Robert C. Plumb
Worcester Polytechnic Institute Worcester, Mossochusetts
Teaching the Entropy Concept
There is widespread feeling among chemistry teachers that students benefit by training in both the classical thermodynamical and statistical mechanical treatments of entropy in undergraduate courses. The statistical mechanical approach to entropy has frequently been neglected. The usefulness of the physical insight gained from a statistical mechanical approach to entropy as a driving force in understanding why reactions go has been pointed out by IMacWood and Verhoek.' Persuasive arguments for introducing statistical mechanical arguments at an early point in the training in chemistry were the subject of an editorial in THIS JOURNAL.% Students easily recognize the principle that system tend to go from high potential energy to low potential energy spontaneously since there are numerous examples of this type of behavior in our macroscopic surroundings (i.e., gravitational, magnetic, and electrostatic effects). It is not difficult for a student to go from a picture of macroscopic particles seeking a minimum of pntential energy to atoms and molecules seeking a minimum of potential energy. Thus, the teacher has only to introdnce the idea that atoms and molecules exert forces upon each other and generalize from the behavior of macroscopic systems. One can explain many aspects of the behavior of matter without inquiring as to the detailed nature of the forces. The student understands, for example, that the potential energy resulting from the forces of attraction between atoms of hydrogen and oxygen in water molecules is lower than the energy between like atoms of hydrogen and oxygen; therefore, hydrogen and oxygen react t o produce water. One might suspect that the main reason that the students do not as readily appreciate entropy as a driving force is that whereas they can think of internal energy in terms of potential energy and kinetic energy which they understand, there are no simple physical models which they can associate with entropy effects. If students understand the behavior of atoms and molecules, they can find numerous examples of entropy as a driving force; but in everyday experience we lack suitable examples of entropy effects on a macroscopic scale. A macroscopic lecture demonstration illustrating both pntential energy driving forces and entropy driving forces and showing their interrelationship has been developed and found useful. It is the subject of this paper. In using the statistical mechanical approach, one first identifies certain mechanical properties of the particles. Particles in a system possess kinetic energy as a result ' MACWOOD, G . E., AND VERHOEK, F. H., THIS JOURNAL, 38, 334 (1961).
KIEFFER, W. F., THIS JOURNAL, 38,333 (1961).
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of their motions and potential energy as a result of the forces which they exert on each other. The sum of these two types of energies (suitably averaged over time) is the internal energy E. Total Energy
=
Potential Enerw
+ Kinetic Energy
Entropy as a mechanical attribute of a system is measured by the randomness in arrangement of particles in the system. By randomness is meant the number of distinguishable ways of arranging particles of the system. (A convenient example of randomness3 is to consider an occupant of a large house containing 10 rooms as in a more random state than an occupant of a 1-room house. The state of being "at home" for the former will mean that he is in 1 of 10 different rooms, whereas in the latter "at home" will imply that he is in a particular room.) When dealing with molecular systems, the randomness is with respect to distinguishable positions of particles in space and with respect to the occupation of energy levels. Designating possible distinguishable arrangements of the system by the subscript 7: and the fraction of the time that the system is in one of these arrangements by ft, entropy is defined as S
=
-kz,f, In j,
In a particular system in which there are W distinct arrangements all having equal a priori probability, the entropy expression takes the particularly simple form S=klnW
Expressed in qualitative terms, the randomness of a state of the system is an important mechanical property since particles will tend to seek the state of a system where randomness is a t a maximum if the particles are free to move so that the system changes from one state to annther.4 It is this mechanical behavior which the student can associate with the entropy concept, and which the entropy demonstration device illustrates. The Device
The device we have developed is shown in Figure 1. It consists of a 1-in. deep by 30-in. wide by 20-in. high box, constrncted of Plexiglas, in which there are movable barriers g and g' and levels f and f' together with a pneumatic device for providing kmetic energy to a plastic ball h. The plastic ball is impelled upward by a a See, for example, the delightful allegory, "Hotel Management JOHN, Physics Today, 15,28 (1962). in Ergodia," TRIMMER, The rigomus proof of this phenomenon is by no means clear. See, for example, MAYER, J. E., J . C h a . Phys., 34, 1207 (1961); "Fundamental Problems in Statistical Mechanics," John Wiley, 1962; or LANDAU, L. D., AND LIPSHITZ,E. M., %atistical Physics," Adison Wesley, 1958, p. 28.
'
brief air blast and comes to rest on one or the other of the two potential energy levels, f or f'. The potential energy levels contain ducts and orifices for conducting the compressed air and each level is divided into segments by low vertical dividingelements. Thenumberof segments available for motion of the ball is determined by adjustable vertical harriers g, g' extending from the potential energy levels to the top of the box. Thus, there are different potential energy levels for the hall determined by different heights in the earth's gravitational field and variable randomness of the upper and lower energy level determined by the number of positions in each level which are availahle to the ball. The motion of the ball is determined primarily by the pressure of the gas pulse which strikes it. It is insensitive to the duration of the pulse except that if the pulse duration is long compared to the time of flight of the ball, the motion becomes complicated. A pulse with exponentially decaying character has been found most suitable. This type of pulse is achieved by bleeding compressed air into a small storage volume, c, for a variable period of time and then discharging the air to the ducts. The charging and discharging of the storage volume is controlled by an electrically operated valve, b. Power to the valve is controlled by a microswitch which is actuated by a cam device turned by a synchronous motor. The cam has surfaces of different lengths in order to produce a variety of pressures in the storage volume. This variety of pressures produces a distribution of pulse heights which simulates a Rlaxwelliin distribution. By bleeding air into the volume a t a given rate, one obtains pulses with a characteristic distribution and average value. By varying the rate a t which air is bled into the volume, the range of pulse heights produced and hence the average kinetic energy of the ball can be varied. The valve d controlling the rate of bleeding air into the volume can be thought of a s the "temperature control."
Figure 1. Schematic diagram of entropy demonstrotion device. la) Moiwellian pulse generator; l b l Normdly closed two woy air control volve; Icl Air storage volume; I d l inlet needle volve; le) To campres9ed air supply; If), lfll Potentiol energy lerelr; lg), lg'l Dividers limiting motions of boll; lh) Boll.
I n operation, the motion of the ball and in particular its motion from a state of one potential energy to a state of another potential energy is obsenred. The time average behavior of the hall can be conveniently recorded by dropping a counter into one or the other of two glass beakers, one representing one energy level and the other the other energy level, each time the ball comes to rest. In order to make effective use of the entropy demonstration device it is desirable that certain elementary concepts be discussed or reviewed for the students. A student should have some understanding of the translational motions of atoms and molecules so that he can appreciate the fact that in a given quantity of material there will be a distrihution of velocities and hence a distrihution of kinetic energy. It helps if a student realizes that a particular molecule undergoes changes in velocity and kinetic energy as a result of collisions with other molecules so that the time average behavior of an individual molecule is the equivalent of the instantaneous average behavior of a collection of molecules. It is necessary that the student under-
Figure 2.
Device in "re
stand that as the temperature of a sample is increased the average value of the kinetic energy increases. Further, a student should have some knowledge of inter-atomic and inter-molecular attraction, and how inter-particle forces produce not only covalent bonds hut what are commonly called physical bonds such as are responsible for holding water molecules together in solid or liquid water or for holding benzene molecules together in liquid benzene. I n explaiiing the operation of the equipment, it is customary to first demonstrate the R'Iaxwellian distribution of kinetic energy by observing the distribution of heights when the hall is actuated. Then it is shown how the average kinetic energy can be varied and how this is analagous to the effect of a variable temperature. The potential energy difference between states is explained in terms of the height of the hall in the earth's gravitational field. It is then shown that the randomness of the different potential energy states may he varied by moving the boundaries which control the number of positions in which the ball may come to rest. With these introductory explanations one can easily demonstrate the role of entropy as a driving force. Entropy as a Driving Force a1 a Given Temperature and Potential Energy Driving Force. The harriers on each potential energy level are adjusted so that the same number of positions on each level are availahle for Volume 41, Number 5, Muy 1964 / 255
the ball and the temperature is adjusted so that the ball spends some time on each potential energy level. A pair of beakers is filled with counters to record the time average distribution of the ball. Forty counters or so are adequate to get satisfactory statistics. The barrier of the upper state is moved so that the upper state is favored by being more random. Another pair of beakers is filled with counters to demonstrate that the ball tends to spend a larger fraction of time in the upper energy state simply as a result of the entropy of that state being increased. It is not difficult to convince the students, at this point, that if the difference in entmpy between the two states is large enough, the ball could spend essentially all of its time in the higher potential energy state. At this stage in the use of the device one can develop a physical insight into the mechanical significance of entropy as a driving force. A particularly simple and perhaps not too nalve way of describing why the ball tends to spend more time in the state of large randomness is to think of the ball "getting lost" so that a large entropy for the state of high potential energy decreases the likelihood that the hall will find its way back, as a result of random motions, to the lower state. Potential Energy as a Driving Force at a Given Temperature for Entropy Driving Force of Zero. The ball is given a particular temperature, preferably a large value in terms of the range available in the demonstration device, the barriers controlling the entropy of the two states are adjusted so that the two states have the same number of positions available, and the average behavior of the ball when there is only a small potential energy diierence between the states is noted. It will be found that the balI tends to spend about the same amount of time in both potential energy states. The difference in potential energy between the two states is then increased. It will be found that the ball tends to spend relatively more time in the state of lower potential energy because only a few of the kinetic energy pulses are sufficient to carry the ball from the lower state to the upper state whereas all of the pulses are sufficient to carry the ball from the upper state to the lower state. The student can readily see that in a collection of a great many molecule3 most of them will he in the state of lowest potential energy if the potential energy difference between the states is large compared to the average value of the kinetic energy of a molecule. The Role of Temperature in the Relationship Between Entropy and Energy as Driving Forces. In order to demonstrate the interrelationships between entropy and potential energy as driving forces, one adjusts the
harriers g and g' so that the state of highest potential energy is favored by entropy and the behavior of the ball is noted as the temperature is varied. The average position of the ball when the temperature is low is found to be controlled by potential energy and at low temperature one finds that the position of the ball is quite independent of the entropy of the upper state. At high temperature (KE 1 AE) the entmpy term is the predominent one and controls the behavior of the hall. It should now be possible for the student to see the physical significance embodied in the equation AA = AE
- TAS
If the driving force, AA, is positive then the reaction does not tend to take place. If AA is negative, theu the reaction does tend to occur. It is apparent that an increase in AS will tend to offset an increase in AE, if the temperature is high enough. Hence, it is possible for systems to go from states of low potential energy to high potential energy spontaneously. With the simple mechanical concept of entropy, which the student has now obtained, it is possible to explain several phenomena which are inexplicable without the use of the entropy concept. Thus, water evaporates from a pool of liquid water (where it is at a given potential energy as a result of the attractive forces between neighboring molecules) to the atmosphere (where the molecules are not bonded together and have a higher potential energy). At the same time, as a result of going from one potential energy level to another, water molecules extract kinetic energy from the surroundings causing the surroundings to he cooled. Similarly, one can explain what happens during a phase change when, for example, a solid initially at absolute zero is heated to a point where it sublimes. At the temperature of sublimation the molecules have acquired suacient kinetic energy to overcome the potential energy driving force tending to make the molecules stick together, and the entropy driving force tends to make them escape. This is easily demonstrated by setting the barriers so that there is only one position for the ball to occupy a t low potential energy and a large number of positions for the ball in the high potential energy state. The kinetic energy of the ball is varied from absolute zero up to a temperature where the hall has acquired sufficient energy to escape from the state of low potential energy. The equipment may also be used to demonstrate other phenomena such as the influence of activation energies upon the tendency of reactions to take place.
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