Research: Science and Education
Pressure–Volume Integral Expressions for Work in Irreversible Processes Eric A. Gislason* Department of Chemistry, University of Illinois at Chicago, Chicago, IL 60680; *
[email protected] Norman C. Craig Department of Chemistry and Biochemistry, Oberlin College, Oberlin, OH 44074
The definition of pressure–volume work (PVW ) in irreversible thermodynamic processes remains a contentious topic. For this reason, many authors restrict the definition of PVW to quasistatic or reversible processes (1–7). Some problems with the definition of irreversible PVW were discussed by Bauman (8) and in follow-up papers (9–12). Historically, there have been two different protocols for defining work, w, in a physico-chemical process (13–15). (In this paper we use the modern convention that w is the work done on the system by the surroundings, so the first law is written ∆E ⫽ q ⫹ w.) One protocol is developed from the perspective of the system and the other from the perspective of the surroundings. We fully developed (15) both protocols and showed that either one can be used to define PVW under appropriate circumstances. However, a major conclusion from our analysis was that the values of w can be different for the two protocols if the process is irreversible. This difference has not been recognized in the past. A recent paper by Bertrand (16) references various integral formulations of PVW for a piston moving irreversibly against a gas. A typical apparatus is shown in Figure 1. In this paper the system is always the gas beneath the piston, and everything else is part of the surroundings. As Bertrand points out, w has sometimes been presented as w = −∫ Pint dV
(1)
and sometimes as
w = −∫ Pext dV
(2)
Here Pint represents the pressure of the gas. (Bertrand writes Pgas in place of Pint.) If the piston is stationary, Pint ⫽ P, the equilibrium value. By contrast, Pext is an external pressure (that is, a pressure defined in the surroundings that opposes the motion of the gas in an expansion and supports the motion in a compression). Bertrand concludes that the proper definition of PVW is eq 1 but written as w = −∫ Pop dV
(3)
where Pop is the instantaneous pressure exerted on the piston by the gas. Pop was denoted Poperating by Bertrand. A number of authors agree with this definition (17–22). The integrals in eqs 1–3 must be evaluated over the actual path taken by the piston. Pop is distinguished from Pint to recognize that the piston is, in general, moving. If the piston is moving rapidly, Pint may not be well defined. By contrast, Pop is always well defined, even though it may not be known. If the instantaneous velocity of the piston during expansion, u, is not too fast (23), then
Pop ≈ Pint 1 −
8 π
u v
+
8 π
u v
2
(4)
Figure 1. A constant-temperature apparatus. The system (an ideal gas) initially has a pressure of 4 atm and is contained beneath the piston held by the catch at height h1. The apparatus is in thermal contact with the calorimeter at temperature T. The region above the piston is open to the atmosphere, which exerts a pressure on the piston of 1 atm. The mass of the piston is such that, when the catch is released but before the piston begins moving, the piston plus the air above the piston exert a total pressure on the gas of 2 atm. The greater pressure of the gas lifts the piston, which then oscillates around h2 until coming to rest. The gas has a final pressure of 2 atm. There is, in general, friction between the piston and the cylinder as the piston moves.
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Here is the average speed of the gas molecules, and Pint is the spatially constant, instantaneous pressure of the gas. The result above is correct through second order in u兾. Note that u > 0 if the gas is expanding, and u < 0 if the gas is being compressed. The second term with the minus sign exerts a drag on the piston and guarantees that a piston oscillating around a fixed final point will come to rest. Our analysis (15) has shown, in agreement with Bertrand, that eq 3 is one proper way to define PVW. Here we examine the related question of the definition of PVW in terms of Pext. Bertrand’s paper (16) suggests that there is no satisfactory way to use eq 2 to define PVW, but we disagree. In this present paper we show that there are two appropriate definitions of Pext. One is the pressure exerted by the piston on the system; this leads to a result for w equivalent to eq 3. A second definition of Pext is the total pressure exerted by the surroundings (i.e., excluding the system and excluding any frictional forces) on the piston; this leads to a result for w in the form of eq 2. This paper also treats the contribution of the atmosphere to the work term in many processes. Finally, we give an example that elaborates on a number of the points made here.
the end. In that case the theorem gives h2
0 =
∫ ( f op
(5)
The integral of mg dh is the change in potential energy of the piston, and, as such, its value is independent of the path of the piston. Hence, eq 5 can be rewritten as h2
(
)
mg (h 2 − h1 ) = ∫ f op + Ffr + f 0 d h
System-Based Definition of Work We now try to write the PVW in the form of eq 2. As stated earlier, there are two different protocols for defining w (13–15). The first defines w as the energy change in the system during the process that is potentially convertible into lifting a mass (if the energy change is negative) or lowering a mass (if the energy change is positive) in the earth’s gravitational field (15, 18). This definition leads to the system-based definition of PVW (15–22): h2
V2
w ( sys-based ) = − ∫ f op d h = − ∫ Pop dV h1
V1
V2
= −mg (h2 − h1) − ∫ P0 dV + V1
h2
∫ Ffr d h
a
Here V1 and V2 are, respectively, the initial and final volumes of the system, and the final equality comes from eq 6. In the last line of eq 7 the first two terms clearly have the form of work terms. By contrast, the last term represents thermal energy produced by the motion of the piston against the frictional force Ffr, and, therefore, does not represent work. We conclude that the integral of ᎑fop dh is potentially convertible into lifting a mass in the earth’s gravitational field, but 100% efficiency is only achieved when Ffr ⫽ 0. If Ffr ⫽ 0 then some of the energy expended by the system is “wasted” by conversion into thermal energy. The system, of course, is not “aware” of any frictional forces. From its point of view it has done an amount of PVW equal to the value in eq 7. As discussed earlier, one common way to define Pext is as the instantaneous pressure exerted by the piston on the system. Newton’s third law states that if fop is the force exerted by the gas on the moving piston, then ᎑fop is the force exerted on the gas by the piston. This relationship is well known (23–26). Thus, the expression in eq 7 is equivalent
System-Based Definition a
Surroundings-Based Definitionb
Pint
Instantaneous pressure of gas on piston ⫽ Pop
Not applicable
Pext
Instantaneous pressure of piston on gas ⫽ Pop
Total pressure of surroundings on piston ⫽ mg/A ⫹ P 0
500
The system is the gas: everything else belongs to the surroundings.
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(7)
h1
Table 1. Appropriate Pressure Definitions for Computing Pressure–Volume Work Pressure Type
(6)
h1
General Analysis of Pressure-Volume Work Consider the simple case of a gas (the system) contained below a piston that rises during a process from height h1 to h2 (see Figure 1). We assume there is a fluid (typically the atmosphere) above the piston whose instantaneous pressure P0 on the piston may vary as the piston height varies. In that case, there are four different forces that act on the piston (13, 15). The first is fop, the instantaneous force exerted by the gas on the piston. Clearly fop > 0, since it is directed upwards. The quantity fop兾A, where A is the cross sectional area of the piston, equals Pop. If the speed of the piston is not too fast, then eq 4 can be used to estimate fop and Pop. The second force is the force due to gravity, ᎑mg, where m is the mass of the piston and g is the acceleration due to gravity. The third force on the piston, f0 < 0, is exerted downward by the fluid and equals f0 ⫽ ᎑P0 A. Finally, the fourth force is Ffr, which includes all frictional forces exerted on the piston by other parts of the surroundings, such as the cylinder. Note that Ffr < 0 when the piston is moving up and Ffr > 0 when the piston is moving down. This property of opposing the motion in both expansion and compression distinguishes frictional forces and pressures (Pfr = Ffr兾A) from Pint and Pext. It is well known (13) that the total work done on the piston by all forces in a process equals the net increase in kinetic energy ∆Ekin of the piston. In this paper we restrict the discussion to processes where the piston is at rest at the start and
)
− m g + Ffr + f 0 d h
h1
b
See the experiment shown in Figure 1 and eq 14.
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to eq 2 with
Pext =
f op
(8) A We conclude that when the system-based definition of work is used, V2
w (sys-based ) = − ∫ Pext dV
(9)
V1
with Pext given by eq 8. In this case both eqs 2 and 3 give the same value of PVW (see Table 1). A number of authors (27– 32) have given eq 9 as the definition of PVW.
This result uses the fact that the change in volume of the system, ∆V ⫽ V2 ⫺ V1, is the negative of the volume change of the fluid, ∆Vi ⫽ V2i ⫺ V1i. The other common arrangement for the fluid is when it is the atmosphere. In that case, we have shown elsewhere (33) that the work term should be viewed as a change in the potential energy of the atmosphere, because the volume change ∆V ⫽ V2 ⫺ V1 of the system requires the lifting (or lowering) of a volume ∆V of air against the pull of gravity. Consequently, eq 10 gives (33) w (surr-based ) = −∆E pot (atm ) V2
= − Patm ∆V = − ∫ P0 dV
Surroundings-Based Definition of Work The second protocol for defining work for the process in Figure 1 identifies w as the negative of the energy change in the surroundings that is potentially convertible into lifting (or lowering) a mass in the earth’s gravitational field (13). This definition leads to the surroundings-based definition of work. In general, several parts of the surroundings, such as the piston and fluid above the piston, contribute to the PVW. The contribution from part i of the surroundings is (13)
where P0 ⫽ 1 atm in this case. Because this work term corresponds to a change in the potential energy of the atmosphere, it is always independent of the path that the system took to reach the final state as well as independent of the piston motion. The total work during the process is the sum of eq 11 and either eq 12 or eq 13. In both cases this can be written as (see Table 1) V2
V2i
w (surr-based )i = − ∆E pot, i +
∫ Pi dV
V1i
V2
= − ∫ mg d h = − ∫ V1
mg dV A
(11)
There are two common experimental arrangements for the fluid above the piston. In one case the combined volume of the fluid plus system is a constant (8, 13), so that an expansion of the system corresponds to a compression of the fluid and vice versa. In that case, the contribution of the fluid to w is obtained from the second term in eq 10 as V2
w (surr-based, fluid ) = −
∫ Pext dV
V1
Pext =
mg A
(14)
+ P0
Two qualifications must be noted. The first is that eq 14 is only valid if the piston is at rest at the beginning and end of the process. The second is that, unlike the case in eq 9, where the integral is done over the actual path of the piston, the integral in eq 14 is done along the appropriate reversible path. A number of authors (13, 15, 34–42) define Pext as the total pressure exerted on the piston by the rest of the surroundings. (Frictional forces are excluded from the definition of Pext.) In that case the PVW is given by eq 14. Sample Calculation: Constant-Temperature Process with Frictional Forces
w (surr-based, gravity ) = −mg (h2 − h1 )
h1
w (surr-based ) = −
(10)
Here ∆Epot,i is the change in potential energy of part i, V1i and V2i are the initial and final volumes, respectively, of the part, and Pi is the pressure of part i. (Here, as before, we have assumed that ∆Ekin,i ⫽ 0 for all parts.) The result in eq 10 is valid for the cases where part i undergoes a constant volume or constant pressure or constant temperature or reversible process (13). Then the integral in eq 10 is taken over the appropriate reversible path (e. g., along a reversible constant temperature path for a constant temperature process). In every case the integral represents the change in some state function of part i (13). The contribution to w of gravity acting on the piston comes from the first term in eq 10 as
h2
(13)
V1
∫ P0 dV
(12)
Here we calculate PVW for the experiment shown in Figure 1. The system is n moles of an ideal gas with an initial pressure of 4 atm contained in a cylinder below a piston locked in place by a catch. The mass m of the piston is such that mg兾A ⫽ 1 atm. The cylinder above the piston is open to the air with pressure 1 atm so that at equilibrium the total pressure down on the gaseous system is 2 atm. The cylinder sits in a calorimeter at temperature T. When the catch is removed, the piston moves up and oscillates about its final position before finally stopping. All parts of the experiment have the same initial and final temperature, T. The initial and final volumes of the system are given by nRT兾(4 atm) and nRT兾(2 atm), respectively, so ∆V ⫽ nRT兾(4 atm).
V1
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The motion of the piston is unknown, so it is not possible to compute the integral of Pop dV in eq 7. Thus, we cannot get a result for w using system-based definitions. This point is noted in Table 2. By contrast it is possible to obtain w using the surroundings-based definitions in eq 14. The result for the effect of gravity on the piston is
used by Kivelson and Oppenheim (18). The last line in eq 7 simplifies to the sum of the values in eqs 15 and 16. Consequently, V2
w (sys-based ) = − ∫ Pop dV = − V1
nR T 2
which equals the surroundings-based result in Table 2.
mg w (gravity ) = −m g ∆ h = − ∆V A nRT nRT = − = − [1 atm ] 4 atm 4
(15)
The PVW term for the air at 1 atm above the piston is (see eq 13) V2
w (air ) = − ∫ P0 dV = − (1 atm ) ∆V
Sample Calculation: Constant-Temperature Process with Known Friction If Ffr ⫽ 0, it is not possible to use the method of the previous section. Thus, we are not able to compute w(sysbased). The result for the surroundings-based PVW in eq 17 remains the same, however. It is of pedagogical interest to evaluate w(sys-based) when an assumed amount of thermal energy is created by friction in the process. We assume for the sake of discussion that
(16)
V1
h2
nR T nRT = − [1 atm ] = − 4 4 atm
∫ Ffr dh
= −
h1
nRT 16
(19)
In this case, the value for the PVW can be obtained from the last line in eq 7 as
Adding the results in eqs 15 and 16 gives
V2
w (sys-based ) = −
V2
nRT w (surr-bassed ) = − ∫ Pext dV = − 2 V
(17)
1
This is recorded in Table 2. Sample Calculation: Constant Temperature, No Friction Process If Ffr ⫽ 0, then the PVW using system-based definitions can be determined even though we cannot do the integral of Pop dV. Instead, we evaluate each term in the last line of eq 7. This indirect method for the friction-free case was first
∫ Pop dV
=
V1
−9 nRT 16
b
Discussion and Summary A number of definitions exist for PVW in a process. It is now agreed (15, 16) that eq 3 is the appropriate way to define w using system-based definitions, where Pop is the in-
w, Surroundings-Based
w, System-Based
Ffr ⫽ 0, unknown
᎑nRT/2
Unknown
Ffr ⫽ 0
᎑nRT/2
᎑nRT/2
᎑nRT/2
᎑9nRT/16
h1
冮F
fr
dh⫽ ᎑nRT/16
c
h2 a
502
The calculations are descibed in the text.
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b
Ffr is the frictional force between the moving piston and the cylinder.
•
(20)
This result is entered in Table 2. We see that the values of w(surroundings-based) and w(system-based) are different. This difference is common whenever there is friction between moving parts of the surroundings (15).
Table 2. Sample Calculationsa for Constant–Temperature Process, by Condition Frictional Force, Ffr
(18)
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c
Assumed value.
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stantaneous pressure exerted by the gas on the piston. By contrast, eq 2 is less often used. Part of the problem is that it is not obvious how to define Pext. Nevertheless, we have shown here that eqs 2 and 3 are equivalent ways of defining PVW, when using system-based definitions, provided that Pext is defined as the instantaneous pressure exerted on the gas by the moving piston. In that case, Pext = Pop . The situation is different when using surroundings-based definitions of work. Then the PVW can be written in the form of eq 2 with Pext defined in eq 14. In general, Pext represents the sum of all external pressures including the “pressure” due to gravity exerted on the piston by the surroundings. This paper has focused on the proper ways to define thermodynamic PVW using eqs 2 and 3. Both system-based and surroundings-based definitions have been used, and both sets of definitions give rise to appropriate expressions summarized in Table 1. These results do not mean that both sets of definitions are equally useful in developing thermodynamics. System-based definitions have the intuitive advantage that they directly relate work and heat to energy changes in the system. When it comes to actually carrying out thermodynamic experiments or calculations, however, the present authors strongly prefer surroundings-based definitions for many reasons (15). One of the most important in the context of this paper is the ability to calculate w(surroundings-based) in many irreversible processes where w(system-based) cannot be determined (see Table 2). Advocates of system-based treatments tend to focus on reversible processes despite the fact that all real processes are irreversible. Surroundings-based treatments, on the other hand, can be applied equally well to irreversible and reversible processes. Literature Cited 1. Planck, M.; Ogg, A. A Treatise on Thermodynamics, 3rd ed.; Dover: New York, 1945; p 55. 2. Klotz, I. M. Introduction to Chemical Thermodynamics; W. A. Benjamin: New York, 1964; p 45. 3. Pippard, A. B. Elements of Classical Thermodynamics for Advanced Students of Physics; Cambridge University Press: Cambridge, 1966; p 20. 4. Zemansky, M. W. Heat and Thermodynamics: An Intermediate Textbook, 5th ed.; McGraw-Hill: New York, 1968; p 55. 5. Callen, H. B. Thermodynamics, 2nd ed.; Wiley: New York, 1985; p 19. 6. Mallinckrodt, A. J.; Leff, H. S. Am. J. Phys. 1992, 60, 356– 365. 7. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002; p 41. 8. Bauman, R. P. J. Chem. Educ. 1964, 41, 102–104. 9. Bauman, R. P. J. Chem. Educ. 1964, 41, 675, 676–677. 10. Chesick, J. P. J. Chem. Educ. 1964, 41, 674–675. 11. Kokes, R. J. J. Chem. Educ. 1964, 41, 675–676. 12. Mysels, K. J. J. Chem. Educ. 1964, 41, 677. 13. Gislason, E. A.; Craig, N. C. J. Chem. Educ. 1987, 64, 660– 668.
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14. Craig, N. C.; Gislason, E. A. J. Chem. Educ. 2002, 79, 193– 200. 15. Gislason, E. A.; Craig, N. C. J. Chem. Thermo. 2005, 37, 954– 966. 16. Bertrand, G. L. J. Chem. Educ. 2005, 82, 874–877. 17. Fermi, E. Thermodynamics; Dover: New York, 1956; p 5. 18. Kivelson, D.; Oppenheim, I. J. Chem. Educ. 1966, 43, 233– 235. 19. Battino, R.; Wood, S. E. Thermodynamics: An Introduction; Academic: New York, 1968; p 96. 20. Reid, C. E. Chemical Thermodynamics; McGraw-Hill: New York, 1990; p 17. 21. DeVoe, H. Thermodynamics and Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; p 45. 22. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002; p 43. 23. Bauman, R. P.; Cockerham, H. L., III. Am. J. Phys. 1969, 37, 675–679. 24. DeVoe, H. Thermodynamics and Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; p 421. 25. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002; p 40. 26. Canagaratna, S. G. Am. J. Phys. 1978, 46, 1241–1244. 27. Steiner, L. E. Introduction to Chemical Thermodynamics, 2nd ed.; McGraw-Hill: New York, 1948; p 38. 28. Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McGraw-Hill: New York, 1961; p 15. 29. Blinder, S. M. Advanced Physical Chemistry: A Survey of Modern Theoretical Principles; Macmillan: New York, 1969; p 257. 30. Rock, P. A. Chemical Thermodynamics: Principles and Applications; Macmillan: New York, 1969; p 21. 31. Redlich, O. Thermodynamics: Fundamentals, Applications; Elsevier: Amsterdam, 1976; p 13. 32. Russell, L. D.; Adebiyi, G. A. Classical Thermodynamics; Saunders: Fort Worth, TX, 1993; p 170. 33. Gislason, E. A.; Craig, N. C. J. Chem. Educ. 2006, 83, 885–889. 34. Moore, W. J. Physical Chemistry, 4th ed.; Prentice Hall: Englewood Cliffs, NJ, 1972; p 31. 35. Nash, L. K. Chemthermo: A Statistical Approach to Classical Chemical Thermodynamics; Addison-Wesley: Reading, MA, 1972; p 51. 36. De Heer, J. Phenomenological Thermodynamics with Applications; Prentice-Hall: Englewood Cliffs, NJ, 1986; p 31. 37. Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: New York, 1995; p 20. 38. Dykstra, C. E. Physical Chemistry: A Modern Introduction; Prentice Hall: Upper Saddle River, NJ, 1997; p 73. 39. McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997; p 767. 40. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford: New York, 2000; p 374. 41. Raff, L. M. Principles of Physical Chemistry; Prentice-Hall: Upper Saddle River, NJ, 2001; p 70. 42. Atkins, P. W.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman: New York, 2002; p 38.
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