W. H. Eberhardt Georgia Institute of Technology Atlanta
Reversible and Irreversible Work A lecture demonstration
I n view of the current trend toward inclusion of increasingly sophisticated treatments of thermodynamics in the freshman course in chemistry, there is a need for quantitatively meaningful demonstrations which will help the student visualize the difficult abstractions of thermodynamics and which will show the application of thermodynamic principles to chemical problems. The demonstration described here has been used in the elementary chemistry course a t Harvard University and was received with considerable enthusiasm, especially by those students who had been exposed previously to the words and notions of reversibility but who had found the concepts rather hard to grasp.
rings on the ring stand are adjusted to coincide with the level of the pan as the load is reduced by successive decrirnents of 200 g. Thus, the large ring in the center of the stand is a t an altitude corresponding to that of the pan with 800 g on it; the large ring a t the top represents the altitude of the pan with no load.
Nature of the Experiment
The system involved in this experiment comprises a pendulum and attached balance pan as indicated in Figure 1. The pendulum consists of a 10 kg mass suspended by a length of clothesline. The mass is attached to a rather substantial balance pan by means of a light cord; Braided Nylon Surf Cord, 36-lh Test, was used in our system. This cord passes over a highquality pulley with small frictional resistance. The length of the pendulum cord and the distance from the pendulum to the pan were both about 20 feet in our experiment, the large distances being desirable to minimiie nonlinear effects as well as to make the apparatus more easily visible in the large lecture hall. The pulley is suspended a t the end of the lecture bench a t a height of about 4 feet above the bench. Situated on the lecture bench next to the pan is a ring stand and attached scale which may be either a meterstick or a scale graduated in units appropriate to the size of the apparatus. The ring stand carries eight rings, two 4-in. diameter rings with plywood platforms a t the top and center and six 2-in. diameter rings iuterspersed as indicated in the figure. The position of these rings on the ring stand is determined by the dimensions and mass of the pendulum as described in the procedure below. The dimensions of the system we used are such that a mass of 1600 g added to the pan pulls the pendulum away from its nearly vertical position until the pan sinks to a level even with the top of the base of the ring stand which is also the origin of the scale of the meterstick. As the weights on the balance pan are removed in 200-g units, the pendulum returns toward its undisplaced position and the pan rises. The positions of the This demonstrstion was designed and used during the Fall of 1963 when the author was Arthur and Ruth Sloan Visiting Professor of Chemistry a t Haward University.
Figure 1 .
Esrentiol components of t h e demonstration.
The experiment involves the measurement of the work done by the system (the pendulum and pan) on the surroundings (everything else, but mostly the weights) as the system goes from a state I corresponding to an equilibrium state with 1600 g on the pan to a second equilibrium state I1 corresponding to zero load on the pan. A second series of experiments involves measurement of the work done to restore the system, i.e., starting with the system in state I1 with no weights on the pan, to return it to state I with the full 1600 g on the pan. The work done by the system on the surroundings is measured bv the distances through which weights are lifted; i.e., if in the course of the-experiment, t h e pan lifts a mass M through an altitude Az, the work done in this step is MgAz. The experiment is conducted as follows: Starting a t state I (1600 g on the pan), the demonstrator holds the pan in place and removes all 1600 g at once, setting them upon the top of the base of the ring stand. The pan is then released (with a bit of care so the system is not immediately destroyed by truly irreversible processes) and allowed to come to rest in state 11, at the top of the scale. Obviously, no weights have been raised in this process; and no work done. The system is restored to state I by the demonstrator, and weights artre now removed in two steps, 800 g each. The first 800 g is removed and set on the base of the ring stand; no work is done. The system then lifts the remaining 800 g and settle^ down
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a t mid-height along the scale. The remaining weights are removed here, holding the pan a t this level until all are removed, and the ~ y s t e mallowed to go to state 11. In this process, the amount of work done by the system clearly corresponds to that required to raise 800 g halfthe height of the scale. The system is again restored to its state I and weights are now removed in four steps of 400 g each, being distributed appropriately on the pans of the ring stand. The system is again restored to atate I, and weights are removed in 200-g steps so that, when the system attains the final state 11, weights are distributed evenly over all the platforms except the uppermost.
The work done in each of these processes can be computed either in proper units (ergs) or in arbitrary (mass X altitude) units, determined by the nature of the scale use. The calculated amounts of work can be plotted, as indicated in the lower curve of Figure 2, against the unit of mass removed in each step, AM. The straight line so obtained is easily extrapolated to the limit and this limit clearly represents the maximum work the system can do in proceeding from state I to state 11.
200
400
600
BOO
1000
I200
A M
Figure 2. Work involved in the proceseu Lower curve: work done by the system in going from state I to state 11; Upper curve: work done on the system in the reverse process. The abscissa represents the mossshonge per step in each process.
The reverse process can now be demonstrated, again carrying through several processes with different increments of mass. At the start, all 1600 g may he set on the uppermost platform and transferred in one operation to the pan. Again some care is required on the part of the demonstrator to ensure that when the pan is released, it does not fall so rapidly as to injure either the apparatus or the demonstrator. The work now done by the surroundings on the system is defined as the change in potential energy of the 1600 g falling from the top of the scale to the bottom. The experiment can be repeated in smaller steps, the smallest being that with 200 g on each platform to start the experiment and added in eight steps. The work invested in these processes can then be Computed and plotted as the upper curve of Figure 2. This curve is the mirror image of that plotted earlier, and again extrapolated to the limit of zero mass increment, AM = 0, intersects the axis at the same point established by the lower curve. Thus, in the limit, the minimum work required to restore the system to its original state I from state 11 is seen to be exactly equal to the m m i m u m work done by the system in going from state I to state 11. 484
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Analysis of the Experiment
The experiment is sufficiently simple that a detailed analysis of it can be carried through without much difficulty by any interested student. The work done by or on the system can be expressed in terms of the mass removed, AM, and the total mass M required to establish equilibrium in state I as follows:
The upper sign inside the brackets refers to the work done by the system in going from state I to state I1 and the lower sign refers to the work done on the system in the reverse process. Z is the difference in altitude corresponding to the two end states. The important simplification of the experiment is that the equilibrium forces involved are linearly related to the altitude or displacement of the system from state I. This simplilication results from the large distances involved in the length of the pendulum and the line connecting the pendulum to the pan. The only serious experimental difficulty to he expected is that the line connecting the pendulum and pan may stretch irreversibly. It is important, therefore, to make this connection with line that will not stretch enough to spoil the quantitative nature of the experiment, and also to avoid sudden shocks upon this line consequent to changes in mass on the pan. The demonstration also poses a rather interesting question relative to the nature of "heat" and its defiuition. The initial and final states are the same, but d i e r e n t amounts of mechanical work are involved in the transitions between the two states. Thus, the change in internal energy of the system, AE,is always the same, but, since different amounts of work W are involved, the quantity Q defined as Q = AE W, must be variable. The question one then poses is: where is the thing called Q? The experiment demonstrates clearly the difficulty of basing rigorous thermodynamic thinking on Q. However, one can compute AE in a restricted adiabatic processes and then define Q in terms of AE and W. The logical excellence and simplicity of this approach is becoming increasingly apparent in recent texts.' Another ramification of the experiment points to the question can an experiment really be carried out reversibly? The answer to this question must be stated in terms of experiment and the ability of the experiment to distinguish between reversible and irreversible paths for the process. Thus, suppose the experiment to be conducted in some carefully defined way, e.g., by removing mass from the pan in units AM, and the work done in going from state I to state 11, W is determined. A corresponding quantity, W,., can be calculated assuming reversibility, defined as the limit as AM + 0, or equivalently, as the integral of the force t i e s the change in distance. If these two quantities, We, and W,-, differ by a number which is less than the experimental error, then one may say
+
..,,
DENBIGH,K., "The Principles of Chemical Equilibrium," Cambridge University Press, Cambridge, England, 1957, pp. 14-21. CALLEN,H. B., "Thermodynamics," John Wiley and P. T., Reu. Sons, Inc., New York, 1955, pp. 10-20. LANDSBERG, Mod. Phys., 28,363 (1956), a rather rigorous but penetrating presentation, "Foundations of Thermodynamics."
that the experiment has been carried out reversibly within the limits qf the experimental error. It is only in this context that the question is meaningful and a meaningful anslrer may be given. Other Experiments Relating to Reversibility
Once the connection between AE and W is established in a restricted adiabatic process, thermochemical measurements are most readily approached through electrical techniques. Thus, the electrical work done by or on a system in a particular process is related to En5 where & is the actual potential concerned, n the number of equivalents of charge transferred in the process, and 5 the value of the faraday. The charging and discharging of an electrical cell makes another example appropriate to discussion of maximum work and reversibility. The lecture demonstration used in this instance consisted of a Zn-Cu cell comprised of a porous cup separating the two half cells in a Cliter beaker. This cell has a moderately large internal resistance. The cell is connected, in series, with a large but sensitive milliammeter across a potentiometer which is a large Ohmite variable resistance. The potential is measured by the voltmeter, the scale of which is projected to be readily visible. The inilliammeter is adjusted to be centered when no current flows so that current flowing in either sense can he seen and measured by it. The experiment consists of measuring the potential of the cell as a function of the charging or discharging current. To a good approximation, this potential is related to the current by the equation
~=&&rn Here, the upper sign refers to the charging process and corresponds to work done on the cell; the lower sign refers to the discharging process and refers to work
done by the cell: R is the internal resistance of the cell. Plots of & versus I are straight lines and intersect in the limit I = 0 a t the value So. The interpretation of these experiments is almost identical with that of the pendulum experiment except for the complication that a chemical reaction occurs in the cell and a precise description of the changes occurring in the system and surroundings is more difficult to obtain. If the system, the cell, is considered completely isolated from the surroundings except for the electrical connections, i.e., if it is imagined iu an adiabatic box with constant volume, then the initial and final states of the system consequent to the charge or discharge iuvolving a definite amount of charge or I X t product involves not only the charge, but also the value of the current since the final temperature of the system will depend on the rate a t which the cell discharges. Only if the cell is operated isothermally, i.e., in a large thermal reservoir, is the change in state of the chemical system related uniquely to the charge transferred. Thus, it is considerably more difficult to abstract the notion of heat and work simply from this system. The electrical work involved is &It, and only in the limit of I = 0 does this work become equal to the reversible work associated with the chemical process, but the heat involved is now associated with the difference in work done and the Gibhs free energy, rather than just the internal energy, and hence, the analysis of the experiment is likely to get ahead of the presentation in the course. Acknowledgment
The author is particularly indebted to Professor Leonard I
