Nearly ideal gas

Although we have explained Bose-Einstein condensation as a characteristic of an ideal or nearly ideal gas, i.e., a system of non-interacting or weakly interacting particles, systems of strongly interacting bosons also undergo similar transitions. Eiquid helium-4, as an example, has a phase transition at 2.18 K and below that temperature exhibits very unusual behavior. The properties of helium-4 at and near this phase transition correlate with those of an ideal Bose-Einstein gas at and near its condensation temperature. Although the actual behavior of helium-4 is due to a combination of the effects of quantum statistics and interparticle forces, its qualitative behavior is related to Bose-Einstein condensation. 

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. 

Heating a sample of matter from 20°C to 40°C at constant pressure causes its volume to increase from 546.0 to 547.6 cm. Classify the material as a nearly ideal gas, a nonideal gas, or condensed. 

Relationships between the principal specific heats for a near-ideal gas... 

Differentiating equation (3.17) with respect to temperature for a near-ideal gas when Z is constant gives ... 

We can proceed further, since the kinetic theory of gases gives us the result for a near-ideal gas that ... 

First we make use of the following thermodynamic relationships for a near-ideal gas, as discussed in Chapter 3, Section 3.3 ... 

While a formal differentiation of the characteristic gas equation, pv = ZR T, gives for a near-ideal gas ... 

In the case we are considering, there is no mechanical power extracted, and so we may set P = 0. In line with our choice of kmol units for the gas mixture, we will make the assumption that it acts as a gas with Z = 1 or a near-ideal gas (Z = constant). We may then substitute... 

Figure 7.1 shows the dependence of the pressure of a nearly ideal gas as a function of the molar volume, Vm With only two axes on our graph, a curve can show the dependence of P on Vrn only for a fixed value of T. The figure shows curves for several members of a family of functions of each for a different value of T. 

Figure 7.1 The pressure of a nearly ideal gas as a function of the molar volume at various fixed temperatures.

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

Consider a nearly ideal gas that is dilute, so the free energy can be expanded in a power series in the density, n. Derive expressions for the free energy up to second order in n for particles that interact with a potential... 

Consider the nearly ideal gas with attractive interactions in the low-density expansion discussed previously. For large enough attractions, the system will undergo a gas-liquid phase transition. The virial expansion for the free energy per unit volume, /, as a function of density, n, to third order is... 

F or an ideal machine, having a thermally perfect regenerator of negligibly small gas volume and no resistance to gas flow, the amount of refrigeration produced per cycle can be shown to be (p - Pq)V where p and pq are supply and exhaust pressure of the refrigerator and V is the volume of the cylinder gas space. In the subsequent analysis the gas is assumed to be helium, or some other nearly ideal gas. 

Gases. - By convention, the activity of the i-th component of a dilute (nearly ideal) gas mixture is defined as... 

At very low densities, such as at nearly ideal gas conditions, Eq. 17.3.8 is valid because the probability that three or more molecules are close enough to interact is very small, compared with that of only two molecules. At higher densities, however, the assumption of pairwise additivity will cause some error. Thus, it is estimated that the nonadditive contributions to the internal energy of liquid argon at its triple point is of the order of 5 to 10 percent (Reed and Gubbins, p.97). 

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