Real gas approximation

The perfect gas is an abstraction to which any real gas approximates according to the nature of the gas and the conditions. For a given temperature and composition, the perfect gas condition is approached when tile density tends to zero. From a molecular point of view, the perfect gas laws correspond to the behavior of a system of molecules whose interactions may be neglected in expressing the thermodynamic equilibrium properties. However, even at a low density, the transport properties depend essentially on the interactions. 

But, generally, such a cycle with adiabatic and isothermal irreversible processes may be realized with real gas (or even liquid). Those with real gas approximate the reversible Carnot cycle with ideal gas by a double limiting process as follows (i.e., we form the ideal cyclic process from set A (and also B and C), see motivation of postulate U2 in Sect. 1.2) running this cycle slower and slower... 

To start, convert the flow to values estimated to be the compressor inlet conditions. Initially, the polytropic head equation (Equation 2.73) will be used with n as the polytropic compression exponent. If prior knowledge of the gas indicates a substantial nonlinear tendency, the real gas compression exponent (Equation 2.76) should be substituted. As discussed m Chapter 2, an approximation may be made by using the linear average ut the inlet and outlet k values as the exponent or for the determination of the polytropic exponent. If only the inlet value of k is known, don t be too concerned. The calculations will be repeated several times as knowledge of the process for the compression cycle is developed. After selecting the k value, u,se Equation 2.71 and an estimated stage efficiency of 15 / to de clop the polytropic compression exponent n. 

The discussion of the performance of gas turbine plants given in this chapter has developed through four steps reversible a/s cycle analysis irreversible a/s cycle analysis open circuit gas turbine plant analysis with approximations to real gas effects and open circuit gas turbine plant computations with real gas properties. The important conclusions are as follows ... 

Avogadro s Hypothesis is consistent with the kinetic theory. Therefore a perfect gas follows Avogadro s Hypothesis. At one atmosphere pressure and 0°C, one mole (6.02 X 10 molecules) of a perfect gas occupies 22.414 liters. How closely real gases approximate a perfect gas at one atmosphere pressure and 0°C is shown by measur-... 

The virial equation is a general equation for describing real gases. The van der Waals equation is an approximate equation of state fora real gas the parameter a represents the role of attractive forces and the parameter b represents the role of repulsive forces. 

Figure 3-2. The pressure of a real gas as a function of temperature and volume approximation based on the van der Waals equation.

Equation (2.2) is sometimes referred to as the ideal gas law. However, for our present purposes, we must recognize that this law [like those summarized in (2.3a-d)] is merely a crude approximation that never describes any real gas exactly, except in the idealized limit of zero pressure (to be discussed in Section 2.3). Hence, we must sharply distinguish between crude empirical laws (which are at most approximate rules of thumb) and true thermodynamic laws as summarized in Table 2.1. A difficulty for the beginning student of thermodynamics is to distinguish those equations that are based on the ideal gas approximation (and thus are practically never true) from those of rigorous thermodynamic quality. We shall often flag equations of the former type with IG (ideal gas), for example... 

As the name implies, the ideal gas law exactly describes the behavior of an ideal gas under all conditions. Unfortunately, we encounter only real gases. Ideal gases don t exist. Fortunately, though, as long as the pressure on the gas isn t too high and the temperature isn t too low, real gases approximate ideal behavior. 

The biggest differences are obtained for the lower (p jC pc) density points at each temperature, where the calculated ratios are up to approximately 20 smaller than the molecular dynamics results. This result is not surprising at the lower densities, the effects of attractive forces become important and cause diffusion in a real gas to be slower. Dymond (25) has found that the theory predicted the experimental self-diffusion coefficients for densities down to about 0.7 Pc for T > Tc. 

Joule s experiments on the free expansion of an ideal gas showed that the internal energy of such a system is a function of temperature alone. For a real gas, this is only approximately true. For condensed phases, which are effectively incompressible, the volume dependence on the change in internal energy is negligible. As a result, the internal energies of liquids and solids are also considered a function of temperature alone. For this reason, the internal energy of a system may loosely be referred to as the thermal energy . 

It may be emphasised that the above postulates are meant for an ideal gas only. These are only approximately valid for a real gas. From the above postulates and classical mechanics, expression... 

Then, for a real gas the isentropic work of compression is approximated by... 

We have seen that a very simple model, the kinetic molecular theory, by making some rather drastic assumptions (no interparticle interactions and zero volume for the gas particles), successfully explains ideal behavior. However, it is important that we examine real gas behavior to see how it differs from that predicted by the ideal gas law and to determine what modifications of the kinetic molecular theory are needed to explain the observed behavior. Since a model is an approximation and will inevitably fail, we must be ready to learn from such failures. In fact, we often learn more about nature from the failures of our models than from their successes. 

For discussion of intramolecular forces it is essential to remove from consideration effects due to intermolecular forces, that is, to have heats of formation referring to the ideal gas state. In general the correction of heats of formation of real gases at 1 atmosphere pressure to the ideal gas state is very small compared with the accuracy to which heats of formation are known for example, approximately 0-002 kcal mole for methane and 0-02 kcal for methyl chloride. This means that for all purposes connected with bond energies, a knowledge of the heat of formation of the real gas is adequate. Thus for substances whose heat of formation is known directly for the liquid or solid, a knowledge of the heat of vaporization at the appropriate temperature is required. Strictly, however, the quantity concerned is the heat of vaporization to the ideal gas state. 

The first assumption means tliat Raoult s law can apply only for low to moderate pressures. The second implies that it can liave approximate validity only when the species that comprise the system are chemically similar. Just as the ideal gas serves as a standard to wliich real-gas... 

We must emphasise the fact that the second law was deduced without the aid of any hypothesis from the experimental observation that heat never goes of its own accord from places of lower to places of higher temperature. The analytical form in which the second law was stated in equations (2) and (3) was, no doubt, deduced from the gas laws, which do not apply rigidly to any real gas. We might conclude from this that the equations (2) and (3), like the gas laws, are only valid approximately. This conclusion would, however, be erroneous. The essence of the second law is the statement that the efficiency of a cycle between any two temperatures must be independent of the working substance. We may therefore imagine the cycle to be performed with a hypothetical gas having the properties defined by the equation pv = RT. 

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. 

As to liquid mixtures, it is even more difficult to predict the p-V-T properties of liquid mixtures than of real gas mixtures. Probably more experimental data (especially at low temperatures) are available than for gases, but less is lcnown bburth estimation of the p-V T properties of liquid mixtures. For compounds with like molecular structures, such as hydrocarbons of similar molecular weight, called ideal liquids, the density of a liquid mixture can be approximated by assuming that the specific volumes are additive ... 

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