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Compression of an Ideal Gas As a Classroom Example of Reversible and Irreversible Processes*

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Compression of an Ideal Gas As a Classroom Example of Reversible and Irreversible Processes* Int. J. Engng Ed. Vol. 16, No. 6, pp. 524±528, 2000 0949-149X/91 $3.00+0.00 Printed in Great Britain. # 2000 TEMPUS Publications. Compression of an Ideal Gas as a Classroom Example of Reversible and Irreversible Processes* DONALD R. OLANDER Department of Nuclear Engineering, University of California, Berkeley, CA 94707, USA. E-mail: [email protected] The distinction between reversible and irreversible compression of an ideal gas in a cylinder with a frictionless piston is analyzed quantitatively.Reversible compression is achieved by moving small masses appropriately distributed vertically onto the piston as it descends and compresses the gas.In the past, this example has been used pedagogically in qualitative form only.Reversibility is demonstrated in several ways, most notably by showing the equality of the work done on the system and the work done by the surroundings. INTRODUCTION receiving heat from a reservoir at some arbitrary temperature To. The work done by the imaginary OF ALL THE CONCEPTS in thermodynamics, heat engine represents the irreversibility of the the one that most distresses students in university actual process. courses on this subject is reversibility of a process The purpose of this paper is to analyze a simple and its counterpart irreversibility. Many of the two process used in many textbooks to illustrate rever- dozen or so textbooks dealing with engineering sible and irreversible paths between fixed initial thermodynamics that have been published in the and final states. Specifically, the process is past three decades treat this topic qualitatively, described quantitatively rather than qualitatively, mainly by listing common sources of irreversibility which has been the approach heretofore. The (friction, unrestrained gas expansion, heat quantitative analysis presented here may be too exchange over a non-zero temperature difference, detailed for undergraduate thermodynamics etc.). Also noted as conditions for reversibility are: courses, but it will at least deepen the instructor's understanding of the process. A limitation on the speed at which a process Wang's extension of the classic Carnot theory of takes place: change must be sufficiently slow heat engines (1) is a different example of an that the system maintains internal equilibrium advanced thermodynamic analysis intended for at all times. pedagogical use. Returning the system from its final state to its initial state by the same path must leave both system and surroundings unchanged. ISOTHERMAL COMPRESSION OF AN . Work done by (or on) the system must be equal IDEAL GAS to the work done on (or by) the surroundings. Other texts describe irreversibility as a feature of a The process referred to above takes place in a process that increases the entropy of the system cylinder with a frictionless piston. The system and its surroundings, with equilibration of an contained in the cylinder consists of a simple initially non-equilibrium vapor/liquid mixture in substance such as water or an ideal gas. The an isolated vessel and free expansion of an ideal surroundings provide a fixed external pressure gas into a vacuum as common examples. A third and a means of changing the force on the piston, method, favored by texts intended for mechanical usually by adding or removing weights. The or chemical engineers, defines irreversibility as the thermal surroundings may be either a temperature difference between the work that a process could reservoir for isothermal processes or thermal have produced had it been conducted reversibly insulation for adiabatic processes. and the work actually produced. The portion of Several authors have used this device as a means the entropy change produced by irreversibilities is of demonstrating the distinction between reversible replaced by reversible entropy added to the system and irreversible processes with common initial and as reject heat from an imagined ideal heat engine final states [2±5]. Figure 1 depicts the reversible and irreversible methods of compressing an ideal gas between specified initial and final volumes. In * Accepted 12 June 2000. [2±5], the discussions of these two versions were 524 Compression of an Ideal Gas as a Reversible and Irreversible Processes 525 Fig. 1. Isothermal compression of an ideal gas. qualitative. The objective of the present paper is to compression process, the conditions of external quantitatively demonstrate the following features equilibrium, namely: of the process: T Tsurr and p psurr jmg=A 1 1. In the reversible version, the work done by the surroundings is identical to the work done on are satisfied. In equations (1), Tsurr and psurr are, the system. respectively, the constant temperature and 2. The work done by the surroundings in the constant pressure of the surroundings. The irreversible process is numerically compared to number of small masses added to the piston is the work performed in the reversible process. denoted by j (0 j N), g is the acceleration of gravity, and A is the cross-sectional area of the In both cases shown in Fig. 1, a total mass M is piston. The p-V work done on the ideal gas during added to the top of the piston. The only difference the reversible, isothermal compression from initial between the two methods is the way that the mass and final volumes Vo and Vf is: is placed on the piston's pedestal. In the reversible V0 process, the mass M is divided into a large number V0 N of small masses m such that M=Nm. The small Wrev pdV nRT ln 2 Vf masses are slid on to the piston one at a time from Vf shelves at different elevations. In the irreversible The work done by the surroundings consists of the version, the entire mass M is placed on the piston sum of the p-V work on the piston by the external at once. pressure and the loss to gravitational potential of In both cases, the cylinder contains n moles of the small masses: and ideal gas and is immersed in a constant- temperature bath. The thermal reservoir removes Wsurr psurr V0 À Vf ÁEp 3 heat as the small weights are added to maintain the ÁEp is calculated by summing the changes in the gas at temperature T during the entire reversible potential energy of each small mass as it is slid process. In the irreversible process, only the horizontally onto the stack on the pedestal. initial and final states are at temperature T; Detailed calculation of ÁE given in the appendix these are equilibrium states of the gas whereas p shows that Wrev =Wsurr, so a criterion of the state depicted in the lower middle panel of reversibility is satisfied for this process. Fig. 1 is not. In addition, the process can be reversed by sequentially sliding the small masses off the pedes- Reversible compression tal at their original elevations. When the initial Sliding a small mass onto the pedestal causes the state is recovered, exactly the same amount of heat piston to very slightly compress the gas in the that was released by the compression process will cylinder. Each addition of a small mass approx- have been absorbed during the expansion process. imates an infinitesimal equilibrium stage so that Both system and surroundings will have been the overall process is reversible. At all points in the restored to their original states. 526 D.Olander IRREVERSIBLE COMPRESSION Comparing this equation with equation (2) for Vo/Vf = 3 (as an example) shows that the ratio When the entire mass M is placed on the piston, of the reversible work of compression to the as in the bottom of Fig. 1, the system is not in irreversible work is: equilibrium with the surroundings. The piston W ln 3 rapidly descends, oscillates as the gas acts as a rev 0:55 spring, and eventually settles down to the final Wirr 3 À 1 elevation. Although equation (2) does not apply to this situation (because the process is not internally That is, the surroundings need to supply only reversible), equation (3) does. In this case, the about one half as much work for the reversible process as it does for the irreversible process. potential energy loss ÁEp is easily determined from the elevation change of the large mass, which Finally, simply sliding the large mass off the is directly related to the volume change of the gas piston in the final state in the lower portion of Fig. 1 returns the system (gas plus piston) to its by Áh =(Vo ±Vf)/A. The work done by the surroundings in this irreversible process is: initial state but leaves the surroundings with the large mass at a lower elevation than it was initially. Wirr psurr Vo À Vf Mg Vo À Vf =A Both of these consequences are characteristic of an irreversible process. (Strictly speaking, the p-V work term in this equation does not apply because of internal irreversibililties in the atmosphere of the surround- ings as the piston oscillates to its final resting CONCLUSIONS position. However, the final results are a reason- able representation of the work done to the extent Reversible compression of an ideal gas in a that pf > psurr.) piston-cylinder apparatus by the distributed At the final equilibrium state, the force balance weight method has been analyzed quantitatively. on the piston gives: The objective was to add an analytical structure to this often-used pedagogical tool to aid students in pf À psurr Mg=A thermodynamics courses to better grasp the Combining these two equations and using the concepts of reversible and irreversible processes. ideal gas law yields: For the reversible compression process, the equality of work done on the system with that done by the surroundings is demonstrated. The V0 Wirr pf V0 À Vf pf Vf À 1 ratio of the reversible and irreversible work Vf requirements for the isothermal version of the V0 process is determined as a function of the nRT À 1 4 compression ratio.
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