Daniel Kivelronl University of California Los Angeles, and Irwin Oppenheim Massachusetts Institute of Technology Cambridge
I
II
Work in irreversible Expansions
W o r k is a rather subtle concept in thermodynamics, particularly for irreversible processes. The precise specification of the system, its boundaries, and the process which it undergoes, is of paramount importance. I n this paper we examine some illustrative examples involving expansions and compressions of gases. We believe that the discussion presented here clarifies the points raised recently by Bauman,2 and by Chesick, Bauman, and ICoke~.~Ambiguities in the expression for the work that appears in the surronndings in a given process disappear when the system and the process are carefully defined and described. We consider an isolated over-all system which consists of a gas in a cylinder with a movable, frictionless piston of mass m. The system consists of the gas which is bounded by the inside walls of the cylinder and the piston (Fig. l a ) . I n some cases we shall extend the system to include catches or stops which are used to hold the piston. For simplicity, we assume that the cylinder and piston are in a vacuum and that the boundaries of the system are adiabatic; therefore, there is no heat flow between the system and its snrroundings. We consider adiabatic compressions and expansions in which the boundary of the system is stationary in both the initial and final states.
Since the over-all system is isolated, the energy change of the system, AE,,,,, is equal to minus the energy change of the surroundings AE,,,,. Since the process is adiabatic, W
= - AE,,,
= AE.,,
(1)
where W is the work that appears in the surroundings. From an operational point of view, work is that part of the energy change of the system which could be completely used to raise weights in the surroundings. Our purpose is to compute W for the examples below. Example 1. Initially the piston is held a t height hl by catch 1 (Fig. l a ) and the system is a t equilibrinm with pressure p, > m g / A where A is the cross-sectional area of the piston, m is the mass of the piston, and g is the acceleration due to gravity. When the catch is released, the piston is accelerated and just before reaching catch 2 (Fig. l b ) a t h2 it attains a velocity v,. At this point, all of this energy has been or can be used to raise a weight in the surroundings. The piston is stopped and held firmly by the catch 2 which i s part of the surroundings. The kinetic energy of the piston is given up to the catch, but A&,, is still given by Eq. (2). The system then comes to equilibrinm at pressure pr without a change in energy. The work that has appeared in the surroundings in this process is given by
+
Figure 1.
Thecatches ore panof thesurroundings.
I n the expansions considered here the piston moves from a height h~ to a height hZ where h~ < h2; correspondingly the gas volume changes from V , to T12.
' National Science Foundation Senior Postdaetoral Fellow.
a B . ! ~ ~R.&P., ~ ,J. CHEM. EDUC.,41, 102 (1964). P C ~ ~J. P., ~ J. s CHEM. ~ ~ ~EDUC., , 41, 674 (1964);BIUMAN, 11. P., J. CHEM.EDUC., 41, 674 (1964); AND KOKES,R. J., J. CHEM.EDUC., 41, 675 (1964).
W = rng(h2 - ht) '/% mvps (3) from Eqs. ( 1 ) and (2). The velocity vz and the pressure p, cannot be calculated from thermodynamics because the system is far from equilibrinm during the process. A detailed study of the hydrodynamics of the gas would be necessary to determine vz and pz. Since the pressure of the system is nonuniform, W cannot be represented by an integral of p,,,, d V . It can, of course, be written in terms of an integral of F,,,, clh where Fa,,, is the force exerted by the gas on the piston. By Newton's third law F,,,, is equal and opposite to the force exerted by the piston on the gas which is given by mg m6(t). I n this example, W is not given by an integral of p,,,d V where p,, is the applied external pressure m g / A . I n fact,
+
where the subscript c specifies the process. Example 2. The process to be analyzed here is identical to that in Example 1 except that the catch 3 (Fig. 2) i s part of the system. When the piston is Volume 43, Number 5, May 1966
/ 233
stopped, its kinetic energy is transferred to the catch and hence to the system. Thus, AE,,,, becomes AESE,,,. = mdhz
- hd
(5)
and W
=
mg(h2 - h,)
=
(6)
I n this case, W can he readily calculated from simple mechanical considerations.
close to equilibrium, the force exerted by the gas on the piston is p,,,,(a' A where p8v.r'") is the equilibrium pressure of the gas during the athexpansion and av = Ash. I n order to obtain expressions for AE,,,, and W for the entire process, Eqs. (7) and (8) must be summed over all of the individual small expansions. We obtain W
=
mg(hz - h,)
=
2 p,.,,(d
6V
+ x= 6
mv2)
(9)
a
If the individual expansions are truly infinitesimal, the sums in Eq. (9) may be replaced by integrals and W = mg(ha - hJ
+
nw'2
'/l
(10)
(4
fbl
Figure 2. Although as drawn the sotcher ore portly in and partly oui of tho system, we will consider an idealized vetup in which they are entirely within the system.
Example 3. This experiment is designed to produce an irreversible quasistatic process in which the system is never far from equilibrium so that its pressure p,,,, is well defined and in which the velocity of the piston is always close to zero. The initial state of the over-all system is the same as that in Example 1. There are a very large number of catches between hl and h2 (Fig. 3). The experiment proceeds as follows: Catch 1 is released and the gas expands until the piston is 2 caught by catch a a t height hl ah where ah is very small. The b\ 0 system is allowed I Itb come to equilibrium, and then catch a is reh1 leased and the gas expands until the piston is caught by catch Figure 3. b a t height h, f 2 ah. This procedure is repeated until the piston reaches the height h2 and the catch 2. All of the catches are assumed to be part of the surroundings. The energy change for the surroundings aE,,,(") during the expansion between the athand the (a 1)st catch is
n
n
.
i
T
= mg6h
+ S ('Iz mvoP)= 6W(d
(7)
where mv-2) is the kinetic energy developed by the piston during this expansion. Since the system is 234
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Journd of Chemical Education
The experimental situation described here is essentially the same as that for an irreversible expansion of a system in which the piston is prevented from attaining finite velocity by frictional forces within the system. We note that for reversible processes, p,,, = p,,,,, the velocity of the piston is always negligible and the expressions for W for Examples (14)all become
+
+
SE,,,(4
where m ~ is' the ~ total ~ kinetic energy developed and dissipated by the piston in moving from hl to hz in the manner described. I n this case, the system behaves exactly as it does when executing a reversible adiabatic expansion from the initial state to the final state characterized by volume V2. Thus, an explicit expression for W can be obtained by making use of the equation of state and thermal properties of the gas. Example 4. The process to be analyzed here is identical to that in Example 3 except that all of the catches are part of the system. As in Example 2, all of the kinetic energy is given up within the system and
Example 5. We consider the situation in which the mass of the piston approaches zero; i.e., free expansion. Without fnrther ado, we see that for Examples 2 and 4, W = 0. As m approaches zero, the quantity mvzZ approaches zero, since v2 remains finite as can he seen from considerations of conservation of energy and momentum. Thus from Eq. (4) it follows that W = 0 for Example 1. It is clear from the first equality in Eq. (10)that W = 0 for Example 3. The second equality in Eq. (10)is not valid when the mass of the piston is too small because the piston is then accelerated by the impacts of very few gas molecules to velocities larger than the speed of sound, and the gas is unable to keep up with the piston. Therefore, even for expansions between closely spaced catches, the force on the piston is not given by p,,,, A unless the terminal piston velocity is small compared to the speed of sound. This force approaches zero as the mass of the piston approaches zero.
We next turn our attention to a discussion of irreversible compressions. I n the compressions considered here the piston moves from the height hl to the height h2, hl > ht, and mg/A is greater than the pressure of the system. The expressions for W for compressions which correspond to Examples (14) are identical to those for the corresponding expansions. The discussion above can be summed up in the following way. If an irreversible expansion or compression takes place in a quasistatic manner, i.e., the piston doesnot develop finite velocity, then
ence of an external gas. If the kinetic energy of the piston is given up to the surroundings, W cannot be calcula.ted without recourse to hydrodynamics. If the kinetic energy is given up to the system
w
=STJW~V
(15)
provided p,,, is due to the mass of the piston; if p,,, arises from the presence of external gases, W cannot be calculafed without recourse to hydrodynamics. For free expansions,
w=o if the kinetic energy is given to the surroundings. Similarly,
if the kinetic energy is given to the system. Both the system and the surroundings remain close to equilibrium in these situations; pa,, and p,,,, are well defined and the force on the piston can be related to these pressures. If the kinetic energy is given up partly to the system and partly to the surroundings, the problem is more difficult to analyze. If the process is irreversible and non-quasistatic, i.e., the piston develops a finite velocity, p,,,, is not well defined since the gas cannot keep up with the piston; p,, is well defined if it arises entirely from the piston mass but is not well specified in the region near the piston if it arises from the pres-
(16)
in all cz3es. The analyses above clarify some ambiguities in the literature. I n particular we have shown that: (1) Work cannot always be measured in terms of the actual weights raised in the surroundings in a given process since, in irreversible processes, some work may be used to accelerate the surroundings; (2) The differential element of work dW cannot always be expressed as 2W = p,,, dV for irreversible processes; (3) Quasistatic processes need not be reversible; (4) The presence of frictional dissipation in a process does not necessarily imply a flow of heat; ( 5 ) If the over-all system consisting of the system and its surroundings undergoes an irreversible process, the system may or may not undergo an irreversible process. Acknowledgment
We should like to acknowledge interesting discussions with R. L. Scott and J. A. Beattie.
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