Ideal gas deviations

The compressibility factor z, defined as pV/RT, is clearly unity for an ideal gas. Deviations from unity are a measure of the nonideality of a gas negative deviations arise if the intermolecular attraction is dominant whereas positive deviations appear if the finite volume effects are paramount By way of specific example, we present in Fig. 3.2 the compressibility factor of methane as a function of pressure at several temperatures. At low temperatures, the attractive deviations are dominant and negative deviations from ideality result. At high temperatures, the excluded volume repulsion is paramount and... 

P-V isotherms (T7 > Tj > > Tj > T,) of a hypothetical pure substance that can exist as a gas and as a liquid. The isotherms for the gas phase are similar to those of an ideal gas. Deviations from ideal behavior become apparent at lower temperatures. The one isotherm (at Tj) shown for the liquid state has the typical pressure-volume behavior of a liquid, a small change in volume corresponding to a large change in pressure. To change smoothly from one type of behavior to the other, there must be a P-V isotherm (Tj) with an inflection (critical) point. 

A close examination of Table 6.5 shows that the values of both a and b increase as the sizes of the molecules increase. The smaller the values of a and h, the more closely the gas resembles an ideal gas. Deviations from ideality, as measured by the compressibility factor, become more pronounced as the values of a and h increase. In Example 6-17 we calculate the pressure of a real gas by using the van der Waals equation. Solving the equation for either norV is more difficult, however (see Exercise 121). 

The above equation is valid at low pressures where the assumptions hold. However, at typical reservoir temperatures and pressures, the assumptions are no longer valid, and the behaviour of hydrocarbon reservoir gases deviate from the ideal gas law. In practice, it is convenient to represent the behaviour of these real gases by introducing a correction factor known as the gas deviation factor, (also called the dimensionless compressibility factor, or z-factor) into the ideal gas law ... 

It was found that that in the case of soft beta and X-ray radiation the IPs behave as an ideal gas counter with the 100% absorption efficiency if they are exposed in the middle of exposure range ( 10 to 10 photons/ pixel area) and that the relative uncertainty in measured intensity is determined primarily by the quantum fluctuations of the incident radiation (1). The thermal neutron absorption efficiency of the present available Gd doped IP-Neutron Detectors (IP-NDs) was found to be 53% and 69%, depending on the thicknes of the doped phosphor layer ( 85pm and 135 pm respectively). No substantial deviation in the IP response with the spatial variation over the surface of the IP was found, when irradiated by the homogeneous field of X-rays or neutrons and deviations were dominated by the incident radiation statistics (1). 

In 1873, van der Waals [2] first used these ideas to account for the deviation of real gases from the ideal gas law P V= RT in which P, Tand T are the pressure, molar volume and temperature of the gas and R is the gas constant. Fie argried that the incompressible molecules occupied a volume b leaving only the volume V- b free for the molecules to move in. Fie further argried that the attractive forces between the molecules reduced the pressure they exerted on the container by a/V thus the pressure appropriate for the gas law isP + a/V rather than P. These ideas led him to the van der Waals equation of state ... 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

From these results, the thennodynamic properties of the solutions may be obtamed within the McMillan-Mayer approximation i.e. treating the dilute solution as a quasi-ideal gas, and looking at deviations from this model solely in temis of ion-ion interactions, we have... 

In the case of bunolecular gas-phase reactions, encounters are simply collisions between two molecules in the framework of the general collision theory of gas-phase reactions (section A3,4,5,2 ). For a random thennal distribution of positions and momenta in an ideal gas reaction, the probabilistic reasoning has an exact foundation. Flowever, as noted in the case of unimolecular reactions, in principle one must allow for deviations from this ideal behaviour and, thus, from the simple rate law, although in practice such deviations are rarely taken into account theoretically or established empirically. 

The raie gas atoms reveal through their deviation from ideal gas behavior that electrostatics alone cannot account for all non-bonded interactions, because all multipole moments are zero. Therefore, no dipole-dipole or dipole-induced dipole interactions are possible. Van der Waals first described the forces that give rise to such deviations from the expected behavior. This type of interaction between two atoms can be formulated by a Lennaid-Jones [12-6] function Eq. (27)). 

By combining Equations (8.4) and (8.6) we can see that the partition function for a re system has a contribution due to ideal gas behaviour (the momenta) and a contributii due to the interactions between the particles. Any deviations from ideal gas behaviour a due to interactions within the system as a consequence of these interactions. This enabl us to write the partition function as ... 

At room temperature and atmospheric pressure, 95% of the vapor consists of dimers (13). The properties of the vapor deviate considerably from ideal gas behavior because of the dimeri2ation. In the soHd state, formic acid forms infinite chains consisting of monomers linked by hydrogen bonds (14) ... 

Liquid solutions are often most easily dealt with through properties that measure their deviations, not from ideal gas behavior, but from ideal solution behavior. Thus the mathematical formaUsm of excess properties is analogous to that of the residual properties. 

The foregoing discussion has dealt with nonideahties in the Hquid phase under conditions where the vapor phase mixes ideally and where pressure-temperature effects do not result in deviations from the ideal gas law. Such conditions are by far the most common in commercial distillation practice. However, it is appropriate here to set forth the completely rigorous thermodynamic expression for the Rvalue ... 

From this equation, the temperature dependence of is known, and vice versa (21). The ideal-gas state at a pressure of 101.3 kPa (1 atm) is often regarded as a standard state, for which the heat capacities are denoted by CP and Real gases rarely depart significantly from ideaHty at near-ambient pressures (3) therefore, and usually represent good estimates of the heat capacities of real gases at low to moderate, eg, up to several hundred kPa, pressures. Otherwise thermodynamic excess functions are used to correct for deviations from ideal behavior when such situations occur (3). 

AH fluids, when compared at the same reduced temperature and reduced pressure have approximately the same compressibiHty factor and deviate from ideal gas behavior to the same extent, giving... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

A key limitation of sizing Eq. (8-109) is the limitation to incompressible flmds. For gases and vapors, density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal-gas-law model. Deviations from ideal behavior are corrected for, to first order, with nommity values of compressibihty factor Z. (See Sec. 2, Thvsical and Chemical Data, for definitions and data for common fluids.) For compressible fluids... 

Compressibility of Natural Gas All gases deviate from the perfect gas law at some combinations of temperature and pressure, the extent depending on the gas. This behavior is described by a dimensionless compressibility factor Z that corrects the perfect gas law for real-gas behavior, FV = ZRT. Any consistent units may be used. Z is unity for an ideal gas, but for a real gas, Z has values ranging from less than 1 to greater than 1, depending on temperature and pressure. The compressibihty faclor is described further in Secs. 2 and 4 of this handbook. 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

A term may now be added to Equation 2.1 to correct it for deviations from the ideal gas or perfect gas law. 

Many process components do not conform to the ideal gas laws for pressure, volume and temperature relationships. Therefore, when ideal concepts are applied by calculation, erroneous results are obtained—some not serious when the deviation from ideal is not significant, but some can be quite serious. Therefore, when data are available to confirm the ideality or non-ideality of a system, then the choice of approach is much more straightforward and can proceed with a high degree of confidence. 

The heart of the question of non-ideality deals with the determination of the distribution of the respective system components between the liquid and gaseous phases. The concepts of fugacity and activity are fundamental to the interpretation of the non-ideal systems. For a pure ideal gas the fugacity is equal to the pressure, and for a component, i, in a mixture of ideal gases it is equal to its partial pressure yjP, where P is the system pressure. As the system pressure approaches zero, the fugacity approaches ideal. For many systems the deviations from unity are minor at system pressures less than 25 psig. 

Deviations from Ideal Gas Laws Compressibility See Figures 12-13A-D. 

For Figure 12-16B, as illustrated by a 24-76% (volume) mixture of nitrogen-hydrogen at around 5,000 psia, the deviation is opposite to that of Figure 12-16A. The actual power requirements are greater than ideal volumetric efficiency exceeds ideal gas laws. 

The solution of the work compression part of the compressor selection problem is quite accurate and easy when a pressure-enthalpy or Mollier diagram of the gas is available (see Figures 12-24A-H). These charts present the actual relationship of the gas properties under all conditions of the diagram and recognize the deviation from the ideal gas laws. In the range in which compressibility of the gas becomes significant, the use of the charts is most helpful and convenient. Because this information is not available for many gas mixtures, it is limited to those rather common or perhaps extremely important gases (or mixtures) where this information has been prepared in chart form. The procedure is as follows ... 

Although real gases deviate from ideal gas behavior and therefore require different equations of state, the deviations are relatively small under certain conditions. An error of 1% or less should result if the ideal gas law were used for diatomic gases whenV> 5 f/ gm-mole (80 ftyib-mole) and for other gases and light hydrocarbon vapors when V > 20 f/gm-mole (320 ftyib-mole) [61, p. 67]. 

All gases deviate from the perfect or ideal gas laws to some degree. In some cases the deviation is rather extreme. It is necessary that these deviations be taken into account in many compressor calculations to prevent compressor and driver sizes being greatly in error. 

Section 5.6 considers the kinetic theory of gases, the molecular model on which the ideal gas law is based. Finally, in Section 5.7 we describe the extent to which real gases deviate from the law. ... 

In this chapter the ideal gas law has been used in all calculations, with the assumption that it applies exactly. Under ordinary conditions, this assumption is a good one however all real gases deviate at least slightly from the ideal gas law. Table 5.2 shows the extent to which two gases, 02 and CO2, deviate from ideality at different temperatures and pressures. The data compare the experimentally observed molar volume, Vm... 

In general, the closer a gas is to the liquid state, the more it will deviate from the ideal gas law. 

From a molecular standpoint, deviations from the ideal gas law arise because it neglects two factors ... 

Notice that in Table 5.2 all the deviations are negative the observed molar volume is less than that predicted by the ideal gas law. This effect can be attributed to attractive forces between gas particles. These forces tend to pull the particles toward one another reducing the space between them. As a result, the particles are crowded into a smaller volume, just as if an additional external pressure were applied. The observed molar volume, Vm, becomes less than V , and the deviation from ideality is negative ... 

Deviation of methane gas from ideal gas behavior. Below about 350 atm, attractive forces between methane (CH4) molecules cause the observed molar volume at 25°C to be less than that calculated from the ideal gas law. At 350 atm, the effect of the attractive forces is just balanced by that of the finite volume of CH4 molecules, and the gas appears to behave ideally. Above 350 atm, the effect of finite molecular volume predominates and V, > 1C... 

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