Chemistry for Everyone
LeChâtelier’s Principle in the Sciences Volker B. E. Thomsen Spectro Analytical Instruments, 160 Authority Drive, Fitchburg, MA 01420
Conceived by a chemist for chemical equilibria, LeChâtelier’s principle is actually a very general statement about systems in equilibrium and their behavior when subjected to external force or stress. It seems strange, therefore, that one almost never finds mention of his name or law in other sciences. In fact, several analogous principles do exist. They simply appear under different names and address the concept of equilibrium within different contexts. In this note we examine some of the interesting forms taken by this chemical principle in the fields of physics, geology, biology, and economics. The discussion is presented to provide additional insight into the nature of LeChâtelier’s principle along with analogies useful in its teaching. LeChâtelier’s Law Henri Louis LeChâtelier (1850–1936), a French chemist, noted the effect of disturbances on equilibrium states in chemical systems in 1884 and first published his principle in 1888: For a system in equilibrium, external changes will be accompanied by changes in the state of the system which will act to reduce the magnitude of the initial change. The changes referred to here may be in concentration, pressure, or temperature. Examples can be found in most introductory texts. The temperature and pressure dependence may be predicted quantitatively by the equations for the change in the Gibbs free energy (1, 2), and
d(∆G )/dP = ∆V
(1)
d(∆G )/dT = ᎑∆S
(2)
It is understood that the temperature is held constant in the first equation and pressure in the second. Some comments on the origin of these statements are important for the discussion that follows. Starting with the first law of thermodynamics, dU = dQ + dW, and substituting the expression for the work, dW = ᎑P dV, we get dU = dQ – PdV Now the second law comes into play with the addition of the entropy variable, defined as dS = dQ/T for a reversible process. This leads us to dU = TdS – P dV Other thermodynamic variables such as the free energy are defined in terms of the variables in this equation. The Gibbs free energy is a thermodynamic state variable with the units of energy, defined as G ≡ U + PV – TS. The alternate form, ∆G = ∆U + P∆V – T∆S, shows the way to eqs 1 and 2 above. Consequently, we may note that LeChâtelier’s principle is a result of both the principle of conservation of energy and the second law of thermodynamics. The van’t Hoff law of equilibrium is a special case of this principle. Jacobus Henricus van’t Hoff (1852–1911), a
Dutch theoretical chemist, was the first recipient of the Nobel Prize for chemistry in 1901. His law of equilibrium states that when the temperature of a system is raised, the reaction goes in the direction that absorbs heat. This is clearly an application of the LeChâtelier principle. However, since the times of discovery or publication are similar, at least one source (3) has suggested the name “Hoff–LeChâtelier law.” The van’t Hoff equation for the variation of the equilibrium constant (K ) with temperature (T ) is d(ln K )/dT = ∆H/RT 2
(3)
where ∆H the enthalpy of reaction and R is the universal gas constant. LeChâtelier’s principle, stated in the most general terms, is: If system in a state of equilibrium is subjected to an external force, action, or stress that disturbs this equilibrium, then the equilibrium is shifted in a direction such that the effect of the applied force is reduced. In other words, a change takes place within the system, opposing the action of the force and tending to restore the initial state of equilibrium. Lenz’s Law in Physics Heinrich Friedrich Emil Lenz (1804–1865) was a German physicist who did most of his work in Russia. His law indicating the direction of electromagnetically induced currents was published in 1834. We start with a coil of wire connected to a galvanometer. Move a magnet toward this coil and a current will be registered by the galvanometer. Stop the motion and the current ceases. Remove the magnet and a current in the opposite direction is observed. In general, an (instantaneous) voltage is induced in a circuit that is equal to the rate of change of magnetic flux through the circuit. This is a statement of Faraday’s law: V = ᎑N (∆Φ/∆t)
(4)
where V is the induced voltage, N is the number of turns in the coil, Φ is the magnetic flux, and t is time. In magnetic induction, Lenz’s law states that the direction of current produced by the induced voltage (emf ) is such as to counter the original change. “The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in the magnetic flux through the loop. That is, the induced current tends to maintain the original flux through the circuit” (4 ). This statement is certainly similar to that of LeChâtelier’s principle. Algebraically, Lenz’s law amounts to the minus sign in Faraday’s law. It should also be noted that this law is a direct consequence of conservation of energy (4, 5). (This is perhaps easiest to see by assuming the opposite and noting that it violates conservation of energy.) Lenz’s law may be characterized as an example of electromagnetic equilibrium, as opposed to the thermodynamic equilibrium of LeChâtelier’s principle. Since the entropy does not enter into this law of physics, it must remain an analogy.
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Chemistry for Everyone
A Geological Application The geological principle of isostatic uplift, sometimes called isostatic rebound, also shows definite similarities to LeChâtelier’s principle. The basic idea of isostasy is that the crustal masses of the earth’s surface layer “float” upon the semiplastic mantle, much like ice cubes in a tub of water. We can equate large ice cubes with mountains and small ones with plains. If material is added to the small ice cube it floats lower in the water. On the other hand, if matter is removed from the large ice cube it will rise in the water. There exists a gravitational equilibrium in the earth’s crust. If this equilibrium is disturbed, then there occurs a compensatory movement in the crust. For example, owing to erosion, mountains become lighter and rise, whereas the areas of deposition of the eroded material become heavier and sink. If the Greenland ice cap were to melt, the crust would rise until a new isostatic balance is achieved (6 ). The British geophysicist John Henry Pratt (1809–1871) and the astronomer Sir George Biddell Airy (1801–1892) introduced this concept in 1855, although the two differed in their consideration of the density of the various crustal “blocks” (7). This principle is also analogous to LeChâtelier’s principle, but it is an example of mechanical rather than thermodynamic equilibrium. It is a consequence solely of conservation of energy, in this case gravitational potential energy. Biological Applications We find an analogy to LeChâtelier’s principle in the biological sciences appearing under the name homeostasis: “Homeostasis...is the term generally applied to the tendency for biological systems to resist change and to remain in a state of equilibrium” (8). The similarity with LeChâtelier’s principle is made even clearer in the following excerpt: “Homeostasis is said to be shown by a (physiological) system if, given a moderate disturbance that tends to displace the system from its normal values, its parts so react and interact that the harmful effects of the disturbance are much diminished” (9). This concept of homeostasis finds application at various levels of biological organization. An example from the ecology of population distributions is provided by the following excerpt from Odum’s classic text on the subject (10): “Lotka (1925) has shown on theoretical grounds that a population tends to develop a stable age distribution, that is, a more or less constant proportion of individuals of different ages, and that if this stable situation is disrupted by temporary changes in the environment or by temporary influx from or egress to another population, the age distribution will tend to return to the previous situation upon restoration of normal conditions. More permanent changes, of course, would result in development of a new stable distribution.” Examples of homeostatic regulation at the physiological level include the maintenance of body temperature (“homeothermy”) and the body’s acid–base balance. “All these examples of regulatory phenomena…are correlated with the general maintenance of a so-called steady state, or HOMEOSTASIS, in the body. The internal environment is maintained in a relatively constant condition by a complex series of 174
Figure 1. Equilibrium and steady state.
mechanisms such as those for preventing marked changes in the volume or composition of the blood, in the temperature of the body, in the normal sugar and salt content of the blood, and so on” (11). Is homeostasis the biological equivalent of LeChâtelier’s principle, or simply another analogy like the previous two examples (Lenz’s law and isostasy)? While the above quotations show a clear similarity to this principle, they also raise some flags. Note the use of the term “equilibrium” in the first quotation of this section and “steady state” in the last. Also, the term “regulation” surfaces. The concept of equilibrium is not applicable to biological systems, except at the molecular level. At all higher levels of organization, cell, organ, organism, and community, there is a constant flow of energy and matter maintaining a steady state. The confusion of these two terms “equilibrium” and “steady state” is not really surprising, as in common (dictionary) usage the term “equilibrium” is usually defined as a “state of balance”, which may be used to describe both conditions. (Note that we have a similar problem with the terms “precision” and “accuracy.”) Figure 1, adapted from ref 12, shows the difference between an equilibrium condition and a steady state. On the left we find an equilibrium between fresh water and salt water. The water is also in equilibrium with the air above it in the closed container. A steady-state geological analogue is shown on the right. Note the feedback loop from the ocean to the lake. One of the few nonchemical references to LeChâtelier’s principle discovered was in the field of biology (13). The editor of this volume states in his introduction that LeChâtelier’s principle is not applicable: “Events in the inorganic world— to which man-made machines do not belong—though ‘governed by laws’ are not conspicuously regulated, claims to the contrary like LeChâtelier’s (1888) principle notwithstanding.” Although in agreement with the inapplicability of LeChâtelier’s principle, it should be noted that the geological example of Figure 1 shows regulation through feedback in the “inorganic world”. In fact, one might conclude that regulation is a steady-state phenomenon, required as a necessary (but not sufficient) condition. (Steady-state heat transfer along a rod is an example without feedback.) There is another theoretical approach to LeChâtelier’s principle, based on non-equilibrium or irreversible thermodynamics (14 ), which ties this all together. In the section entitled “Stationary States and Biological Systems”, this reference mentions LeChâtelier’s principle explicitly and shows that it is a special case of the entropy production condition,
Journal of Chemical Education • Vol. 77 No. 2 February 2000 • JChemEd.chem.wisc.edu
Chemistry for Everyone
Conclusions
Figure 2. Supply and demand curves.
valid only for the equilibrium situation: “It is a fact, however, that the increase of flux is always attended by a diminution of the force conjugate to it, that the flux δJm will tend to nullify the perturbation δXm, and thus restore the unperturbed stationary state. This conclusion was first formulated by LeChâtelier (1888) for thermostatic equilibrium (k = 0); the extension of his principle to k = 1 was made by Prigogine (1947) and to stationary states of higher order by de Groot (1951).” The more general case is that of a steady-state system, with application to biological systems (15, 16 ). Therefore, while the biological concept of homeostasis provides another interesting analogy to LeChâtelier’s principle, we must conclude that it is not identical. In fact, because of its steady-state nature, it would seem to be more general. Economics and LeChâtelier’s Principle The “Law of Supply and Demand” has been called LeChâtelier’s principle applied to economics (17). And indeed, when we examine a statement of this law, the analogy becomes apparent. The supply of and demand for a particular commodity establish its market price. The market is said to be in equilibrium when the price remains constant. A disturbance of the market equilibrium due to a change in supply or demand will result in a shift of the equilibrium to a new price level in a direction such as to offset the disturbance. In modern economic theory there is no supply or demand; rather, there are supply and demand curves. In Figure 2 the amount of commodity available (y-axis) is graphed versus the price per unit quantity (x-axis) (18). The supply curve has a positive slope, whereas the slope of the demand curve is negative. The intersection of the two curves establishes the equilibrium market price. With a shift in the demand curve to the right, for example, a shift in the equilibrium occurs such that the new market price is greater than before, tending to counteract the increased demand. This is a regulatory mechanism in a steady-state system, similar to the concept of homeostasis already discussed. The use of the term “equilibrium” is inappropriate, though perhaps understandable (see comments above), and should be replaced in the discussion above with “steady state”. Therefore, this law would also seem to be more general than LeChâtelier’s principle. Nevertheless, this example from the field of economics can provide a useful analogy in introductory chemistry classes.
LeChâtelier’s principle is a very general statement and we find analogies in various fields of scientific endeavor that may well be pedagogically useful in introductory chemistry classes. Each of the four examples presented above shows an obvious similarity to the statement of this principle. In fact, at first glance, they all appear identical. A closer look, however, reveals that Lenz’s law in physics and the geological principle of isostatic uplift both depend only on conservation of energy and thus might be considered more restrictive (less general) than the thermodynamic principle of LeChâtelier. The biological concept of homeostasis and the economic law of supply and demand are examples of steady-state systems regulated by feedback mechanisms. They too must also be regarded as analogies, albeit of a higher order, with LeChâtelier’s principle as the special case. It may, however, be more appropriate to consider all these principles, laws, and concepts as “principles of stability” within the applicable context of mechanical, electromagnetic, or thermodynamic equilibrium, or the non-equilibrium steady-state systems discussed above. In each case we find mechanisms in place that serve to maintain the equilibrium or steady-state condition—that is, the “state of balance”. Viewed in this way, any characterization as “less general” or “more general” disappears in favor of the context of application of the principle: All show that regulation occurs broadly in the natural universe. Finally, it is also important from a pedagogical viewpoint to identify LeChâtelier’s principle as a consequence of the more general law of conservation of energy and the second law of thermodynamics. Only in this way can the attention of the student be diverted from the many individual laws to the few truly important general principles of science. Acknowledgment I am indebted to the anonymous reviewer who pointed out several flaws in the initial manuscript and generously provided suggestions to improve the revision. Literature Cited 1. Krauskopf, K. Introduction to Geochemistry; McGraw-Hill: New York, 1967; p 546. 2. Bauman, R. P. Modern Thermodynamics with Statistical Mechanics; Macmillan: New York, 1992; p 293. 3. Dictionary of Scientific Biography; Gillispie, C. C., Ed.; Charles Scribner’s Sons: New York, 1980; Vol. 13, p 578. 4. Serway, R. A.; Faughn, J. S. College Physics, 2nd ed.; Saunders: Philadelphia, PA, 1989; p 552. 5. Bueche, F. Introduction to Physics for Scientists and Engineers; McGraw-Hill: New York, 1969; p 519. 6. Zumberge, J. H. Elements of Geology; Wiley:London,1958;p73. 7. Mears, B. Jr. The Changing Earth: Introduction to Geology; Van Nostrand: New York, 1977; p 202. 8. Odum, E. P. Fundamentals of Ecology; Saunders: Philadelphia, PA, 1971; p 34. 9. Arbib, M. A. Brains, Machines, and Mathematics; McGrawHill: New York, 1964; p 106. 10. Odum, E. P. Op. cit., p 176.
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Chemistry for Everyone 11. Woodruff, L. L.; Baitsell, G. A. Foundations of Biology, 7th ed.; Macmillan: New York, 1951; p 217. 12. Casey, E. J. Biophysics: Concepts and Mechanisms; Reinhold: New York, 1962; p 195 (see also Chapter 8, Speeds of Some Processes in Biological Systems, pp 192–233). 13. Kalmus, H. Regulation and Control in Living Systems; Wiley: London, 1966; p 4. 14. Yourgrau, W.; van der Merwe, A.; Raw, G. Treatise on Irreversible and Statistical Thermophysics; Dover: New York, 1982; pp 48–53; originally published by Macmillan: New York, 1966. 15. Katchalsky, A.; Curran, P. Non-Equilibrium Thermodynamics
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in Biophysics; Harvard University Press: Cambridge, MA, 1965; pp 37, 74–75, 231–235. 16. Katchalsky, A. In Biology and the Physical Sciences; Devons, S., Ed.; Columbia University Press: New York, 1969; pp 267– 298. This article provides a short introduction to non-equilibrium thermodynamics and discusses energy conversions in equilibrium and non-equilibrium conditions applied to biological systems. 17. Chem. Eng. News 1994, 72(28 Feb), 2. 18. McConnell, C. R. Economics; McGraw-Hill: New York, 1987; pp 50–64.
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