Real gas molar volumes

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 ... 

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

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) ... 

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... 

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 ... 

The presence of intermolecular forces also accounts for the variation in the compression factor. Thus, for gases under conditions of pressure and temperature such that Z > 1, the repulsions are more important than the attractions. Their molar volumes are greater than expected for an ideal gas because repulsions tend to drive the molecules apart. For example, a hydrogen molecule has so few electrons that the its molecules are only very weakly attracted to one another. For gases under conditions of pressure and temperature such that Z < 1, the attractions are more important than the repulsions, and the molar volume is smaller than for an ideal gas because attractions tend to draw molecules together. To improve our model of a gas, we need to add to it that the molecules of a real gas exert attractive and repulsive forces on one another. 

Compressing a gas brings the particles into close proximity, thereby increasing the probability of interparticle collisions, and magnifying the number of interactions. At this point, we need to consider two physicochemical effects that operate in opposing directions. Firstly, interparticle interactions are usually attractive, encouraging the particles to get closer, with the result that the gas has a smaller molar volume than expected. Secondly, since the particles have their own intrinsic volume, the molar volume of a gas is described not only by the separations between particles but also by the particles themselves. We need to account for these two factors when we describe the physical properties of a real gas. 

Using the a Function. A typical molar volume-pressure isotherm for a real gas is illustrated in Figure 10.6, together with the corresponding isotherm for an ideal gas. From Equation (10.34) we can write... 

Figure 10.6. Comparison of molar volume-pressure isotherms for a possible real gas and an ideal gas.

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed 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 van der Waals parameters for carbon dioxide area = 3.640L2-atnvmol-2and b = 0.042 67L-mol 1. For carbon dioxide confined in a 1.00-L vessel at a constant temperature of 27°C, calculate the pressure of the gas by using the ideal gas law and the van der Waals equation for 0.100 to 0.500 mol C02 at 0.100-mol increments. Calculate the percentage deviation of the ideal value from the real value at each point. Under these conditions, which term has the larger effect on the real pressure of C02, the intermolecular attractions or the molar volume ... 

As a caution to the reader Of course, Avogadro s hypotheses and the gas laws assume that gases are all ideal gases. The gases in the real world are not all ideal gases the molar volume at S.T.P. is most often a little below the 22.414 L indicated here. In the remainder of this chapter, the rounded value 22.4L/mol will be used for all gases and, if not so identified, all the gases are ideal. 

JK mol-1 the value V°L = 0.91 cm3mol is obtained. An interpretation of the Hildebrand/Trouton Rule is that this free volume, V°L, allows for the freedom of movement of molecules (particles) necessary for the liquid state at the temperature Th. The explanation of the constant entropy of evaporation is that it takes into account only the translational entropy of the vapor and the liquid. It has to be pointed out that V°L does not represent the real molar volume of a liquid, but designates only a fraction of the corresponding molar volume of an ideal gas Vy derived from the entropy of evaporation. The real molar volume VL of the liquid contains in addition the molar volume occupied by the molecules V0. As a result the following relations are valid VL -V°L + V0 and Vc=Vq + V0. However, while V] < V0 and VL is practically independent of the pressure, V0 VaG in the gaseous phase. Only in the critical phase does VCIVL = 1 and the entropy difference between the two phases vanishes. 

Table 12.1 Molar Volume of Several Real Gases at STP 8. A certain set of conditions allows 4.0 mol of gas to be held in a 70 L container. What volume do 6.0 mol of gas need under the same conditions of temperature and pressure 9. At STP, a container holds 14.01 g of nitrogen gas, 16.00 g of oxygen...

You know that ideal gases have a volume of 22.4 L at STP. Do real gases have the same volume The volumes of several real gases at STP are given in Table 12.1. All the volumes are very close to 22.4 L/mol, the molar volume of an ideal gas. Scientists have decided that 22.4 L/mol is an acceptable approximation for any gas at STP when using gas laws. Although the volumes are the same, one mole of a gas will have a different mass and density than one mole of another gas. (See Figure 12.7.)... 

Look back at Table 12.1. It gives more accurate molar volumes for several gases. Using what you have learned about the way real gases behave, explain why these molar volumes are slightly different from the molar volume of an ideal gas. 

The second virial coefficient is thus approximately equal to the difference between the molar volume of a real gas, and the ideal molar volume at the same temperature and pressure. 

In the gas law for real gases, the molar volume can be expressed with one or two virial coefficients according to the equations from Redlich-Kwong or Prausnitz [18, 19], With low pressures, the dependency of the fugacity coefficient can be neglected. 

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... 

We have noted in section 7.6 that V lg is in general not zero. This is an example of the difference between the theoretical ideal gas and the ideal gas limit of real gas as P—> 0. In this appendix we shall first derive a general expression for the partial molar volume at a fixed position then apply the result to the case of ideal gas. 

Because of the deviations from ideal behavior (and thus from Avogadro s hypothesis) shown by real gases, the actual observed molar volume of a gas at S.T.P. may be slightly different from 22.414 L, usually lower. In the rest of this chapter, the rounded value 22.4 L will be used for all real gases. 

Equations 9.3-3 to 9.3-5 resemble those obtained in Sec. 9.1 for the ideal gas mixture. There is an important difference, however. In the present case we are considering an ideal mixture of fluids that are not ideal gases, so each of the pure-component properties here will not be an ideal gas property, but rather a real fluid property that must either be measured or computed using the techniques described in Chapter 6. Thus, the molar volume Vj is not equal to RT/P, and the fugacity of each species is not equal to the pressure. 

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