The Isothermal Expansion and Compression of an Ideal Gas

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. 

Initially, assume that the gas occupies a volume Vx at pressure Pu where Pi is just balanced by a mass Mj on the pan. Thus 

One-Step Expansion—No Work. If mass Ml is removed from the pan, the gas will expand, moving the piston to the right end of the cylinder. After expansion the gas occupies a volume V2 = 4Vj and pressure P2 = Pi/4. 

When the process goes from state 1 (Pu Vj) to state 2 (Pi/4, 4Vi) with no mass on the pan, no heat flows into or out of the gas because T is constant and no work is done (no mass is lifted). Thus work = wo - 0. This is called a free expansion. 

---

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.

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

What we have accomplished here is to use the definition of entropy in terms of probability to derive an expression for AS that depends on volume, a macroscopic property of the gas. We can now relate the change in entropy to heat flow by noting the striking similarity between the above equation for AS and the one derived in Section 10.2 describing qrev for the isothermal expansion-compression of an ideal gas. Compare... 

It will be recalled from the statements in 9d that in an isothermal, reversible expansion of an ideal gas the work done is exactly equal to the heat absorbed by the system. In other words, in this process the heat is completely converted into work. However, it is important to observe that this conversion is accompanied by an increase in the volume of the gas, so that the system has undergone a change. If the gas is to be restored to its original volume by reversible compression, work will have to be done on the system, and an equivalent amount of heat will be liberated. The work and heat quantities involved in the process are exactly the same as those concerned in the original expansion. Hence, the net result of the isothermal expansion and compression is that the system is restored to its original state, but there is no net absorption of heat and no work is done. The foregoing is an illustration of the universal experience, that it is not possible to convert... 

Use this relationship and Boltzmann s relationship between entropy and number of arrangements (Equation 19.5) to derive the equation for the entropy change for the isothermal expansion or compression of u moles of an ideal gas. 

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,... 

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... 

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. 

COMPRESSION/EXPANSION OF AN IDEAL GAS Consider an ideal gas enclosed in a piston-cylinder arrangement that is maintained at constant temperature in a heat bath. The gas can be compressed (or expanded) reversibly by changing the position of the piston to accomplish a specified change in volume. In Section 12.6, the heat transferred between system and bath when the gas is expanded (or compressed) isothermally and reversibly from volume Vi to Vi is shown to be... 

It is easy to calculate entropy changes for isothermal processes, because T is constant and comes outside the integral to give AS = q ev/T. A specific example is the isothermal compression or expansion of an ideal gas, for which AS = nR InlVf/V ). A second example is any phase transition at constant pressure for which q, y = The entropy change is then AS ang =... 

A Carnot cycle, in which the initial system consists of 1 mole of an ideal gas of volume 7, is carried out as follows (i) isothermal expansion at 100 C to volume 37, (ii) adiabatic expansion to volume 67, (iii) isothermal compression, (iv) adiabatic compression to the initial state. Determine the work done in each isothermal stage and the efficiency of the cycle. 

What might a reversible isothermal expansion of an ideal gas be This process will occur only if initially, when the gas is confined to half the qdinder, the external pressure acting on the piston exactly balances the pressure exerted by the gas on the piston. If the external pressure is reduced infinitely slowly, the piston will move outward, allowing the pressure of the confined gas to readjust to maintain the pressure balance. This infinitely slow process in which the external pressure and internal pressure are always in equilibrium is reversible. If we reverse the process and compress the gas in the same infinitely slow manner, we can return the gas to its original volume. The complete cy cle of expansion and compression in this hypothetical process, moreover, is accomplished without any net change to the surroundings. 

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. 

In the remaining part of this AppendixA.1, we obtain the important result (A.9) using an ideal cyclic process from subset C of Sect. 1.2, namely the Carnot cycle [1, 2, 4, 5]. Carnot cycle is a cyclic process with (fixed number of mols, n, of) uniform ideal gas composed from isothermal and adiabatic (no heat exchange) expansions followed by isothermal (at lower temperature) and adiabatic compressions back to the starting state. All these processes pass the equilibrium (stable) states and they are reversible (cf. definition in Sect. 1.2), see also Rem. 48 in Chap. 3. 

Suppose that we would like to calculate the following volume-related properties of a pure substance (not necessarily an ideal gas)—the gas molar volume, liquid molar volume, isothermal compressibility factor (kappa, k), thermal expansion coefficient (alpha, a), and compressibility factor (Z)—while inputting both the pressure in atmosphere and temperature in Kelvin. The calculation is based on the van der Waals equation of state. Let us design a GUI that handles such a duty. Details on how to add controls to a blank GUI and how to customize them are shown in previous sections. However, more features are explored here that have not been covered before. Given that a database exists in the form of an Excel sheet, which lists the name of a substance, its chemical formula, its critical pressure, and its critical temperature, we would like the user to search for the substance of interest via a keyword that is based either on name or chemical formula. The search results shall be presented and the user will then decide on the substance of interest via selecting the corresponding... 

The expansion process is adiabatic and isentropic for an ideal or reversible expander, and the compression process is isothermal for an ideal or reversible compressor. By assuming that the perfect gas law holds, the ideal work of expansion is... 

If the working fluid in an ideal Stirling cycle is helium and can be assumed to behave as a perfect gas, determine the ideal refrigeration effect and the fraction of refrigeration lost if the regenerator effectiveness is reduced to 0.96. The refrigerator operates between 300 and 90 K. The ratio of the specific volume of the gas before the isothermal compression to that after the compression is 1.6. Likewise, the ratio of the specific volume of the gas after the isothermal expansion to that before the expansion is 1.6. 

---