Virial equation of state pressure

There is another commonly used series equation of state, sometimes called the pressure virial equation of state ... 

The pressure virial equation of state was shown in Eq. (1.3-4), and it was shown in an example that A2, the second pressure virial coefficient, is equal to B2, the second virial coefficient. Find an expression for (dS/dP)j- for a gas obeying the pressure virial equation of state truncated at the A2 term. 

Derive the expression for the entropy change for an isothermal pressure change of a gas described by this truncated pressure virial equation of state. 

Calculate AS for the expansion of 1.000 mol of argon from 10.00 atm to 1.000 atm at 298.15 K, assuming the truncated pressure virial equation of state. Compare your result with that obtained... 

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture. 

The virial equation of state is a power series in the reciprocal molar volume or in the pressure ... 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

Virial Equations of State The virial equation in density is an infinite-series representation of the compressiDility factor Z in powers of molar density p (or reciprocal molar volume V" ) about the real-gas state at zero density (zero pressure) ... 

A. Milchev, K. Binder. Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions Monte Carlo simulations of a bead-spring model. Macromol Theory Simul 5 915-929, 1994. 

The next level of approximation is valid to higher pressures. It assumes that the gas mixture obeys the virial equation of state, with the third, fourth and higher, virial coefficients equal to zero. Thus... 

Hm for steam + n-heptane calculated by the above method is shown by the dashed lines in figure 6. Considering the simplicity of the model and the fact that no adjustable parameters have been used, agreement with experiment is remarkable. For mixtures of steam + n-hexane, benzene and cyclohexane agreement with experiment is much the same. At low densities the model reproduces the curvature of the lines through the results better than the virial equation of state. The method fails to fully reproduce the downward turn of the experimental curves at pressures near saturation, but does marginally better in this region than the P-R equation with k. = -0.3. At supercritical temperatures the model seems to... 

Flow calorimetric measurements of the excess enthalpy of a steam + n-heptane mixture over the temperature range 373 to 698 K and at pressures up to 12.3 MPa are reported. The low pressure measurements are analysed in terms of the virial equation of state using an association model. An extension of this approach, the Separated Associated Fluid Interaction Model, fits the measurements at high pressures reasonably well. 

At moderate pressures, the virial equation of state, truncated after the second virial coefficient, can be used to describe the vapor phase. As suggested by Hirschfelder, et. al. (1 3) the temperature dependence of the virial coefficients is expressed... 

The virial equation of state represents the pressure as a polynomial series in the inverse molar volume as... 

The mathematical relationship between pressure, volume, temperature, and number of moles of a gas at equilibrium is given by its equation of state. The most well-known equation of state is the ideal gas law, PV=RT, where P = the pressure of the gas, V = its molar volume (V/n), n = the number of moles of gas, R = the ideal gas constant, and T = the temperature of the gas. Many modifications of the ideal gas equation of state have been proposed so that the equation can fit P-V-T data of real gases. One of these equations is called the virial equation of state which accounts for nonideality by utilizing a power series in p, the density. 

The mutual dependence of the pressure, volume, and temperature of a substance is described by its equation of state. Many such equations have been proposed for the description of the actual properties of substances (and mixtures) in the gaseous and liquid states. The van der Waals expression is just one of these and of limited applicability. The virial equation of state ... 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The virial equation of state discussed in Section 7.2 is applicable to gas mixtures with the condition that n represents the total moles of the gas mixture that is, n = f= l n,. The constants and coefficients then become functions of the mole fractions. These functions can be determined experimentally, and actually the pressure-volume-temperature properties of some binary mixtures and a few ternary mixtures have been studied. However, sometimes it is necessary to estimate the properties of gas mixtures from those of the pure gases. This is accomplished through the combination of constants. 

The Clapeyron equation can be simplified to some extent for the case in which a condensed phase (liquid or solid) is in equilibrium with a gas phase. At temperatures removed from the critical temperature, the molar volume of the gas phase is very much larger than the molar volume of the condensed phase. In such cases the molar volume of the condensed phase may be neglected. An equation of state is then used to express the molar volume of the gas as a function of the temperature and pressure. When the virial equation of state (accurate to the second virial coefficient) is used,... 

For an accurate description of the PVT behavior of fluids over wide ranges of temperature and pressure, an equation of state more comprehensive than the virial equation is required. Such an equation must be sufficiently general to apply to liquids as well as to gases and vapors. Yet it must not be so complex as to present excessive numerical or analytical difficulties in application. 

Two gram-moles of nitrogen is placed in a three-liter tank at -150.8 C Estimate the tank pressure using the ideal gas equation of state and then using the virial equation of state truncated after the second term. Taking the second estimate to be correct, calculate the percentage error that results from the use of the ideal gas equation at the system conditions. 

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