taking into account that P = 10 bar is lower than P (300 K) = 12.1 bar. the vapor solution is relevant. The solution is... 

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