Coefficients for an ideal gas

This definition implies that the fugacity coefficient for an ideal gas is always 1. 

For ideal gases, the partial derivative on the left side of Equation 3.51 is zero which means that the Joule-Thomson coefficient for an ideal gas is zero. We can see this by differentiating H=U + PV with respect to P at constant T. 

Coefficients of Rihani s and Doraiswamy s method (1965) for calculating enthalpy, entropy, and the for an ideal gas. 

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, l/t)(3 0/3T)p for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (3p/3T),, should be calculated either from the equation of state or from tabulated PVT data. 

Phase Equilibria Models Two approaches are available for modeling the fugacity of a solute in a SCF solution. The compressed gas approach includes a fugacity coefficient which goes to unity for an ideal gas. The expanded liquid approach is given as... 

H (MPa) (Eq. (13)) and HA (MPa m3 mor1) (Eq. (14)) are often referred to as Henry s constant , but they are in fact definitions which can be used for any composition of the phases. They reduce to Henry s law for an ideal gas phase (low pressure) and for infinitely dilute solution, and are Henry s constant as they are the limit when C qL (or xA) goes to zero. When both phases behave ideally, H depends on temperature only for a dilute dissolving gas, H depends also on pressure when the gas phase deviates from a perfect gas finally, for a non-ideal solution (gas or liquid), H depends on the composition. This clearly shows that H is not a classical thermodynamic constant and it should be called Henry s coefficient . 

For an ideal gas the diffusion coefficient is related to the mean free path of the gas molecule, 2, which represents the mean distance between collisions for that molecule. 

The first method, which is the more flexible, is to use an activity coefficient model, which is common at moderate or low pressures where the liquid phase is incompressible. At high pressures or when any component is close to or above the critical point (above which the liquid and gas phases become indistinguishable), one can use an equation of state that takes into account the effect of pressure. Two phases, denoted a and P, are in equilibrium when the fugacity / (for an ideal gas the fungacity is equal to the pressure) is the same for each component i in both phases ... 

The physics of the problem under study is assumed to be governed by the compressible form of the Favre-filtered Navier-Stokes energy and species equations for an ideal gas mixture with constant specific heats, temperature-dependent transport properties, and equal diffusion coefficients. The molecular Schmidt, Prandtl, and Lewis numbers are set equal to 1.0, 0.7, and 1.43, respectively [17]. 

The fugacity coefficients are a function of pressure, temperature and the equilibrium mole fractions, so at given pressure and temperature eq. (2.4-20) can be solved for s and the equilibrium mole fractions can be calculated. Table 2.4-1 gives the calculated equilibrium composition of the reaction mixture at different pressures for an ideal gas mixture and in case the gas is described with the Redlich-Kwong equation of state. 

We recognize that the equation of state for an ideal gas does not describe adequately tire behavior of gases at temperatures and pressures normally encountered in petroleum reservoirs. However, Equation 6-6 does illustrate that we can expect the coefficient of isothermal compressibility of a gas to be inversely proportional to pressure. Equation 6-6 can be used to determine the expected order of magnitude of gas compressibility. 

FIGURE 9.4 The variation of the molar free energy of a real gas with its partial pressure (orange line) superimposed on the variation for an ideal gas. The deviation from ideality is expressed by allowing the activity coefficient to vary from 1. 

The results of early experiments showed that the temperature did not change on the expansion of the gas, and consequently the value of the Joule coefficient was zero. The heat capacity of the gas is finite and nonzero. Therefore, it was concluded that (dE/dV)Tn was zero. Later and more-precise experiments have shown that the Joule coefficient is not zero for real gases, and therefore (dE/dV)Ttheoretical concepts of the ideal gas. 

Most of the plate height terms we have discussed are velocity sensitive. For gases, any term containing the mobile (gas) phase diffusion coefficient Dm (Dg for gases) is also pressure dependent, since Dg varies in inverse proportion to pressure p. For an ideal gas... 

The symbol i is used to denote the enthalpy instead of the customary h to avoid confusion with the heat-transfer coefficient.) The subscripts 1 and 2 refer to entrance and exit conditions to the control volume. To calculate pressure drop in compressible flow, it is necessary to specify the equation of state of the fluid, viz., for an ideal gas,... 

Isothermal compressibility for an ideal gas mixture k is given by HP, whereas for liquids the compressibility is negligible. The activities can be introduced to describe the deviations from the ideal behavior of solutions the activities are expressed in terms of the activity coefficients. 

Natural convection occurs when a fluid is in contact with a solid surface of different temperature. Temperature differences create the density gradients that drive natural or free convection. In addition to the Nusselt number mentioned above, the key dimensionless parameters for natural convection include the Rayleigh number Ra = p AT gx3/ va and the Prandtl number Pr = v/a. The properties appearing in Ra and Pr include the volumetric coefficient of expansion p (K-1) the difference AT between the surface (Ts) and free stream (Te) temperatures (K or °C) the acceleration of gravity g(m/s2) a characteristic dimension x of the surface (m) the kinematic viscosity v(m2/s) and the thermal diffusivity a(m2/s). The volumetric coefficient of expansion for an ideal gas is p = 1/T, where T is absolute temperature. For a given geometry,... 

The denominator on the right side of Eq. (4) is the heat capacity at constant pressure Cp. The numerator is zero for an ideal gas [see Eq. (1)]. Accordingly, for an ideal gas the Joule-Thomson coefficient is zero, and there should be no temperature difference across the porous plug. Eor a real gas, the Joule-Thomson coefficient is a measure of the quantity [which can be related thermodynamically to the quantity involved in the Joule experiment, Using the general thermodynamic relation ... 

In this chapter, we have considered natural convection heat transfpf where any fluid morion occurs by oatural means such as buoyancy. The volume expansion coefficient of a substance represents ihe variation of the density of that substance wilh temperanjre at constant pressure, and for an ideal gas, it is expressed as j8 = VT, where T is the absolute temperature in KorR. 

Equation (11.6) is quite general and should apply to any gas/ for its derivation is based entirely on the first la>y of thermodynamics without assuming any specific properties of the system. However, for an ideal gas, (dE/dV)r is zero, as seen earlier, and since PV = /2T, it follows that td PV)/dP ]T is also zero hence, since Cp is finite, it is seen from equation (11.6) that for an ideal gas mj.t. must be zero.f The Joule-Thomson coefficient of an ideal gas should thus be zero, so that there should be no change of temperature when such a gas e.xpands through a throttle. J... 

For an ideal gas, satisfying the equation PV = RT under all conditions, dV/dT)p is equal to V/T it follows, therefore, from equation (22.2), that the Joule-Thomson coefficient is always zero. For a real gas, however, this coefficient is usually not zero even at very low pressures, when ideal behavior is approached in other respects. That this is the case may be seen by making use of an equation of state for a real gas. 

The fugacity of the vapor phase can be accurately represented in terms of a fugacity coefficient defined by Eq. (12). For an ideal gas, tpi = the... 

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