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The Efficiency of Reversible Heat Engines The Possible Misinterpretation of a Corollary to Carnot's Theorem Kurt Seidman and Thomas R. Michallk Randolph-Macon Woman's College, Lynchburg, VA 24503
A Carnot cycle employs four stages: two isotherms and two adiabats. and all staees are reversihle. Carnot's theorem states that, if we con&uct a heat engine that is to operate between two temperature reservoirs, then that engine will have the maximum possible efficiency if it operates via a Carnot cvcle. If the hieh temperature reservoir is a t a tempemture-of TH and the lowtemperature reservoir is a t a temperature of TL, then the efficiency of an engine that employs a Carnot cycle and operates between those reservoirs is given by the standard expression,
Equation 1 presumes the temperatures TH and TLare exmessed with the absolute temoerature scale. This theorem is straightforward and is seldoisubject t o confusion or misinterpretation. There is a corollary t o this theorem that, unfortunately, may be subject to a great deal of confusion and misinterpretation. Fermi ( I ) states the corollary as follows: If there are several cyclic heat engines, some of which are reversible, operating around cycles between the same temperatures TH and TI,, all the reversible ones have the same efficiency, while the nonreversible ones have efficiencies which can never exceed the efficiency of the reversihle engines. For later reference, let us call this statement A. A statement very similar to this can be found in one of the more popular general physics texts (2) as well as a number of standard phvsical chemistrv texts (3.4). From this statement. it is quite easy to conclude that any reversihle engine thatoperates between the same two temperatures as an engine that operates with a Carnot cycle musr have the same e?ficiency as the Carnot engine. This conclusion l'ollows directly from the statement that all reversible engines operating between the same two temperatures have the same efficiency. This conclusion is technically correct when properly interpreted, but i t is subject to misinterpretation, not only by students, but also bv those who teach them. No less an authoritv on thermodynamics than Atkins makes just such a mistake when he asserts that a reversible Stirling cycle has the same efficiency as that of a Carnot cycle operating between the same temperatures (5).We will now consider two examnles to illustra& the problems that can arise. Example 1 Consider the followingthree-stage cycle in which all stages are reversible. Srope I l b o mdes ofa monatomic ideal gas ((: = 1.5 XI undergo an isothermnl compression at a temperature of RUO K from a pres. sure oi 1.00 atm to a pressure of 2.00 ntm. It can be readily deter208
Journal of Chemical Education
Figure 1. P-Vdiagram for a threestage reversible cycle.
mined that the volume of the gas will decrease from 49.2 L to 24.6 L durine the - ~ comnression. Stage 2: The temperature of the gas is isobarically raised from 300 K to 424 K. The volume of the gas at the end of this isobaric process will be 34.8 L. Stage 3: The gas is brought back to its initial state along a reversible path defined by PV = k, where k is a constant.
- ~~~~~.
AP-Vdiagram for this cycle is shown in Figure 1.Since this cycle is reversihle and appears to operate between two temperatures (TH = 424 K and TL = 300 K) we might be tempted to jump t o the conclusion that this cycle should have the same efficiency as that of aCarnot engine operating between the same two temperatures. From eq 1 we would conclude that the efficiency of this cycle should be 0.292. However, suppose that we determine the efficiency of this cycle using the alternative equation
which is applicahle to any engine that operates in a cyclic fashion. Here W is the work produced during one cycle of the engine's operation, while Q represents the total transfer of
thermal energy from the surroundings t o the engine during each cycle of operation. Let us calculate Q and W for each stage of the above cycle. Stage I: Since the process is isothermal and the working fluid of the engine is an ideal gas, AU for this stage is zero, and Q = W. We can obtain W from the standard formula that is applicable to any ideal gas undergoing an isothermal process,
Substituting for n (2.00 mol), T (300 K),V2 (24.6 L), and VI (49.2 L), we obtain W , = Q I = -3.46 X lo3joules. Stage 2: The work performed during this stage is easily evaluated since this stage is isobaric. Therefore,
and converting t o joules, we obtain W2 = 2.07 X lo3joules. The heat interaction for this stage can he obtained from the first law of thermodynamics: Since the gas is ideal, AU = CJT, - T,) = 3.0R(424 - 300) joules = 3.09 X lo3joules
Substitution of this result and thevalue of W2 into eq 4 gives Q2 = 5.16 X lo3joules. Stage 3: T o calculate the work performed during this stage we must evaluate the integral
The constant k is equal to P V . If we evaluate this constant using the values of P and V a t the beginning of this stage, then we obtain k
= (2.00) (34.8Y L-atm = 2.42 X
lo3L-atmz
Substituting this result into eq 5 and evaluating the integral yields
Substitntion of 34.8 for V, and 49.6 for Vz, followed by conversion to joules yields
We can solve for Q3 by once again making use of eq 4, where in this case AU = 3.0R(TL- TH)= -3.09
X
lo3J
and Q3 = -1.02 X lo3J. Of the three stages, i t is only during stage 2 that thermal energy is transferred to the engine from the surroundings, and so Q = Qz = 5.16 X 10" J. The total work performed during the cycle is W=Wj+W2+W~=68OJ
Substitntion of these results into eq 2 yields 0.132 for the efficiency, well below that of a Carnot engine. Does this mean that Fermi's statement of the corollary to Carnot's theorem is incorrect? The answer is no. But, as we will soon observe, i t could be stated in a less ambiguous fashion. The mistake of assuming the same efficiencies for both the-above three-stage reversible cycle and an engineer that employs a Carnot cycle and operates between thesame two temmratures is one that arises out of confusion brought on by the ambiguity of Fermi's statement. The ambiguity in Fermi's statement arises primarily from the phrase "operating around cycles between two tempera-
tures". However, the difficulties that are caused by this phrase are exacerbated by the opening sentence in Fermi's statement, which might lead one to believe that there are other types of reversible cycles that have the same efficiency as a Carnot cycle. We will consider this additional source of confusion later. I t appears that the above three-stage cycle does operate between two temperatures, an upper temperature of 424 K and a lower temperature of 300 K. Consider an alternative statement of the corollary to Carnot's theorem: All reversible engines operating between the same two temperature reservoirs have the same efficiency; all irreversible engines have a Lower efficiency.
For later reference, let us call this statement B. The major difference between this statement, which can he found in a number of sources (6,nand statement A is the use of the phrase "temperature reservoir" in place of temperature. Even so, amajority of students, and more than afew of those who teach them, would assume that the above three-stage cycle operates between two temperature reservoirs, but such is not the case! During a Carnot cycle, transfers of thermal energy to and from the engine occur isothermally. Thermal energy is transferred to the engine at a constant high temperature, while the eneine exnels thermal enerm -. to the surroundings a t a constant low tkmperature. As the temperature of the engine moves hack and forth between the two reservoirs, there is no transfer of thermal energy; these stages of the cycle are adiabatic. In other words, only two temperatures reservoirs (or "two temperatures") are required to operate an engine that uses a Carnot cycle. With the three-stage cycle that was considered above, thermal energy is expelled to the surroundings at a constant low temperature during the first stage, but during the second and third stages transfers of thermal energy occur as the temperature of the engine moves back and forth between the low and high tem~eratures.The only way that these stages can he reversible isif the transfers of thermal enerw occur in a sequence of infinitesimal steps. During each of these steps a veiy small quantity of thermal energy is transferred to or from the engine. But such a sequence of small transfers can only occur reversibly if there are a large numher of intermediate temperature reservoirs, each reservoir differing only slightly in temperature from the one that precedes it. Inother words, the second and third stages of the above cycle require large numbers of temperature reservoirs. This is the reason that the efficiency of the cycle is less than that of a Carnot cycle. If such a cycle required only two temperature reservoirs, stages two and three would involve thermal energy transfers via finite temperature changes, and these stages would then have to be irreversible. Example 2 T o reinforce this conclusion, consider another reversible cycle that "operates between two temperatures". The cycle is illustrated below. TI = 424 K VI = 6.96 L PI= 10.0 atm
stage 4
isothermal expansion st%* 1
1
lsochoric warming
T4=300K V4 = 6.96 L P4= 7.07 atm
-
T2 = 424 K V2= 27.8 L Pz = 2.50 atm
cw1ing isothermal 4
stsgc 3
T3 = 300 K V3 = 27.8 L P3 = 1.77 atm
This cvcle, called the Stirling cycle, consists of four reversible stages; and uses 2.00 mof"fa monatomic ideal gas as the working fluid. A P-Vdiagram for rhis cycle appears in FigVolume 68
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209
quire many more than two temperature reservoirs to operate reversibly. Stages two and four of this Stirling cycle involve nonisothermal transfers of thermal energy, and this can only be accomplished reversibly via a sequence of very small steps as outlined above. After considering these two reversible cycles, one might well ask whether any other reversihle cycle that operates between two temperature reservoirs can be constructed that will be as efficient as a Carnot cycle. This question goes right t,o heart of the secondarv cause of confusion in Fermi's .. the .-----statement (statement A from ahove), a source of coufusion that is not eliminated with the alternative statement (statement B from ahove). The answer to the question is no! In fact. i t is imoossihle to construct a reversible cvcle that will ope;ate between only two reservoirs and yet differ from a Carnot cycle. If one is to avoid the difficulties illustrated by the threestaee cvcle and the Stirline cycle, then the stages in which thetemperature moves back and forth between the temperatures of the reservoirs must be adiabatic. Transfers of thermal energy must he avoided during stages which involve finite temperature changes, since such transfers require the presence df many temperature reservoirs if the stages are to be reversible. Needless to say, transfers of thermal energy must take place a t the high and low temperature reservoirs and must take place isothermally. Only a Carnot cycle operates in this manner. Of course, many different kinds of irreversible cycles can be constructed that will operate between two reservoirs. Indeed, each of the above two example cycles can he constructed to operate irreversibly between two temperature reservoirs. Naturally, these cycles will have lower efficiencies than that of a Carnot cycle. With this background, onemight now ask why the corollarv to Carnot's theorem is so often stated in textbooks with either statement A or statement B. Fermi's statement of this .-~ - --~corollary (statement A) and the less ambiguous alternative (statement B) are both technically correct, but the ways in which they are stated must naturally lead to the misinterpretations outlined ahove. Finally, consider the following alternative: ~~~~
Figure 2. P-Vdiagram
for a reversible Stirling cycle.
ure 2. If wemake the same error that wemade with the threestage cycle, then we would predict that the efficiency of a reversihle Stirling engine operating between these temperatures (the same as those used with the three-stwe reversihle engind) should be 0.292, the same as that of &engine that emolovs . . a Carnot cvcle. What does a more careful analysis
-- -
reveal? . -..
We now calculate Q and W for each stage of the cycle. Stage 1: Since this is an isothermal reversible expansion we can apply eq 3, to obtain
Since the working fluid is an ideal gas and the process is isothermal, AU = 0 and QI = W I . Stage 2: Since there is no change in volume during this stage W 2 = 0, and AU = Q, = C,(T3 - Tz)= -3.09
X
lo3joules
Stage 3: Since this is an isothermal reversible compression we can apply eq 3 once again W , = nRT In
(9 -
= -6.91 X
lo3joules
AU for this stage is also 0, and so Q3 = W 3 . Stape 4: There is no volume change . for this stage, and W
4
AU = Q, = C,(TI - T4)= 3.09 X 1O~oules
Thermal energy is transferred to the engine during the first and last stages of the cycle, so
Q = Q, + Q4= 12.86 X 10"joules The total work performed during the cycle is =
W , + W3= 2.85 X lo3joules
and the efficiency of the reversible Stirliug cycle is
This cycle has an efficiency that is also lower than that of a Carnot cycle, and the reason for the lower efficiency is the same as that for the lower efficiency of the three-stage cycle illustrated in the first example. This Stirliug cycle will re-
210
Journal of Chemical Education
~
~
~
~
All reversible cycles that operate between the same two, and only two, temperature reservoirs must be Carnot cycles that have the same efficiencv. and this remesents the maximum efficiency that between these reservoirs. All em he obtainedfor anv-evcle.ooeratine , . ~,thrrcyrleithat operate brrwern the two reserwirs must, bv nerea. shy, be irreversihle and, therciore, h a w a lower dficienry. ~~~~
= 0. Therefore,
W
~~~~
~
~~
~
~
Such a statement or its equivalent seems to be less common and appears less frequency in texts (8-10). This statement makes it quite clear that there is only one possible class of reversiblerycle that can operate between only two temperature reservoirs, a fact that would clear up the confusion that can arise from the more commonly encountered statements of the corollary. Indeed, one must wonder how these statements arose in-the first place. Literature Cited 1. Ferrni. E. Thermodynomies; Dover: New York. 1936: p 39. 2. Ha1lidav.D.: Resnick, RPhvsics: Wiley: New York, 1978: ~ 5 4 7 . 3. he^^, J. P. Physical chPmislry,2nd ed.;Allyn and Bacon: Bmton, 19% p 126. 4. Barrow,C. M.Physico1 Chemistry,Sthed.:McCraw-Hill: New York, 1 9 8 8 ; ~187. i Atkini P. W. The SerondLow:SciontificAmerican: New York. 1984;o92. 163. . A" Introduction to Thrrmodynomics. the Kindic T h m y ofcorm and 7 . sears, F. W StoiisiicolM~chonics,2nd ad.: Addinon-Wesley: Reading, MA. 1951: p 114. 8. Dickerson, R. E. Maleculor Thermodynamics: Benjamin: Monlo Park. CA. 1969: p
.""-.".. ,'A
,c,
9 . Zemsnrky. M. W. Heal ond Thermodynomica.5thod.; McCraw-Hill: New York, 1968: p 222. LO. lae,J. F.: Sears, F. W. Thermodynomics:Addiso~-Wwwley:Reading, MA, 1963.
