Expansion reversible

Figure A2.1.2. Reversible expansion of a gas witli the removal one-by-one of grains of sand atop a piston.

If a gas obeys Boyle s law the diminution of potential on isothermal reversible expansion is equal to the diminution of free energy, and both are equal to the maximum work. 

The Reversible Process Process (iv) is a representation of a hypothetical reversible expansion in which p and / exl never differ by more than an infinitesimal dp. To carry out the process, pexl is decreased by an amount dp. An expansion then occurs so that V goes to V +dV). This causes the internal pressure to decrease by an amount dp and equilibrium is re-established. This process is repeated an infinite number of times until all of the d V changes add up to AV and V2 = V + AV is reached.g In each step, p never differs from pexl by more than an infinitesimal amount. The process is under control and can be reversed at any time by increasing the pressure by an infinitesimal amount. Hence, the name reversible" is applied to this process. 

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... 

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

The entropy changes ASa and ASB can be calculated from equation (2.69), which applies to the isothermal reversible expansion of ideal gas, since AS is independent of the path and the same result is obtained for the expansion during the spontaneous mixing process as during the controlled reversible expansion. Equation (2.69) gives... 

E3.13 A gas obeys the equation of state PVm RT + Bp and has a heat capacity Cy m that is independent of temperature. Derive an expression relating T and Vm in an adiabatic reversible expansion. 

In general the conditions under which a change in state of a gas takes place are neither isothermal nor adiabatic and the relation between pressure and volume is approximately of the form Pvk = constant for a reversible process, where k is a numerical quantity whose value depends on the heat transfer between the gas and its surroundings, k usually lies between 1 and y though it may, under certain circumstances, lie outside these limits it will have the same value for a reversible compression as for a reversible expansion under similar conditions. Under these conditions therefore, equation 2.70 becomes ... 

Path A is an isothermal, reversible expansion. Path B has two steps. In the first step, the gas is cooled at constant volume to 1.19 atm. In the second step, the gas is heated and allowed to expand against a constant external pressure of 1.19 atm until the final volume is 7.39 L. Calculate the work for each path. 

In Section 6.3 we saw how to calculate the work of reversible, isothermal expansion of a perfect gas. Now suppose that the reversible expansion is nor isothermal and that the temperature decreases during expansion, (a) Derive an expression for the work when T = Tinitja — c( V — Vjnitia ), with c a positive constant, (b) Is the work in this case greater or smaller than that of isothermal expansion Explain your observation. 

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. 

Reversible expansion work (achieved by matching the external to the internal pressure) is given by the infinitesimal version of Eq. 3 of Chapter 6 (itzL.xpanNlon = — PKKAV, which becomes dtfcxpansion = —PexdV) and setting the external pressure equal to the pressure of the gas in the system at each stage of the expansion ... 

By modifying the procedure described above to explode a wire in the water sphere while the system was under compression, they did attain explosions. Measuring the rebound of the cylinder and the loss of aluminum, they could estimate the work produced by the event. Assuming the maximum energy transfer to the water would occur by constant volume heating to the aluminum temperature, foUowed by an isothermal, reversible expansion, they estimated an efficiency of about 25%. Clearly the exploding wire led to an immediate and effective dispersal of the water. 

Let us look now at vapor-liquid systems with more than one component. A liquid stream at high temperature and pressure is flashed into a drum, i.e., its pressure is reduced as it flows through a restriction (valve) at the inlet of the drum. This sudden expansion is irreversible and occurs at constant enthalpy. If it were a reversible expansion, entropy (not enthalpy) would be conserved. If the drum pressure is lower than the bubblepoint pressure of the feed at the feed temperature, some of the liquid feed will vaporize. 

Let Figure 3.3 represent a plane surface (such as a soap film between wires) that is being expanded in the direction indicated. (The symbol A must be replaced by a symbol L for the length of the wire, with A being the area of the surface.) Show that the work of reversible expansion is given by the expression... 

Any finite expansion that occurs in a finite time is irreversible. A reversible expansion can be approximated as closely as desired, and the values of the thermodynamic changes can be calculated for the limiting case of a reversible process. In the limiting case, the process must be carried out infinitely slowly so that the pressure P is always a well-defined quantity. A reversible process is a succession of states, each of which is an equilibrium state, in which the temperature and pressure have well-defined values such a process is also called a quasi-static process. 

Figure 5.1. Schematic representation of an isothermal reversible expansion from pressure Pi to pressure P2. The external pressure is maintained only infinitesimally less than the internal pressure.

So far we have not specified whether the adiabatic expansion under consideration is reversible. Equations (5.40), (5.42), and (5.44) for the calculation of the thermodynamic changes in this process apply to the reversible expansion, the free expansion, or the intermediate expansion, so long as we are dealing with an ideal gas. However, the niunerical values of W, AU, and AH will not be the same for each of the three types of adiabatic expansion because T2, the final temperature of the gas, will depend on the type of expansion, even though the initial temperature is identical in aU cases. 

For the reversible adiabatic expansion, we can see from Equation (5.42) that the final temperature T2 must be less than Ti, because W is negative and Cy is always positive. Thus, the adiabatic reversible expansion is accompanied by a temperature drop, and W, AU, and AH can be calculated from the measured initial and final temperatures using Equations (5.42) and (5.43). 

For an irreversible adiabatic expansion in which some work is performed, the work performed is less in magnimde than that in the reversible process because the external pressure is less than the pressure of the gas by a finite amount. Thus, if the final volume is the same as in the reversible process, the final temperature will not be as low in the irreversible process, because, according to Equation (5.47), the temperature drop depends directly on the work performed by the expanding gas. Similarly, from Equations (5.42) and (5.44), AC7 and A//, respectively, also must be numerically smaller in the intermediate expansion than in the reversible expansion. In the adiabatic expansion, from a common set of initial conditions to the same final volume, the values of Af7 and A//, as well as the values of the work performed, seem to depend on the path (see summary in Table 5.2). At first glance, such behavior seems to contradict the assumption that U and H are state functions. Careful consideration shows that the difference occurs because the endpoints of the three paths are different. Even though the final volume can be made the same, the final temperature depends on whether the expansion is free, reversible, or intermediate (Table 5.2). 

Isothermal. The procedure used to calculate the work and energy quantities in an isothermal reversible expansion of a real gas is similar to that used for the ideal gas. 

From the first law of thermodynamics, we now can calculate the heat absorbed in the isothermal reversible expansion ... 

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