Deviation from Ideal Behavior

Since the ideal gas law does not accurately represent the behavior of real gases, we shall now attempt to f ormulate more realistic equations of state for gases and explore the implications of these equations. 

If measurements of pressure, molar volume, and temperature of a gas do not confirm the relation pV = RT, within the precision of the measurements, the gas is said to deviate from ideality or to exhibit nonideal behavior. To display the deviations clearly, the ratio of the observed molar volume V to the ideal molar volume Vi (=RT/p) is plotted as a function of pressure at constant temperature. This ratio is called the compressibility factor Z. Then, 

For the ideal gas, Z = 1 and is independent of pressure and temperature. For real gases Z = Z(r, p), a function of both temperature and pressure. 

How can the ideal gas law be modified to yield an equation that will represent the experimental data more accurately We begin by correcting an obvious defect in the ideal gas law, namely the prediction that under any finite pressure the volume of the gas is zero at the absolute zero of temperature V = RT/p. On cooling, real gases liquefy and ultimately solidify after liquefaction the volume does not change very much. We can arrange the new equation so that it predicts a finite, positive volume for the gas at 0 K by adding a positive constant b to the ideal volume  

According to Eq. (3.2) the molar volume at 0 K is b, and we expect that b will be roughly comparable with the molar volume of the liquid or solid. 

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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). 

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... 

When we consider equilibrium between two phases at high pressure, neither phase being dilute with respect to one of the components, we can no longer make the simplifying assumptions made in some of the earlier sections. We must now realistically describe deviations from ideal behavior in both phases for each phase, the effect of both pressure and composition must be seriously taken into account if we wish to describe vapor-liquid equilibria at high pressures for a wide range of conditions, including the critical. 

Every gas shows deviations from ideal behavior at high pressure. Figure 11-6 shows PV/nRT for He, F2, CH4, and N2, all of which are gases at room temperature. Notice that PV/nRT for helium increases steadily as pressure increases. Interatomic forces for helium are too small to reduce the ratio below 1, but the finite size of the helium atom generates deviations from ideality that become significant at pressures above 100 atm. 

All deviations from ideal behavior, which are directly (electrostatically) or indirectly (through the solvent) caused by ion-ion interactions are usually treated by scaling the concentration with a so-called activity coefficient fj, which leads to the activity =fiCi [Atkins, 1990]. Inserting the activity into (5.2) gives... 

If one completely vaporizes a measured amount of a volatile liquid, the molecular weight of the liquid can be determined by measuring the volume, temperature, and pressure of the resulting gas. When one uses this procedure, one uses the ideal gas equation and assumes that the gas behaves ideally. However, if the sample is slightly above the boiling point of the liquid, the gas deviates from ideal behavior. Explain the postulates of the ideal gas equation, and explain why, when measurements are taken just above the boiling point, the calculated molecular weight of a liquid deviates from the true value. 

C—Deviations from ideal behavior depend on the size and the intermolecular forces between the molecules. The greatest deviation would be for a large polar molecule. Sulfur tetrafluoride is the largest molecule, and it is the only polar molecule listed. 

The theory proposed by Debye and Huckel dominated the study of aqueous electrolytes from around 1920 to near the end of the 1950 s. The Debye-Huckel theory was based on a model of electrolyte solutions in which the ions were treated as point charges (later as charged spheres), and the solvent was considered to be a homogeneous dielectric. Deviations from ideal behaviors were assumed to be due only to the long range electrostatic forces between ions. Refinements to include ion-ion pairing and ion... 

Henry s Law is obeyed with organic pollutants of low solubility provided the pressures are not high or temperatures too low - conditions under which one might expect deviations from ideal behavior. Experimental values for Henry s Law constant may be obtained by equilibrating a pollutant between the solvent and vapor phase and measuring its concentration in those two phases. Providing the solubility is low (PA< 0.1) Henry s Law constant can be calculated from the equilibrium vapor pressure (PA) and solubility (S) ... 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

Equations 5.15 and 5.16 give only approximate values for the volumes of olivine compounds. More accurate study of binary mixtures outlines important deviations from ideal behavior, which result in slight curves in cell parameter vs. composition plots, as shown in figure 5.7. The greatest deviations regard cell edges and are particularly evident for the (Mg,Ni)2Si04 mixture. 

Any deviations from ideal behavior may be attributable to a high site distortion, represented in linear terms by parameter A ... 

As we have already seen, the universal chart of gases assumes that all gaseous species exhibit the same sort of deviation from ideal behavior at the same values of 7, and V. This fact, known as the principle of corresponding states, is analytically expressed by the deviation parameter (or compressibility factor ) Z. For n =, ... 

Another type of nonideal SEC behavior, which will not be covered in this chapter, is related to the use of mixed mobile phases (multiple solvents). Because solute-solvent interactions play a critical role in controlling the hydrodynamic volume of a macromolecule, the use of mixed mobile phases may lead to deviations from ideal behavior. Depending on the solubility parameter differences of the solvents and the solubility parameter of the packing, the mobile phase composition within the pores of the packing may be different from that in the interstitial volume. As a result, the hydrodynamic volume of the polymer may change when it enters the packing leading to unexpected elution results. Preferential solvation of the polymer in mixed solvent systems may also lead to deviations from ideal behavior (11). 

The second type of interaction possible for adsorbed molecules is direct adsorbate-adsorbate interaction. Interactions of this sort are expected to lead to deviations from ideality in the two-dimensional phase just as they lead to deviations from ideal behavior for bulk gases. In this case surface equations of state, which are analogous to those applied to nonideal bulk gases, are suggested for the adsorbed molecules. The simplest of these allows for an excluded area correction (see Equation (7.23)) ... 

The behaviour in Fig. 20 was analyzed on the basis of chemical reaction during cure and the physical state of the system as a function of curing. During the early stages of curing, products may form from the epoxide groups reacting with H20, alcohol or HX to form products other than the oxymethylene units. Side reactions may also be caused by impurities. The by-products are felt to be less than 3 % and should not be responsible for the deviation from ideal behavior. 

Deviations from ideal behavior arise from three causes ... 

Interionic attraction in dilute solutions al.su leads to an effective ionic concentration or activity that is less than flic stoichiometric value The oriiriiy of an ion species is its thermodynamic concentration. i.e., the ion concentration corrected for the deviation from ideal behavior. For dilute solutions lire activity id ions is less than one. for concentrated solutions it may be greater than one. It is the ionic activity that is used in expressing the variation of electrode potentials, und other electrochemical phenomena, with composition. 

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