Reversible isothermal expansion of an ideal gas

During reversible expansion or compression, the temperature and pressure remain uniform. If we substitute p = nRT/ V from the ideal gas equation into Eq. 3.4.10 and treat n and T as constants, we obtain 

In this expression the amount n appears as a constant for the process, so it is not necessary to state as a condition of validity that the system is closed. 

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Calculation of AS for the Reversible Isothermal Expansion of an Ideal Gas Integration of equation (2.38) gives... 

We have found that the work of reversible, isothermal expansion of an ideal gas from the volume Vlnitia to the volume Vflna is... 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

It has been shown [Equation (6.72)] that in the reversible isothermal expansion of an ideal gas... 

So far, we have been looking at systems for which the external pressure was constant. Now let s consider the case of a gas that expands against a changing external pressure. In particular, we consider the very important case of reversible, isothermal expansion of an ideal gas. The term reversible, as we shall see in more detail shortly, means that the external pressure is matched to the pressure of the gas at every stage of the expansion. The term isothermal means that the expansion takes place at constant temperature. In an isothermal expansion, the pressure of the gas falls as it expands so to achieve reversible, isothermal expansion, the external pressure must also be gradually reduced in step with the change in volume (Fig. 6.9). To calculate the work, we have to take into account the gradual reduction in external pressure. 

The change in entropy AS for a reversible isothermal expansion of an ideal gas from its initial volume Vj to a volume V2 is AS - R In(V2IVj) and therefore V2IVj = exp(ASIR). By setting V2IVt equal to the ratio between the molar volume Vq =... 

In a reversible process, Pe t = Pgas = but the relation w = -Pext from Section 12.2 cannot be used to calculate the work, because that expression applies only if the external pressure remains constant as the volume changes. In the reversible isothermal expansion of an ideal gas. 

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. 

Entropy is sometimes said to he a measure of disorder. According to this idea, the entropy increases whenever a closed system becomes more disordered on a microscopic scale. This description of entropy as a measure of disorder is highly misleading. It does not explain why entropy is increased by reversible heating at constant volume or pressure, or why it increases during the reversible isothermal expansion of an ideal gas. Nor does it seem to agree with the freezing of a supercooled hquid or the formation of crystalline solute in a supersaturated solution these processes can take place spontaneously in an isolated system, yet are accompanied by an apparent decrease of disorder. 

Consider a reversible, isothermal expansion of an ideal gas. A schematic of a piston-cylinder assembly undergoing such a process is shown in Figure 2.15. The gas is kept at constant temperature by keeping it in contact with a thermal reservoir. A thermal reservoir contains enough mass so that its temperature does not noticeably change during the process. Can you predict the signs of AU, Q, and W ... 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

A reversible isothermal expansion of the ideal gas is made from an initial volume V to a volume Vz at an absolute (ideal gas) temperature 73. The amount of pressure-volume work in done by the system is obtained by substituting into Equation (2.16). The result is... 

STRATEGY Because entropy is a state function, the change in entropy of the system is the same regardless of the path between the two states, so we can use Eq. 3 to calculate AS for both part (a) and part (b). For the entropy of the surroundings, we need to find the heat transferred to the surroundings. In each case, we can combine the fact that AU = 0 for an isothermal expansion of an ideal gas with AU = w + q and conclude that q = —tv. We then use Eq. 4 in Chapter 6 to calculate the work done in an isothermal, reversible expansion of an ideal gas and Eq. 9 in this chapter to find the total entropy. The changes that we calculate are summarized in Fig. 7.21. 

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... 

Because the expansion is isothermal, implying that T is constant, we can use Eq. 1 directly. However the formula requires q for a reversible isothermal expansion. We find the value of qrev from the first law, AU = q + w. We know (Section 6.3) that AU = 0 for the isothermal expansion of an ideal gas, so we can conclude that q = —w. The same relation applies if the change is carried out reversibly, so we can write qiev = wiev. Therefore, to calculate qrcv, we calculate the work done when an ideal gas expands reversibly and isothermally, and change the sign. For that, we can use the following expression, which we derived in Section 6.7 ... 

Comparison of a reversible and an irreversible isothermal expansion of an ideal gas for the same initial and final states (see text Example 7.12) ... 

Step A "5 B is the reversible isothermal expansion at Th. On p. 172 we showed that for the isothermal expansion of an ideal gas there is no change of internal energy ... 

Both the processes in this section can be considered polytropic. The isothermal expansion of an ideal gas follows Equation (2.49) with 7 = 1 while the reversible, adiabatic expansion of an ideal gas with constant heat capacity has y = k = c lc . Can you think of another example of a polytropic process ... 

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. 

One specific reversible expansion of an ideal gas that will be of particular interest to us is the one in which the system remains at constant temperature (by being immersed in a thermostat). Such a process is called isothermal. For this case, we can use Eq. (6), with P = nRT/V ... 

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 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... 

If an ideal gas is used as the working substance in a Carnot engine, the application of the first law to each of the steps in the cycle can be written as in Table 8.3. The values of and FF3, which are quantities of work produced in an isothermal reversible expansion of an ideal gas, are obtained from Eq. (7.6). The values of AC are computed by integrating the equation dU = C dT. The total work produced in the cycle is the sum of the individual quantities. 

Thus, we expect the spontaneous expansion of a gas to result in an increase in entropy. To see how this entropy increase can be calculated, consider the expansion of an ideal gas that is initially constrained by a piston, as in the rightmost part of Figure 19.5. Imagine that we allow the gas to undergo a reversible isothermal expansion by infinitesimally decreasing the external pressure on the piston. The work done on the surroimdings by the reversible expansion of the system against the piston can be calculated with the aid of calculus (we do not show the derivation) ... 

Now let s consider another example, the expansion of an ideal gas at constant temperature (referred to as an isothermal process). In the cylinder-piston arrangement of Figure 19.5, when the partition is removed, the gas expands spontaneously to fill the evacuated space. Can we determine whether this particular isothermal expansion is reversible or irreversible Because the gas expands into a vacuum with no external... 

Example 7.2 illustrates the use of Equation 7.7 for calculating the work done in the isothermal, reversible expansion of an ideal gas. Because q - -w for an ideal gas. Equation 7.7 can also be written as follows ... 

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