Ideal gases compression/expansion

Calculate the amounts of heat and work and the change in energy of an ideal gas during expansions and compressions (Section 12.4, Problems 17-22). 

Real fluids are neither ideal gases nor are they composed of hard spheres. But if the density is low, a gas might be nearly ideal, or if the temperature is high, a gas might behave somewhat like a fluid of hard spheres. In such cases the ideal-gas or hard-sphere models may serve as references in expansions that approximate real behavior. In this section we consider Taylor expansions (see Appendix A) of the compressibility factor Z about that for the ideal gas. The expansions may be done using either density or pressure as the independent variable we introduce the density expansion first. 

It suffices to carry out one such experiment, such as the expansion or compression of a gas, to establish that there are states inaccessible by adiabatic reversible paths, indeed even by any adiabatic irreversible path. For example, if one takes one mole of N2 gas in a volume of 24 litres at a pressure of 1.00 atm (i.e. at 25 °C), there is no combination of adiabatic reversible paths that can bring the system to a final state with the same volume and a different temperature. A higher temperature (on the ideal-gas scale Oj ) can be reached by an adiabatic irreversible path, e.g. by doing electrical work on the system, but a state with the same volume and a lower temperature Oj is inaccessible by any adiabatic path. 

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. 

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

In an adiabatic expansion or compression, the system is thermally isolated from the surroundings so that q = 0. If the change is reversible, we can derive a general relationship between p, V, and T, that can then be applied to a fluid (such as an ideal gas) by knowing the equation of state relating p, V, and T. 

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.

The entropy change accompanying the isothermal compression or expansion of an ideal gas can be expressed in terms of its initial and final pressures. To do so, we use the ideal gas law—specifically, Boyle s law—to express the ratio of volumes in Eq. 3 in terms of the ratio of the initial and final pressures. Because pressure is inversely proportional to volume (Boyle s law), we know that at constant temperature V2/Vj = E /E2 where l is the initial pressure and P2 is the final pressure. Therefore,... 

If the compression or expansion is isothermal (at constant temperature) then for unit mass of an ideal gas ... 

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. 

As we noticed in Table 5.1, AC/ = 0 both for the free expansion and for the reversible expansion of an ideal gas. We used an ideal gas as a convenient example because we could calculate easily the heat and work exchanged. Actually, for any gas, AC/has the same value for a free and a reversible expansion between the corresponding initial and final states. Furthermore, AC/ for a compression is equal in magnitude and opposite in sign to AC/ for an expansion no indication occurs from the first law of which process is the spontaneous one. 

A hypothetical cycle for achieving reversible work, typically consisting of a sequence of operations (1) isothermal expansion of an ideal gas at a temperature T2 (2) adiabatic expansion from T2 to Ti (3) isothermal compression at temperature Ti and (4) adiabatic compression from Ti to T2. This cycle represents the action of an ideal heat engine, one exhibiting maximum thermal efficiency. Inferences drawn from thermodynamic consideration of Carnot cycles have advanced our understanding about the thermodynamics of chemical systems. See Carnot s Theorem Efficiency Thermodynamics... 

Recall (see any standard physical chemistry text) that for adiabatic expansions or compressions of an ideal gas, there are several relationships between P, V, and T that hold e.g., PVy = constant, where y is the ratio of the heat capacities at constant pressure and volume, i.e., y = cp/cv. Most useful in the context of potential temperature is TPy/y 1 = constant. Applying this latter relationship,... 

TABLE 11.2 Measured Thermodynamic Properties (in SI Units) of Some Common Fluids at 20° C, 1 atm Molar Heat Capacity CP, Isothermal Compressibility jS7, Coefficient of Thermal Expansion otp, and Molar Volume V, with Monatomic Ideal Gas Values (cf. Sidebar 11.3) Shown for Comparison... 

Determine the entropy change for the isothermal expansion or compression of an ideal gas, Example 7.3. 

Since the Eq must describe states ranging from the ideal gas to the dense compressed state, the G factor must reduce to the virial expansion at low density and must approach the value determined by the repulsive potential at the high density limit. Dr. Jacobs took a semiempirical approach to this problem and used the results of Monte Carlo (MC) and Lennard-Jones Devonshire (LJD) calculations to determine unknown parameters in theoretical expressions for p0(v) and G(v,T). ... 

We now consider the isothermal, quasistatic expansion and compression of the gas for several processes. The curve AE in Figure 3.1 represents a pressure-volume isotherm of an ideal gas at a given temperature. The... 

Consequently, the energy of the gas is constant for the isothermal reversible expansion or compression and, according to the first law of thermodynamics, the work done on the gas must therefore be equal but opposite in sign to the heat absorbed by the gas from the surroundings. For a reversible process the pressure must be the pressure of the gas itself. Therefore, we have for the isothermal reversible expansion of n moles of an ideal gas between the volumes F and V... 

The expansion of an ideal gas in the Joule experiment will be used as a simple example. Consider a quantity of an ideal gas confined in a flask at a given temperature and pressure. This flask is connected through a valve to another flask, which is evacuated. The two flasks are surrounded by an adiabatic envelope and, because the walls of the flasks are rigid, the system is isolated. We now allow the gas to expand irreversibly into the evacuated flask. For an ideal gas the temperature remains the same. Thus, the expansion is isothermal as well as adiabatic. We can return the system to its original state by carrying out an isothermal reversible compression. Here we use a work reservoir to compress the gas and a heat reservoir to remove heat from the gas. As we have seen before, a quantity of heat equal to the work done on the gas must be transferred from the gas to the heat reservoir. In so doing, the value of the entropy function of the heat reservoir is increased. Consequently, the value of the entropy function of the gas increased during the adiabatic irreversible expansion of gas. 

Example 1. Find expressions for the thermal expansion coefficient and the isothermal compressibility of an ideal gas. 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this... 

In this section we will lay the groundwork for several fundamental concepts of thermodynamics by considering the isothermal expansion and compression of an ideal gas. An isothermal process is one in which the temperatures of the system and the surroundings remain constant at all times. Recall that the energy of an ideal gas can be changed only by changing its temperature. Therefore, for any isothermal process involving an ideal gas,... 

To illustrate the work and heat effects that accompany the expansion or compression of an ideal gas, consider the apparatus shown in Fig. 10.5. Assume that the pulley is frictionless and that the cable and pan have zero mass. 

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