Volume of real gases

We will now consider in turn the effect of these two factors on the molar volumes of real gases. 

Several methods have been developed for calculating fugacities from measurements of pressures and molar volumes of real gases. 

REAL GASES DEVIATIONS FROM IDEAL BEHAVIOR (SECTION 10.9) Departures from ideal behavior increase in magnitude as pressure increases and as temperature decreases. Real gases depart from ideal behavior because (1) the molecules possess finite volume and (2) the molecules experience attractive forces for one another. These two effects make the volumes of real gases larger and their pressures smaller than those of an ideal gas. The van der Waals equation is an equation of state for gases, which modifies the ideal-gas equation to account for intrinsic molecular volume and intermolecular forces. 

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 ... 

We can assess the effect of intermolecular forces quantitatively by comparing the behavior of real gases with that expected of an ideal gas. One of the best ways of exhibiting these deviations is to measure the compression factor, Z, the ratio of the actual molar volume of the gas to the molar volume of an ideal gas under the same conditions ... 

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. 

Since both the osmotic pressure of a solution and the pressure-volume-temperature behavior of a gas are described by the same formal relationship of Equation (25), it seems plausible to approach nonideal solutions along the same lines that are used in dealing with nonideal gases. The behavior of real gases may be written as a power series in one of the following forms for n moles of gas ... 

Dalton s Law of Partial Pressures—Amagat s Law of Partial Volumes—Apparent Molecular Weight of a Gas Mixture — Specific Gravity of a Gas Behavior of Real Gases 104... 

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. 

To explain the very different behavior of real gases, the model must be modified. Suppose the molecular volume is small but not negligible. In stales of high compression, where the total molecular volume becomes of the order of the volume available 10 the gas. the free space available to the molecules is only a Traction of what it would be in a perfect gas. and thus the real gas is much harder to compress than Ihe perfect gas. This explains the low compressibility of dense gases and liquids (diagram). 

Such intermolecular forces also account for the deviations of real gases from the ideal behaviour required by the equation PV — RT. Deviation arises from two causes appreciable intermolecular attraction and the finite volume occupied by the molecules themselves, which is another way of saying that repulsive forces come into play when two molecules approach one another closely. In van der Waals equation, these effects are respectively covered by the additional terms in (P+a/V2)(V—b)=RT Because of their relationship to the a/V2 term, secondary attractive forces are often referred to collectively as van der Waals forces. 

Many equations have been suggested to express the behavior of real gases. In general, there are those equations that express the pressure as a function of the volume and temperature, and those that express the volume as a function of the pressure and temperature. These cannot usually be converted from one into the other without obtaining an infinite series. The most convenient thermodynamic function to use for those in which the volume and temperature are the independent variables is the Helmholtz energy. The... 

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. 

The Van der Waal s equation takes into account the deviations of real gases from the kinetic molecular theory of gases (nonzero molecular volume and nonelastic collisions). 

The molecules of real gases have both volume and mutual attraction. Pressure depends on the number of molecules and temperature. 

This equation gives us the means for determining the molecular weight of a gas sample from the measurements of mass, temperature, pressure, and volume. Most real gases at low pressures (1 atm or less) and high temperatures (300 K or more) behave as an ideal gas and thus, under such conditions, the ideal gas law is applicable for real gases as well. 

Of course, no gas is really ideal. The ideal gas theory ignores certain facts about real gases. For example, an ideal gas particle does not take up any space. In fact, you know that all particles of matter must take up space. Gas particles are small and far apart, however. Thus the space occupied by the particles is insignificant compared to the total volume of the container. You will learn more about the behaviour of real gases in Chapter 12. 

Eq. 3.44 is more accurate than the ideal gas equation PV=nRT for expressing the P -V - T behaviour of real gases. Thus, if we take one mole of carbon dioxide at 47 C and compress it to different pressures, the volume, as observed by experiment, is found to be closer to that calculated from the van der Waals equation than to that calculated from the ideal gas equation. The departure from ideal gas equation becomes more and more wide as the pressure increases. 

We will examine the experimentally observed behavior of real gases by measuring the pressure, volume, temperature, and number of moles for a gas and noting how the quantity PV/nRT depends on pressure. Plots of PV/nRT versus P are shown for several gases in Fig. 5.22. For an ideal gas PV/nRT equals 1 under all conditions, but notice that for real gases PV/nRT approach -... 

The notion of excluded volume dates back to the van der Waals formulation for the behavior of real gases [15], the equation of which reads... 

Sj. Mixtures of Real Gases Additive Pressure Law.—The rule that the total pressure of a mixture of gases is equal to the sum of the pressures exerted by each gas if it alone occupied the whole of the available volume ( 5b) does not apply to real gases. The total pressure is thus not equal to the sum of the partial pressures defined in the usual manner. However, for some purposes it is convenient to define the partial pressure of a gas in a mixture by means of equation (5.8), i.e., p == n P, where p is the partial pressure and n is the mole fraction of any constituent of the mixture of gases of total pressure P. 

A simple modification of the law of partial pressures as applied to ideal gases has been proposed for mixtures of real gases (E. P. Bartlett, 1928). If PJ is the pressure which would be exerted by a constituent of a gas mixture when its molar volume is the same as that of the mixture, then it is suggested that the total pressure P is given by... 

Sk. Additive Volume Law.— The additive pressure law, as given by equation (5.26), is useful for the calculation of the approximate pressure exerted by each gas, and the total pressure, in a mixture of real gases, when the volume is known. If the total pressure is given, however, the evaluation of the volume is somewhat more complicated, involving a series of trial solutions. An alternative approxi-... 

The cause-effect relationship is fundamental in the scientific approach. Its identification is essential both for conceptual understanding and for the interpretation of experimental information. Errors in the expression of cause-effect relationships are symptoms of serious misconceptions, as illustrated by the examples below the presence of intermolecular interactions is the cause (not the result) of the different behaviour of real gases (26) our descriptions, like equations (27), or the fact that we determine certain values experimentally (28), cannot be the cause of the behaviour of a system work is done in the reaction between iron and HCl (29) because there is the formation of a gas and, therefore, a volume change ... 

In thermodynamics, the pressure-volume temperature relationships of real gases are described by the equation of state. The compressibility factor from the reduced pressure and temperature can be rearranged from Redlich and Kwong [9] to two constant state equations in the form... 

In the case of real gases, the terms in the expansion (6.83) correspond to successive corrections to the ideal-gas behavior, due to interactions among pairs, triplets, quadruplets, etc., of particles. One of the most remarkable features of the theory is that the coefficients Bj depend on the properties of a system containing exactly jparticles. For instance, B2(T) can be computed from a system of two particles in a volume V at temperature T. 

This law is the basis of the gas thermometer, and the scale on which the temperature is defined is known as the ideal-gas or absolute temperature scale. If either the volume or the pressure of a fixed quantity of gas is held constant, then the measurement of the other determines the temperature directly. No gas is ideal, however, so that the gas thermometer is actually based on an approximately-ideal gas and the assumption that departures from ideality can be accurately measured and taken into account. By this means, the thermometer is made to be independent of the properties of real gases and, thus, becomes effectively ideal. 

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