Frank J. Bockhoff
College Cleveland, Ohio
A Model for Introducing
Fenn
the Entropy Concept
O n e of the more difficult tasks in teaching introduct,ory thermodynamics is the presentation of a physical concept of entropy. A specific model system, such as that shown in Figure 1, proves very helpful. To develop the entropy concept, one must consider the fate of the system when it undergoes (1) reversible isothermal expansion, ( 2 ) free isothermal expansion, and (3) isothermal expansion intermediate between reversihle and free expansion. I t can he shown, for (2) and (A), that the system and surroundings cannot he restored to their init,ialstate because of the theoretical impossibility of quantitatively converting heat to work. The initial state of the system is shown in Figure 1. The ideal gas, piston, and cylinder constitute the system, which is surrounded by a vacuum. The weight and energy reservoir represent the surroundings, equivalent to the remainder of the universe. The variable mechanical advantage of the cone and V-pulley arrangement provides for a confining force which decreases as the gas expands. I n the model shown, the drive diameter of the V-pulley is directly proportional to the volume of the gas. Thus, if the weight is oonstant,, the confining force on the gas is inversely proportional to its volume, a condition required for rrvcrsihle expansion and compression of the ideal gas. If the we,ight is chosen so that the confining pressure due to the piston is exactly equal to the internal pressure of the gas under the initial conditions of Figure 1, the confining pressure will equal thc internal pressure at any position of the piston. That is, using thc mechanism shown, expansion and compression are reversihle throughout the entire piston stroke.
The gas in the cylinder, initially a t a volume VI and an absolute temperature T, absorbs an arbitrary amount of heat, for example, 4 units, from the energy reservoir. does maximum work in expanding to V2,and converts the heat quantitatively to potential energy by lifting the weight an equivalent of 4 energy units. The state of the system after expansion is shown in Figure 2. To restore the system to its original state, a l l m the weight to drop an equivalent of 4 enrrgy units, which represents the minimum work required to compress the gas from V 2 to V,. During this compression, 4 units of energy are transferred back to thr reservoir, because the change is isothermal and Q . ,
=
W,.,
=
4 umts
The system and surroundings are thus restored to initial state (Fig. 1) with no permanent changes.
Figure 2.
System after reversible irothermal exponrion
Isothermal Free Expansion
Isothermal free expansion, however, presents an additional problem. To effect free expansion in the model system, starting from the condition shown in Figure 1, lift the cone off the V-pulley during expansion of the gas from V1 to V 2 . The gas does no work and thus absorbs no energy from the reservoir. ENERGY RESERVOIR,l
I
END
VIEW
HEAT
CARNOT WEIGHT
Figure 1.
Syrtem in initial state,
Isothermal Reversible Expansion
Let us first examine isothermal reversible expansion. 340
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Journol o f Chemical Education
Q.,,".,
=
0,
W,t".l
= 0
The state of the system after isothermal free expansion is shown in Figure 3. Now, let us restore the system to its initial state. To compress the gas in the cylinder from Vz to VI, one must perform at least the minimum work of compression, which is equal to the maximum work of expansion, 4 units, performed in the reversible isothermal expansion. To restore the system, allow the weight to drop a distance equivalent to 4 units in potential energy, thus compressing the gas. During its isothermal compression, the gas evolves 4 units of heat
ENERGY RESERV0IR.T
CARNOT WORK T O L l F T WElGHT
Figure 3.
Figure 5.
System ofter isothermal free exponrion
to the reservoir, because Q is equal to W. The state of the system at this point is shown in Figure 4. Although the pist,on is restored to its initial position, the weight and reservoir are not. There are 4 units of excess energy in t,he reservoir, and the weight is an equivalent of 4 energy units below its initial position. The system and surroundings could be completely restored t,o initial state if the 4 units of energy in the reservoir could br quantitatively converted to work by lifting the wcight to its original position. The total amount of heat which must be converted to work to restore the system and surroundings to initial stat,e is given by Q,*" - Qdotusl = 4
S y t e m after intermediate irreversible expansion.
surroundings after compression are shown in Figure 0. There are 2 units of excess energy in the reservoir and the weight is now an equivalent of 2 energy unitb below its initial position. The system and surroundings could be completely restored to initial stlte if the 2 units of heat in the reservoir (surroundings) could he quantitatively converted to 2 units of work by raising the weight to its initial position. The net amount of heat which must be converted to work to restore the system and surroundings to original state is given by Q,,, - QmtUal = 4 - 2 = 2 units
The General Case
- 0 = 4 units
It can easily be shown with the model, that for all other intermediate cases of irreversible expansioil -
(&.,").,*,
>0
(1)
(Qaot"sl)ay.trm
I n contrast, in a reversible expansion (Qrer).ylrt.m - ( Q s r t u a l ) l r y . ~ c= m
1 HEAT
A]: I
WEIGHT SINK.T,
Figure 4.
But, if one defines Q,. as the heat absorbed by the surroundings (or energy reservoir) during the actual change,
-I1$ -3
$g
-4
Piston restored ofter isothermal free exponrion.
Intermediate Irreversible Expansion
Before discussing the mechanism for converting this heat to work, let us examine case (3), an isothermal irreversible expansion intermediate between reversible a,nd free expansion. To effect this expansion in the model system, (Fig. I), allow the cone to slip on the V-pulley during the expansion stroke, so that the weight is raised an equivalent of 2 energy units, rather than 4 energy units had the expansion been performed reversibly. During the expansion, 2 units of heat are absorbed by the gas from the energy 'reservoir, since Q = W . Thr system after expansion is shown in Yigure 5. To restore the system to its original state, again allow the weight to drop an equivalent of 4 energy units, because this represents the minimum work of compression. During compression, the gas transfers 4 units of heat to the reservoir. The system and
(2)
0
(Q~%"ar)ayn~em = -
(3)
Q*"==
Thus, for any process, combining equations (I), (2), and (3), (Qr.v).,t., Q..m 2 0 (4) The model illustrates that for an isothermal change, the amount of heat which must he quant,itatively converted to work if one is to restore the universe
+
1
I i1z
HEAT
-2 WORK TO L l F T WEIGHT
-3
2
g
SINK,T,
Figure 6.
Piston restored after intermediate irreversible expansion.
Volume 39, Number 7, July 1962
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341
(that is, the system and surroundings) to its initial state, is equal to the sum of the heat which would have been absorbed by the system if it had undergone the given change reversibly, plus the heat absorbed by the rest of the universe (surroundmgs) during the actual change. This quantity of heat to be converted to work is always positive for all irreversible isothermal processes, or, more simply, for all natural processes. Can Heat be Quantitatively Converted to Work?
A basic question must still be answered. Can the Q . , be converted quantity of heat, ( Q ) quantitatively to work? A cyclic device must be chosen to convert the heat to work. Otherwise, one would be faced with the additional problem of restoring the heat-work device to its initial state. Theoretically, the most efficient cyclic device for converting heat to work is the Carnot engine. The maximum work, W-, which can be obtained from a quantity of heat Q,, introduced to the Carnot engine a t a temperature, T, the temperature of the previous isot,hermal system and energy reservoir, is given by
+
W-.
=
QdT - TI)/T
For our consideration, QT = ( Q r e & m
+
Qm"
(5)
(6)
A quantity of heat, Q1, is exhausted, unconverted t o work, a t the lower operating (or sink) temperature, T I ,of the engine. The quantity of heat which cannot be converted to the required work necessary to restore the system and surroundings is given by QI = QT - Wms.
(7)
QI = QT - QT(T - T , ) / T
(8)
41 = Q P T L / T (9) Inspection of equation (9) reveals that for a given quantity of heat, Q,, introduced to the Carnot engine a t temperature T , the amount of heat remaining unconvertible to work in the most efficient case is dependent on the sink temperature of the Carnot engine. If T I were O°K, one could completely restore to initial condition t,he system and surroundings
(universe) considered in cases (2) and (3). However, for any value of T,other than zero, the universe cannot be restored to its initial state after any portion thereof has undergone an irreversible change. This situation results not from a loss of energy, but rather from a loss of energy which is available to do work at that temperature at which the process occurred! The heat Q,, exhausted at the temperature T I , is still available for work at some lower temperature provided it can be introduced to another Carnot engine which operates with a sink temperature below T I . Entropy and the Second Law
On rearranging equation (9), QdTr = QTIT
it becomes obvious that, although Ql varies with TI, the ratio, &,/TI, which is the unavailable energy per remaining degree of absolute temperature, depends ~ n l y on Q,, as we have previously referred it to the system and surroundings, and T , the temperature at which the isothermal changes occur within the system. The ratio, Q J T , is thus a measure of increase in unavailable energy per degree which results from a change which occurs irreversibly. It is now convenient to define that thermodynamic function whose increase is being measured as AS..i,,.
=
QP/T
where S is the entropy or total unavailable energy per degre?. It also follows that
+
ASUntvarae= ( Q p e v ) a y 8 t d T Q d T
.,.,.,,,
''Special Sources of Information on Isotopes," TII> 4563 (3rd rev.) is now available. A brief textual introduction is followed by an extensive bibliographic listing of all major sources of information about radioactive iaotopes-availability, use, deteotion, safety considerations, ete. Single copies may be obtained free from the Information Section, Division of Technical Informat,ionExtension, USAEC, P. 0. Box 62, Oak Ridge, Tennessee.
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Because all natural processes are irreversible, AS is always larger than zero. Thus, the entropy of the universe tends to a maximum. As interesting variations, examples may be easily developed for isothermal reversible compression and isothermal irreversible compression, the net results being analogous to those above.
AEC Issues Third Revision of Isotope Information
342
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