Theoretical ideal gas

A theoretical ideal gas is defined as a system of strictly noninteracting particles. The RDF for such a system can be obtained directly from definition (2.48). With Un= 0 for all configurations, the integrations in (2.48) become trivial and we get 

As is expected, g(R) is practically unity for any value of R. This behavior reflects the basic property of an ideal gas i.e., the absence of correlation follows from the absence of interaction. The term N 1 is typical of a closed system. At the thermodynamic limit N- oo, V oc, N/V= const., this term, for most purposes, may be dropped. Of course, in order to get the correct normalization of g(R), one should use the exact relation (2.53), i.e., 

It should be clear that the pair correlation function has, in general, two contributions. One is due to interaction, which in this case is unity. The second arises from the closure condition with respect to N. Placing a particle at a fixed position changes the conditional density of particles everywhere in the system from N/V into (N— )/V. Hence, the pair correlation due to this effect is 

More on this aspect of the pair correlation can be found in Appendix G. 

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Kinetic energy is the energy of motion. Gas particles have a lot of kinetic energy and constantly zip about, colliding with one another or with other objects. The picture is complicated, but scientists simplified things by making several assumptions about the behavior of gas pcirticles. These assumptions are called the postulates of the kinetic molecular theory. They apply to a theoretical ideal gas ... 

It should be noted that for any gas with any intermolecular interactions, when p — 0, we obtain the ideal-gas behavior. For instance, the equation of state has the typical and well-known form. One should distinguish between the ideal-gas behavior of a real gas as p — 0, and a theoretical ideal gas which is a model system, where no interactions exist. Such a system does not exist however, the equation of state of such a model system is the same as the equation of state of a real system as p —> 0. 

In this section, we have seen that in the limit p —> 0, the pair correlation is (2.55). This is different from the theoretical ideal gas case obtained in section (2.5.1). There, the form of g(R) is valid for any density provided that all... 

There exist several reference states of solutions referred to as ideal state, for which we can say something on the behavior of the thermodynamic functions of the system. The most important ideal states are the ideal-gas mixtures, the symmetric ideal solutions and the dilute ideal solution. The first arises from either the total lack of interactions between the particles (the theoretical ideal gas), or because of a very low total number density (the practical ideal gas). The second arises when the two (or more) components are similar. We shall discuss various degrees of similarities in sections 5.2. The last arises when one component is very dilute in the system (the system can consist of one or more components). Clearly, these are quite different ideal states and caution must be exercised both in the usage of notation and in the interpretations of the various thermodynamic quantities. Failure to exercise caution is a major reason for confusion, something which has plagued the field of solution chemistry. 

As in the case of a one-component system, we should also make a distinction between the theoretical ideal gas, and the practical ideal gas. The former is a system of noninteracting particles the latter applies to any real system at very low densities. Occasionally, the former serves as a model for the latter. For instance, to obtain the equation of state of an ideal gas... 

The theoretical ideal-gas partition function for a system of c components of composition N=NU N2,..., Nc contained in a volume V at temperature T is... 

When there are no interactions in the system (theoretical ideal gas), we must have... 

Note that this is not a theoretical ideal gas. We take the limit P —> 0 for a real gas mixture. Uy R) is the pair potential for the ij pair. We assume that the pair potential has a square-well form, i.e.,... 

We start with a (theoretical) ideal gas, i.e., when there are no intermolecular interactions between the particles. For such a system, the pair correlation function may be calculated exactly from the corresponding partition functions. 

The aforementioned arguments are exact for a theoretical ideal gas at any concentration. Of course, such a system does not exist. We now turn to a real gas, which in the limit of p — 0 behaves as an ideal gas. 

For a theoretical ideal gas, U(R) = 0, equation (G.24) is also the total correlation as in (G.3). But for real gas as p — 0, we have an additional contribution due to the intermolecular interaction as in (G.9). The latter, is in general of short range, hence the former is referred to as the long-range correlation. As we have noted before, this part of the correlation for an ideal gas holds true at any distance, not necessarily for R —> oo. Therefore, we feel that the term closure correlation is more appropriate for gc(CC). 

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. 

If we switch off all interactions (i.e., take a theoretical ideal gas), then Saa(R) = 1 and GAA = 0 (recall that GAA is taken here in the open system, where... 

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 ideal gas law is an example of a correlating expression that comes direcdy from experimental observations, but has theoretical significance. Despite its simplicity, the ideal gas law is an excellent estimation tool. Often, it is the first approximation in systems involving real gases of all types. Unfortunately for Hquids and soHds no laws of such general utility are available. 

The total energy of a vessel s contents is a measure of the strength of the explosion following rupture. For both the statistical and the theoretical models, a value for this energy must be calculated. The first equation for a vessel filled with an ideal gas was derived by Brode (1959) ... 

A computer program was developed based upon theoretical considerations. The results of a parameter study were used to compose a diagram (Figure 9.7) for use in determining initial velocity for vessels tilled with an ideal gas (Baker et al. 1978a and 1983). The scaled pressure P on the horizontal axis of Figure 9.7 is determined by... 

For nearly oxygen-balanced expls equilibrium (1) will dominate and control the compn of the detonation products. As already stated this equilibrium is expected to be independent of pressure if the gases behave ideally. But even for ideal gas behavior and an oxygen-balanced expl, no direct comparison can be made between theoretical detonation product calcns and observed products. This is so because measurements are made at temps much lower than detonation temps, and the products reequilibrate as the temp drops. Further complications arise because the reequilibration freezes at some rather high temp. This is a consequence of re-, action rates. At temps below some frozen equb... 

The theoretical equation of state for an ideal rubber in tension, Eq. (44) or (45), equates the tension r to the product of three factors RT, a structure factor (or re/Eo, the volume of the rubber being assumed constant), and a deformation factor a—l/a ) analogous to the bulk compression factor Eo/E for the gas. The equation of state for an ideal gas, which for the purpose of emphasizing the analogy may be written P = RT v/Vq) Vq/V), consists of three corresponding factors. Proportionality between r and T follows necessarily from the condition dE/dL)Ty=0 for an ideal rubber. Results already cited for real rubbers indicate this condition usually is fulfilled almost within experimental error. Hence the propriety of the temperature factor... 

Calculate the theoretical work required to compress 1 kg of a diatomic ideal gas initially at a temperature of 200 K adiabatically from a pressure of 10000 Pa to a pressure of 100000 Pa in (i) a single stage, (ii) a compressor with two equal stages and (iii) a compressor with three equal... 

Numerous representations have been used to describe the isotherms in Figure 5.5. Some representations, such as the Van der Waals equation, are semi-empirical, with the form suggested by theoretical considerations, whereas others, like the virial equation, are simply empirical power series expansions. Whatever the description, a good measure of the deviation from ideality is given by the value of the compressibility factor, Z= PV /iRT), which equals 1 for an ideal gas. 

Because the volume of a gas decreases with falling temperature, scientists realized that a natural zero-point for temperature could be defined as the temperature at which the volume of a gas theoretically becomes zero. At a temperature of absolute zero, the volume of an ideal gas would be zero. The absolute temperature scale was devised by the English physicist Kelvin, so temperatures on this scale are called Kelvin (K) temperatures. The relationship of the Kelvin scale to the common Celsius scale must be memorized by every chemistry student ... 

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