Ideal gas defined

Experience (and also most constitutive models in this book, e.g., models A, B in Sects. 2.1, 2.2 and in Chap. 3) shows that the internal energy of (uniform) fluids (namely real gases) are functions only of V and (denoted later as T, see (1.30) below). For the special case of ideal gas (defined by i., ii. in Appendix A. 1, cf. end of Sect. 3.7) the internal energy is a rising function of temperature t only... 

Pressure-area isotherms for many polymer films lack the well-defined phase regions shown in Fig. IV-16 such films give the appearance of being rather amorphous and plastic in nature. At low pressures, non-ideal-gas behavior is approached as seen in Fig. XV-1 for polyfmethyl acrylate) (PMA). The limiting slope is given by a viiial equation... 

The values of the thermodynamic properties of the pure substances given in these tables are, for the substances in their standard states, defined as follows For a pure solid or liquid, the standard state is the substance in the condensed phase under a pressure of 1 atm (101 325 Pa). For a gas, the standard state is the hypothetical ideal gas at unit fugacity, in which state the enthalpy is that of the real gas at the same temperature and at zero pressure. 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

An ideal gas is a model gas comprised of imaginary molecules of zero volume that do not interact. Each chemical species in an ideal gas mixture therefore has its own private properties, uninfluenced by the presence of other species. The partial pressure of species i (i = 1,2,... , N) in an ideal gas mixture is defined by equation 142 ... 

Heat Capacity, C° Heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree typical units are J/kg-K or J/kmol-K. There are many sources of ideal gas heat capacities in the hterature e.g., Daubert et al.,"" Daubert and Danner,JANAF thermochemical tables,TRC thermodynamic tables,and Stull et al. If C" values are not in the preceding sources, there are several estimation techniques that require only the molecular structure. The methods of Thinh et al. and Benson et al. " are the most accurate but are also somewhat complicated to use. The equation of Harrison and Seaton " for C" between 300 and 1500 K is almost as accurate and easy to use ... 

Enthalpy of Formation The ideal gas standard enthalpy (heat) of formation (AHJoqs) of chemical compound is the increment of enthalpy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard states, defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.3 kPa). Sources for data are Refs. 15, 23, 24, 104, 115, and 116. The most accurate, but again complicated, estimation method is that of Benson et al. " A compromise between complexity and accuracy is based on the additive atomic group-contribution scheme of Joback his original units of kcal/mol have been converted to kj/mol by the conversion 1 kcal/mol = 4.1868 kJ/moL... 

To understand the role of solute-solvent interac tions on solubilities and selectivities, it is instructive to define an enhancement factor, E, as the ac tual solubility, y9, divided by the solubility in an ideal gas, so that E = where P is the vapor pressure. This factor is a normahzed... 

The f, fg, f, and fp are determined for the pure gas at the pressure of the mixture and depend on the pressure and the temperature. In gaseous mixtures, the quantity Kp as defined by Equation 2-38 is used. Eor an ideal gas reaetion mixture, Kf = Kp. Eor a non-ideal system. Equation 2-39 ean be used to ealeulate Kp from the measured equilibrium eompositions Ky using Equation 2-42. The eomposition... 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

Unfortunately, there is no consensus on the measure for defining the energy of an explosion of a pressure vessel. Erode (1959) proposed to define the explosion energy simply as the energy, ex,Br> must be employed to pressurize the initial volume from ambient pressure to the initial pressure, that is, the increase in internal energy between the two states. The internal energy 1/ of a system is the sum of the kinetic, potential, and intramolecular energies of all the molecules in the system. For an ideal gas it is... 

Figure 6.33 can be used to calculate the initial velocity Vj for bursting pressurized vessels filled with ideal gas. The quantities to be substituted, in addition to those already defined (p, po, and V), are... 

The physical laws of thermodynamics, which define their efficiency and system dynamics, govern compressed-air systems and compressors. This section discusses both the first and second laws of thermodynamics, which apply to all compressors and compressed-air systems. Also applying to these systems are the ideal gas law and the concepts of pressure and compression. 

It would appear at first sight necessary to define an ideal gas as one which strictly obeys all the gas laws. As a matter of fact we can prove that if it conforms to two conditions it will conform to all the conditions we shall take as defining an ideal gas. 

Ideal Gases.—The state of unit mass of an ideal gas, undergoing adiabatic compression or expansion, is completely defined by the equations... 

We will more thoroughly define the properties of the ideal gas later. 

In the next chapter, we will return to the Carnot cycle, describe it quantitatively for an ideal gas with constant heat capacity as the working fluid in the engine, and show that the thermodynamic temperature defined through equation (2.34) or (2.35) is proportional to the absolute temperature, defined through the ideal gas equation pVm = RT. The proportionality constant between the two scales can be set equal to one, so that temperatures on the two scales are the same. That is, 7 °Absolute) = T(Kelvin).r... 

What we must consider now is the generality of the result obtained for the special case of the ideal gas. We define a new thermodynamic system that is the... 

We have already shown that the absolute temperature is an integrating denominator for an ideal gas. Given the universality of T 9) that we have just established, we argue that this temperature scale can serve as the thermodynamic temperature scale for all systems, regardless of their microscopic condition. Therefore, we define T, the ideal gas temperature scale that we express in degrees absolute, to be equal to T 9), the thermodynamic temperature scale that we express in Kelvins. That this temperature scale, defined on the basis of the simplest of systems, should function equally well as an integrating denominator for the most complex of systems is a most remarkable occurrence. 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

Equations 4.55 and 4.57 are the most convenient for the calculation of gas flowrate as a function of Pi and Pi under isothermal conditions. Some additional refinement can be added if a compressibility factor is introduced as defined by the relation Pv -= ZRT/M, for conditions where there are significant deviations from the ideal gas law (equation 2.15). 

The terms space time and space velocity are antiques of petroleum refining, but have some utility in this example. The space time is defined as F/2, , which is what t would be if the fluid remained at its inlet density. The space time in a tubular reactor with constant cross section is [L/m, ]. The space velocity is the inverse of the space time. The mean residence time, F, is VpjiQp) where p is the average density and pQ is a constant (because the mass flow is constant) that can be evaluated at any point in the reactor. The mean residence time ranges from the space time to two-thirds the space time in a gas-phase tubular reactor when the gas obeys the ideal gas law. 

A gas will obey the ideal gas equation whenever it meets the conditions that define the ideal gas. Molecular sizes must be negligible compared to the volume of the container, and the energies generated by forces between molecules must be negligible compared to molecular kinetic energies. The behavior of any real gas departs somewhat from ideality because real molecules occupy volume and exert forces on one another. Nevertheless, departures from ideality are small enough to neglect under many circumstances. We consider departures from ideal gas behavior in Chapter if. 

Whereas liquids and solids have well-defined densities, the density of a gas varies strongly with the conditions. To see this, we combine the ideal gas equation and the mole-mass relation and rearrange to obtain an equation for... 

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