Nonideal gas law

The results of the last section showed that, for any macroscopic container at normal pressures, it is not reasonable to conclude that the molecules proceed from wall to wall without interruption. However, if the interaction potential energy between molecules at their mean separation is small compared to the kinetic energy, the speed distribution and the average concentration of gas molecules is about the same everywhere in the container. In this limit, the only real effect of collisions is the excluded volume occupied by the molecule, which effectively shrinks the size of the container. At 1 atm, only about 1/1000 of the space is occupied (remember the density ratio between gas and liquid), so each additional molecule sees only 99.9% of the container as free space. On the other hand, if the attractive part of the interaction potential cannot be totally neglected, the molecules which are very near the wall will be pulled slightly away from the wall by the other molecules. This tends to decrease the pressure. 

Corrections to the ideal gas law can be introduced in many different ways. One well-known form is the van der Waals equation for a nonideal gas  

We can compare a quite nearly ideal gas (He, a = 0.034 L2- atm/mole2, b =. 0237 L/mole with a much less ideal gas (CO2, a = 3.59 L2- atm/mole2, b =. 0427 L/mole). The b term reflects the excluded volume and does not change by much. The a term, reflecting intermolecular attractions, can change dramatically as the gas is changed. 

Another modified form of the ideal gas law is the virial expansion  

This expansion in principle also includes terms proportional to (n/ V)2 and all higher powers of (n/V). However, when the density n/V is much smaller than the density of a solid or liquid, so that most of the container is empty space, this expansion converges rapidly and the higher terms can be ignored. B(T) is called the second virial coefficient, and is a function of temperature. 

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The starting point for developing the equivalence is the nonideal gas law, or equation of state, which more generally can be made applicable to any single phase or single-phase region, whether represented as gas or liquid. This relationship is representable as... 

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 ... 

For gas-phase reactions, the molar density is more useful than the mass density. Determining the equation of state for a nonideal gas mixture can be a difficult problem in thermod5mamics. For illustrative purposes and for a great many industrial problems, the ideal gas law is sufficient. Here it is given in a form suitable for flow reactors ... 

A theory close to modem concepts was developed by a Swede, Svante Arrhenins. The hrst version of the theory was outlined in his doctoral dissertation of 1883, the hnal version in a classical paper published at the end of 1887. This theory took up van t Hoff s suggeshons, published some years earlier, that ideal gas laws could be used for the osmotic pressure in soluhons. It had been fonnd that anomalously high values of osmotic pressure which cannot be ascribed to nonideality sometimes occur even in highly dilute solutions. To explain the anomaly, van t Hoff had introduced an empirical correchon factor i larger than nnity, called the isotonic coefficient or van t Hoff factor,... 

All known gases, called real gases, are nonideal, which means that they do not obey the fundamental gas laws and the equation pv =RT [See under "Detonation (and Explosion), Equations of State , in this Volume]. Specific heats of "real gases vary with temperature and the product composition depends upon both temperature and pressure. 

Equation (23) obviously gives the two-dimensional ideal gas law when a > a2 and with the o2 term included represents part of the correction included in Equation (15). This model for surfaces is, of course, no more successful than the one-component gas model used in the kinetic approach however, it does call attention to the role of the substrate as part of the entire picture of monolayers. We saw in Chapter 3 that solution nonideality may also be considered in osmotic equilibrium. Pursuing this approach still further results in the concept of phase separation to form two immiscible surface solutions, which returns us to the phase transitions described above. 

While an ideal-gas law serves very well under many circumstances, there are also circumstances in which non-ideal behavior can be significant. A compressibility factor Z is an often-used measure of the extent of nonideality,... 

One of the limitations in the use of the compressibility equation of state to describe the behavior of gases is that the compressibility factor is not constant. Therefore, mathematical manipulations cannot be made directly but must be accomplished through graphical or numerical techniques. Most of the other commonly used equations of state were devised so that the coefficients which correct the ideal gas law for nonideality may be assumed constant. This permits the equations to be used in mathematical calculations involving differentiation or integration. 

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 ideal gas law is valid only for ideal gases. All gases can be liquefied at sufficiently high pressure and low temperature and gases become nonideal as they approach liquefaction. However, almost all gases behave nearly ideally at temperatures significantly above their boiling points and at pressures of a few atmospheres or less. 

There are many equations that correct for the nonideal behavior of real gases. The Van der Waals equation is one that is most easily understood in the way that it corrects for intermolecular attractions between gaseous molecules and for the finite volume of the gas molecules. The Van der Waals equation is based on the ideal gas law. 

Since many of the atmospheric gases deviate from the ideal gas law, nonideal gas behavior is estimated using the van der Waals equation of state, which is defined as follows ... 

Boltzmann [3]. Boltzmann was led to thiB generalized formulation of the problem by some attempts he had undertaken (1866) 11] to derive from kinetic concepts the Camot-Clausius theorem about the limited convertibility of heat into work. In order to carry through such a derivation for an arbitrary thermal system (Boltzmann [5], (1871)) it was necessary to calculate, e.g., for a nonideal gas, bow in an infinitely slow change of the state of the system the added amount of heat is divided between the translational and internal kinetic energy and the various forms of potential energy of the gas molecule. It is just for this that the distribution law introduced above is needed. 

The equations derived in Section III for idealized fluidization may be applied to nonideal G/S systems with suitable corrections, much in the manner of the van der Waals corrections to the ideal gas law, as will be shown in Section IX. Direct use of the equations in Section III is for L/S systems, as exemplified by fluidized leaching and washing. 

In this definition, the activity coefficient takes account of nonideal liquid-phase behavior for an ideal liquid solution, the coefficient for each species equals 1. Similarly, the fugacity coefficient represents deviation of the vapor phase from ideal gas behavior and is equal to 1 for each species when the gas obeys the ideal gas law. Finally, the fugacity takes the place of vapor pressure when the pure vapor fails to show ideal gas behavior, either because of high pressure or as a result of vapor-phase association or dissociation. Methods for calculating all three of these follow. 

Note that the elevated value of Cv for a gas whose particles are molecules is not caused by nonideal behavior. That is, it does not depend on whether the gas obeys the ideal gas law. Rather, it is simply that the internal structure of the molecules enables them to absorb energy for processes other than translational motions. 

Parameters of the Bingham Model from Measurements of Pressure Drops in a Line 107 Pressure Drop in Power-Law and Bingham Flow 110 Adiabatic and Isothermal Flow of a Gas in a Pipeline Isothermal Flow of a Nonideal Gas 113 Pressure Drop and Void Fraction in Liquid-Gas Flow Pressure Drop in Flow of Nitrogen and Powdered Coal 120... 

D is correct. An ideal gas has a PV/RT equal to one. Real volume is greater than predicted by the ideal gas law, and real pressure is less than predicted by the ideal gas law. Volume deviations are due to the volume of the molecules, and pressure deviations are due to the intermolecular forces. Thus, a negative deviation in this ratio would indicate that the intermolecular forces are having a greater affect on the nonideal behavior than the volume of the molecules, (see the graph on page 27)... 

Nonideal gas behavior (deviation from the predictions of the ideal gas laws) is most significant at high pressures and/or low temperatures, that is, near the conditions under which the gas liquefies. 

The laws discussed above are strictly valid only for ideal gases. The very fact that all gases can be liquefied if they are compressed and cooled sufficiently is an indication that all gases become nonideal at high pressures and low temperatures. The ideal properties are observed at low pressures and high temperatures, conditions far removed from those of the liquid state. At pressures below a few atmospheres practically all gases are sufficiently dilute for the application of the ideal gas laws with a reliability of a few percent or better. 

For gases at relatively low pressures, the ideal-gas law is often sufficient to calculate the density. The ideal-gas heat capacity (and its integrated form, the ideal-gas enthalpy) are used not only for calculating the properties of gases assumed to be ideal, but also as a starting point in many calculations of the heat capacity and enthalpy of nonideal fluids. 

C02 is considered to be a nonideal gas. That is, it obeys the real gas law, PV = znRT. Using a mass balance, the mass remains constant both at... 

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