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Thermodynamic Calculation of Work for Some Irreversible Processes Gary L. Bertrand Department of Chemistry, University of Missouri–Rolla, Rolla, MO 65409-0010;
[email protected] Over 40 years ago, Robert P. Bauman published an article in this Journal (1) regarding the work of isothermally compressing a perfect gas. His discussion was based on the statement, “Work is defined to be the integral w = ∫f dl where f represents the force opposing the motion.” (added by GLB: “The displacement of the force is represented by dl. Bauman used the convention in which positive work is done by the system.”) Bauman concluded, “In any compression at speeds small compared to the speed of sound, the force opposing the motion is very clearly the pressure of the gas itself (added by GLB: ‘integrated over the area of the piston’).” He further states that w = ∫PsurrdV (note reference to sign above) is correct only when Psurr opposes the motion, which would apply to expansion. Later that year, there were exchanges of letters on this subject by John P. Chesick (2), Richard J. Kokes (3), and Bauman (4, 5), moderated by Karol J. Mysels (6). Chesick and Kokes opposed Bauman’s presentation, concluding that the calculation of work should involve Psurr for both compressions and expansions. Kokes raised the point that the motion of the piston during a compression causes an excess pressure on the face of the piston as a result of imparting a greater change in momenta to molecules striking the face and to an increase in the number of collisions of molecules with the face of the piston. Bauman’s analysis of this effect suggests that a modified value for the pressure should be used in calculating the work of compressing a perfect gas Pcompression = Pgas 1 +
u vx
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These articles published in three issues of this Journal, suggest a broad range of relationships for calculating incremental work for an irreversible process on a perfect gas. In fairness, the reader is reminded that the authors of the articles in this Journal were only discussing isothermal compressions. The general formulation of protocols based on their considerations should be attributed this author, not to the original authors.
Protocol 1 w = −∫ Popp dV in which Popp is the pressure opposing the process. This is the definition given by Rossini (8). This gives
w = −∫ Pgas dV for compression, and
w = −∫ Pext dV for expansion. For compression, Bauman suggested a correction based on the velocity of the piston relative to the average velocity of the molecules.
Protocol 2 If the process is stopped by factors within the system
2
w = −∫ Pext dV
in which u is the velocity of the piston, vx is the average velocity of the gas molecules moving toward the piston,1 and Pgas is the “static pressure”, which is apparently the pressure the gas would have if the piston were not moving. This is clearly compatible with Bauman’s original statement. At this point Mysels halted the discussion, thanking the authors for their contributions to an interesting discussion and essentially stated that the matter remained unresolved. Two years later, Kivelson and Oppenheim (7) re-analyzed this matter by considering a quasi-static process, in which the piston is allowed to move in infinitesimal steps, allowing time for a return to thermal equilibrium with a surrounding bath between steps.2 The authors noted that in an irreversible process, the piston develops a minute quantity of kinetic energy between stops. They concluded that if this kinetic energy is given to the surroundings (external stops) differential work is related to the pressure of the system (Pgas ) multiplied by the change in volume. If the kinetic energy is given to the system (internal stops), differential work is related to the external pressure (Pext, which has been called Psurr above) multiplied by the change in volume. 874
Protocols for Calculating Thermodynamic Work
and if the process is stopped by external factors
w = −∫ Pgas dV
Protocol 3 w = −∫ Pext dV irrespective of the direction of the process.
Protocol 4 One chemical engineering text (9) suggests yet another protocol
w = −∫ Pgas dV However, in at least one example (9, p 69), calculations are presented in terms of Protocol 3 above.
Protocol 3 is used in most physical chemistry texts (10, 11) and some chemical engineering texts (12, 13). Protocol
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1 is used in some chemical engineering texts. In all of these protocols, Pgas is taken as the “static” pressure, which might be calculated from the properties (n, U, V) of the gas. All of these protocols agree for the calculation of reversible work. Differences arise in the application to irreversible processes. Conceptual Consideration Physical chemistry texts usually discuss the Joule experiment in introducing the statement that for a perfect gas, the energy depends only on temperature (and the amount of gas) U = U(T, n). When a perfect gas expands into a vacuum, there can be no work. Conceptually, the expansion of a gas against a weightless, frictionless piston is equivalent to expanding into a vacuum (Pext = 0), with the movement stopped by external factors. Protocols 1 and 3 are consistent with this concept, while protocols 2 and 4 appear to be inconsistent. Experimental Considerations The discussions above address isothermal processes on perfect gases. Experimental verification will require performing a quasi-static process while measuring the heat that must be transferred to the bath to maintain constant temperature. The performance of this experiment would be extremely difficult and well beyond our ability to demonstrate the proper relationships in the classroom or undergraduate laboratory. However, there is an experiment that is capable of shedding some light on the matter. The adiabatic gas law apparatus of Pasco Scientific allows a gas to be rapidly compressed to approximately half of its initial volume while simultaneously recording temperature, pressure, and volume. Because of the speed of the compression, the process is assumed to be adiabatic. The equipment manual describes the calculations as if this were a reversible process. The manual also describes the corresponding expansion experiment, but cites complications such that “the expansion data do not give good quantitative results”. McNairy (14) describes experiments with this apparatus, “The adiabatic process is simulated by rapidly compressing gas in a cylinder by forcefully pushing down on a lever that is attached to a piston in the cylinder, ...”. He refers to the “approximately 100-ms period of the stroke” with which the gas is compressed from approximately 240 cm3 to 100 cm3. While the author does not report temperatures for this experiment and there is no measurement of the applied pressure, the experiment is clearly performed in an irreversible manner. There is no indication that the applied pressure is constant, but the nature of the experiment leads one to expect that the applied pressure is probably much closer to the final pressure than to the pressure of the gas throughout the compression. The results obtained for argon, nitrogen, and carbon dioxide are shown to be consistent with the expected result for a reversible adiabatic compression. The experimental results show argon compressed from 83 kPa and 2.32 × 10᎑4 m3 to 334 kPa and 1.00 × 10᎑4 m3, in excellent agreement with the calculations for reversible compression. The final volume calculated for compression at constant external pressure of 334 kPa is 1.27 × 10᎑4 m3 . These results are seen to be consistent with protocols 1 and 4 above and inconsistent with protocol 3. For compariwww.JCE.DivCHED.org
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son with protocol 2, the reader must decide whether the process is stopped by something inside or outside of the system. Since the movement of the piston is stopped when the pressure of the gas becomes equal to the applied pressure, it is difficult to attribute the stoppage to an external agency. On this basis, the results appear to be inconsistent with protocol 2. This apparatus cannot be used to differentiate between the protocols above when applied to expansion, owing to inadequacy of the design. The experiment clearly demonstrates the temperature drop associated with the expansion, but frictional heat raises the temperature of the cylinder walls so that the full temperature drop in the absence of friction is not realized. However, extensive experiments have been conducted in the cloud chamber at UMR’s Cloud and Aerosol Science Laboratory. In this chamber, argon gas (either alone or saturated with the vapor of a liquid within the chamber) is rapidly expanded from near 2 atm pressure to about 1 atm pressure by a mechanically driven piston in about 200 to 300 milliseconds. Observations in the center of the chamber are consistent with calculations for a reversible expansion. This observation is consistent with protocols 2 and 4 (the movement of the piston is completely controlled by forces outside of the system), and inconsistent with protocols 1 and 3 above. A Suggestion for Resolution of the Problem In these conceptual and experimental examples, each of the four protocols appears to fail in at least one case. Protocol 1 can be identified as due to Rossini and protocol 2 as due to Kivelson and Oppenheim specifically for a quasi-static process. Protocol 3 appears in most physical chemistry texts in which irreversible compressions and expansions are discussed. This author has only found the author cited in ref 9 to espouse protocol 4. To resolve this matter, we must consider the definition of work. Castellan (15) cites J. A. Beattie, “In thermodynamics, work is defined as any quantity that flows across the boundary of a system during its change in state and is completely convertible into the lifting of a weight in the surroundings.” In the case of mechanical work, this is generally considered in terms of a force operating across the boundary w = −∫ f operating dl In the case of a mass in the surroundings connected by a fiber over an arrangement of pulleys to a mass in the system, the operating force ( foperating ) is equal to the tension on the fiber at the point that it contacts the system while the masses are in motion. If the pulleys and the fiber are weightless, and the arrangement is free of friction, the same operating force is acting on the mass in the surroundings. The imbalance of the gravitational forces on the unequal masses is compensated by the equal and opposite accelerations, such that if a mass (m´ ) in the surroundings lifts or lowers a mass (m) in the system, the operating force is
f operating = 2
m⬘m g ( m⬘ + m)
in which g is the gravitational constant (see the Supplemental MaterialW).
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In the case of a fluid
w = −∫ Popeerating dV in which Poperating is the pressure exerted on the face of the piston while it is moving. Levine (16) uses this definition for irreversible work with Psurf (the pressure exerted on the face or surface of the piston) instead of Poperating. DeVoe (17) gives a similar equation, with Pb representing the pressure at the moving boundary. Following the reasoning of Bauman and Kokes, the operating pressure of a compression of a perfect gas may be approximated as 2
u vx
Poperating = Pcompression = Pgas 1 +
Conclusions
Application of the same considerations to an expansion gives
Poperating = Pexpansion = Pgas 1 −
u vx
2
For most processes on gases, the velocity of the piston is very small (less than 1% in the experiments cited) compared to the average velocity of the molecules, and the operating pressure may be approximated as the pressure of the gas. In the case of the conceptual experiment of expansion against zero external pressure, the weightless frictionless piston would be quickly accelerated to a velocity at least as great as the average velocity of the molecules. Carrera-Patiño (18) has provided a detailed analysis for adiabatic expansion of monatomic gases at constant piston velocity. The operating pressure would rapidly drop to zero, and no work would be done. This definition of work in terms of the operating pressure is thus consistent with the cited experiments and the conceptual experiment. The Experiment of Desormes and Clement The issue of reversible versus irreversible work was addressed (19) in terms of the classic experiment, which has been performed in physical chemistry laboratories for decades to determine the Cp兾Cv ratio for gases. A gas is compressed in a large container fitted with a stopper and connections for measuring the internal pressure. The container is equilibrated to ambient temperature, then the stopper is quickly removed and then replaced, allowing the gas to quickly expand against the atmosphere. The gas is then trapped at atmospheric pressure and the lowered temperature, then the pressure increases as the gas returns to ambient temperature. The difference between the final pressure and atmospheric pressure can be used to calculate the temperature drop achieved in the experiment. The question addressed was whether this should be treated as a reversible (Poperating = Pgas ) adiabatic process or as an (irreversible) constant pressure (Poperating = Psurr ) process. The matter was shown to be purely academic, since the two calculations yield almost identical results for the pressure difference (less than 0.1 bar) used in the experiment.
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This is a case in which the imaginary boundary between the gas and the surrounding area moves with a velocity that is significant in comparison to molecular velocities. Calculations have been performed using operating pressures based on reasonable approximations of the piston velocity (see the Supplemental MaterialW). Calculated temperatures fall between those predicted for the reversible (Poperating = Pgas ) and constant-pressure (Poperating = Psurr ) cases above. By extending these calculations to pressure differences of 1 bar or more, some interesting differences may be seen. For air or argon expanding against air, the calculated temperature is near the average of the temperatures calculated for the reversible and constant-pressure processes. For helium expanding against air, the calculated temperature is closer to that calculated for the reversible process.
The idea that thermodynamic work depends on the speed of the process is not new. The concept of a reversible process carried out in a series of infinitesimal steps suggests an extremely slow process. It is generally recognized that the effective voltage of a battery decreases with increasing current, with the reversible voltage represented by the value measured (or extrapolated) at zero current. The same principle can be applied to mechanical processes. This study suggests that the definition of mechanical work should be w = −∫ f operating dl with foperating representing the force that is exerted on the system at its boundary while the movement is occurring. The application of this definition to a system confined by a piston gives w = −∫ Poperating dV with Poperating representing the pressure exerted on the face of the piston as the piston moves. For processes involving the compression or expansion of perfect gases, these considerations lead to the approximation Poperating = Pgas 1 ±
u vx
2
; vx 2 =
RT M
The positive sign applies to compressions and the negative sign to expansions, and M represents the molar mass of the gas and u is the absolute value of the velocity of the piston. This approximation differs only slightly from that of Bauman and Cockerham (20). The normal treatment of reversible processes is clearly retained in the limit of zero velocity of the piston. This treatment retains the relationship that work performed at a finite rate is always greater than reversible work, though the difference may be negligible. The relationship for a free expansion is retained in that the velocity of the (imaginary) piston approaches the average velocity of the molecules and the operating pressure is effectively zero. For most prob-
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lems involving piston speeds of a few meters兾sec, the approximation that the operating pressure differs only slightly from the pressure of the gas will suffice. For high-speed compressions, this approximation is expected to fail near the speed of sound, and possibly well below this point. Acknowledgments The author thanks Howard DeVoe of the University of Maryland for many helpful suggestions and discussions. Helpful discussions were contributed by colleagues at the University of Missouri–Rolla from departments of physics, mathematics, chemical engineering, mechanical engineering, and aerospace engineering. Supplemental Material Additional equations for lifting and lowering masses and for calculations on the experiment of Desormes and Clement are available in this issue of JCE Online.
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complished. It is important to note that this is an irreversible process if there is a pressure difference across the piston, irrespective of the speed of the process.
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Notes 1. Bauman estimated that the effect of the movement of the piston on the change of momentum per molecule would contribute a factor of (1 + u兾vx ) to the pressure on the face of the piston, and that the effect on the number of collisions would add a “similar factor”. He used the relationship
13. 14. 15.
Pcompression = Pgas(1 + 2u兾vx ) for piston velocities that are much smaller than the velocity of the molecules. In a later article with Cockerham (20) the correction was given as [1 − (8α兾π) + (8α2兾π)] in which α is the ratio of the piston velocity to the average velocity of the gas molecules. This was applied to both compressions and expansions with the algebraic sign of the velocity of the piston positive for expansion and negative for compression. 2. While not stated by the authors, this appears to be the only way that a truly isothermal compression or expansion could be ac-
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16. 17. 18. 19. 20.
Bauman, R. P. J. Chem. Educ. 1964, 41, 102. Chesick, J. P. J. Chem. Educ. 1964, 41, 674. Kokes, R. J. J. Chem. Educ. 1964, 41, 675. Bauman, R. P. J. Chem. Educ. 1964, 41, 675. Bauman, R. P. J. Chem. Educ. 1964, 41, 676. Mysels, K. J. J. Chem. Educ. 1964, 41, 677. Kivelson, D.; Oppenheim, I. J. Chem. Educ. 1966, 43, 233. Rossini, F. D. Chemical Thermodynamics; John Wiley & Sons: New York, 1950. Sandler, S. I. Chemical and Engineering Thermodynamics, 3rd ed.; John Wiley & Sons: Newark, NJ, 1999; p 31. Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley & Sons: New York, 2001; p 30. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 3rd ed.; Houghton Mifflin: Boston, 1999; p 59. Smith, J. M.; van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 6th ed.; McGraw-Hill: New York, 2001; pp 35–36. Winnick, J. Chemical Engineering Thermodynamics; John Wiley and Sons: New York, 1997; pp 20–21. McNairy, W. W. The Physics Teacher 1996, 34, 178. Castellan, G. W. Physical Chemistry, 3rd ed.; Addison-Wesley: Reading, PA, 1983; p 104. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995; p 42. DeVoe, H. Thermodynamics and Chemistry; Prentice-Hall: Upper Saddle River, NJ, 2001; p 45. Carrera-Patiño, M. E. J. Chem. Phys. 1988, 89, 2271. Bertrand, G. L.; McDonald, H. O. J. Chem. Educ. 1986, 63, 252. Bauman, R. P.; Cockerham, H. L., III. Am. J. Phys. 1969, 37, 675.
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