Ideal gas isotherms

Figure 3.4 Carnot cycle for the expansion and compression of an ideal gas. Isotherms alternate with adiabats in a reversible closed path. The shaded area enclosed by the curves gives the net work in the cyclic process.

As discussed in Chapter 2, the change in entropy associated with taking 1 mol of an ideal gas isothermally from pressure Pt to a pressure P2 is given by... 

Figure 4.10 Ideal gas isotherms. The work done by a constant pressure reversible compression from 100°Cto 600 Cat 600 bars is the area under the P constant arrow.

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 ideal gas law equation of state thus leads to a linear or Henry s law isotherm. A natural modification adds a co-area term ... 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

Equations of state are also used for pure components. Given such an equation written in terms of the two-dimensional spreading pressure 7C, the corresponding isotherm is easily determined, as described later for mixtures [see Eq. (16-42)]. The two-dimensional equivalent of an ideal gas is an ideal surface gas, which is described by... 

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

When a Mollier chart is available for the gas involved the first method, which is illustrated by Figure 12-12A is the most convenient. On the abscissa of Figure 12-12A four enthalpy differences are illustrated. (Hg — Hj) is the enthalpy difference for the isentropic path. (Hg — Hi°) is the ideal gas state enthalpy difference for the terminal temperatures of the isentropic path. The other AH values are the isothermal pressure corrections to the enthalpy at the terminal temperatures. A generalized chart for evaluating these pressure corrections was presented previously. 

Using generalized isothermal effects of pressure and the ideal gas state S and H values, the final temperature required to satisfy the condition AS = 0 is found, and the value of AH is determined for the path. 

Such a fluid is an ideal gas undergoing isothermal change of volume (cf. 77). 

Illustrative Example.—Consider the isothermal expansion of an ideal gas between fixed limits of volume ... 

The isothermal expansion of an ideal gas is an aschistic process.— If a mass of gas expands isothermally, the heat absorbed is equal to the external work done. 

The work done by an ideal gas of constant specific heat in passing from one isotherm to another is the same for all adiabatic paths, is independent of the initial or final pressures or volumes, and is proportional to the difference of temperature between the isotherms. 

Cohesive energy density is the energy of isothermal vaporization per unit volume to the ideal-gas state. It is the square of the Hildebrand solubility parameter. 

Thermodynamic consistency tests for binary vapor-liquid equilibria at low pressures have been described by many authors a good discussion is given in the monograph by Van Ness (VI). Extension of these methods to isothermal high-pressure equilibria presents two difficulties first, it is necessary to have experimental data for the density of the liquid mixture along the saturation line, and second, since the ideal gas law is not valid, it is necessary to calculate vapor-phase fugacity coefficients either from volumetric data for... 

If a relationship is known between the pressure and volume of the fluid, the work can be calculated. For example, if the fluid is the ideal gas, then pV = nRT and equation (2.14) for the isothermal reversible expansion of ideal gas becomes... 

Figure 2.6 Comparison of the work obtained from the two isothermal expansions at 300 K of ideal gas following path (i) in (a) and path (ii) in (b). In each instance, the initial pressure is 0.200 MPa and the final pressure is 0.100 MPa.

Example 2.1 Compare the work obtained from the two isothermal expansions at 300 K of one mole of ideal gas following the paths shown in Figure 2.6. 

The volumes VmA and Vm 2 can be calculated from their positions along the isotherm. For the ideal gas shown as curve (ii) in Figure 2.6,... 

Example 2.3 Calculate q for the isothermal reversible expansion of the ideal gas under the conditions given in Example 2.1. 

Solution AC = 0 for the isothermal expansion of the ideal gas. Hence, from equation (2.33)... 

In summary, in the isothermal expansion of ideal gas, work flowing out of a system is balanced by heat flowing into the system so that AC = 0. 

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