In the Classroom
Apparent Paradoxes and Instructive Puzzles in Physical Chemistry Lawrence S. Bartell Department of Chemistry, University of Michigan, Ann Arbor, MI 48109;
[email protected] Posing puzzles that challenge the imagination of students makes lectures more lively and conveys ideas in a way that makes them stick with students longer than direct statements of fact and principle. Some examples are presented in the following. The answers are given at the end of the paper. The Puzzles
1. Can mass be converted into energy? Suppose a calorimeter could be constructed that was so sturdy an atomic bomb could be exploded in it without any observable change on the outside. No particles, radiation, or even noise would escape. Would the weight of the calorimeter plus contents decrease, increase, or remain the same after the explosion? 2. A violation of the conservation of energy? Suppose N photons of frequency ν are emitted by a laser aimed at a stationary black body. The black body absorbs the energy Nh ν and converts it to thermal energy (heat). Compare this with the energy Nh ν′ absorbed if the black body is moving away from the source. By the Doppler effect, ν′ is less than ν and consequently the absorber sees and absorbs photons of lower energy than emitted by the laser. Where did the extra energy go? The answer to this simple question eludes many physical chemistry professors. It does, however, yield some important results.
3. A valid perpetual motion machine? First, a brief review of osmotic pressure is appropriate. It should be recalled that the osmotic pressure Π is the pressure at equilibrium exerted by a solution on a semipermeable membrane separating the solution from a body of pure solvent. If a pressure exceeding Π is brought to bear on the solution, the solvent in the solution is forced through the membrane, leaving the solute behind. Such a phenomenon is called reverse osmosis. (This is one method of desalinating sea water at the cost of pushing the piston that presses the solvent [pure water] through the membrane.) It appears to be simple to apply reverse osmosis to devise what seems to be a perpetual motion machine of the second kind (violating the second law of thermodynamics). Place a semipermeable membrane designed to pass water but not salt at the end of a long tube and lower the tube, membrane first, into the ocean. Suppose the tube is first filled to the sea level with fresh water (immaterial to the problem but it helps to visualize what is to follow). By taking advantage of the difference in density between fresh water and sea water, one can make the difference between the hydrostatic pressure on the outside of the tube and that of the fresh water filling the inside of the tube as great as one wishes by pushing the tube as far down as is needed. If this pressure difference is made so great that it substantially
exceeds the osmotic pressure, then the fresh water on the inside of the tube is forced by reverse osmosis to flow out of the tube above the ocean surface, desalinating the water at no energetic cost (once things get started). And as the water flows out the top it could even run a paddle wheel, producing work. What is wrong with the idea? This puzzle is one I used in class after it was posed in the Scientific American magazine (1) without an answer. A pedagogically unsatisfactory answer subsequently appeared in a letter to the editor. This puzzle stumps even most professors of physical chemistry, though biochemists often see the answer. Moreover, the machine would probably actually work in practice! Why?
4. Can heat be converted completely into work? Such a conversion is not forbidden by the first law. Is it forbidden by the second law? Show how it can be done. 5. Is entropy of mixing a valid concept? Can gases be mixed reversibly? And unmixed? Reversibly? If the answers were no, in principle, there would be no valid way in classical thermodynamics of determining the entropy of mixing. 6. Must entropy increase when gases are spontaneously mixed? This deceptively simple question raises the Gibbs paradox (2), which was satisfactorily resolved only after the advent of quantum mechanics. What happens when two volumes of, say, nitrogen are mixed isobarically and isothermally? 7. A violation of the third law? There appears to be a violation if the classical law of equipartition of energy is invoked. The third law of thermodynamics is a striking manifestation of quantum effects, as can be illustrated by the following exercise. Have the students calculate the third law entropy of a crystal at T1 > 0 K in class if the classical heat capacity of monatomic crystals, 3R, is assumed to hold. Although the calculation is elementary, it isn’t quite trivial to average students and the answer often surprises them. 8. A violation of Henry’s law for hydrogen dissolved in Pd, or HCl dissolved in water? The amount dissolved is not proportional to partial pressure of the vapor but rather to the square root of the partial pressure. Although the answer is obvious to the teacher it isn’t at first to all students. 9. Is the Gibbs–Poynting relation wrong? This law relates the vapor pressure of a liquid to the external pressure imposed on the liquid. It is brought up here because its application is illustrated badly in most textbooks.
JChemEd.chem.wisc.edu • Vol. 78 No. 8 August 2001 • Journal of Chemical Education
1067
In the Classroom
Most books apply the external pressure by introducing an “inert gas” at high pressure to a cell containing the liquid. It turns out that this method fails to accomplish what it is supposed to. For example, if you apply air pressure to water and then determine water’s resultant “vapor pressure” from the number of vapor moles of water per unit volume by using P = nRT/V, the pressure effect will be off by roughly an order of magnitude (3). If you apply ethylene pressure over naphthalene, the discrepancy will be even greater. Why? Is the Gibbs–Poynting relation wrong?
10. Are common-sense expectations really valid? Do you always have to add heat to a crystal to melt it? Is the entropy of a crystal always lower than that of the liquid in equilibrium with it? Answers to the Puzzles and Paradoxes in Physical Chemistry
1. Can mass be converted into energy? This question used to raise spirited arguments with students not long after World War II and the atomic bomb because so many books were written so badly, asserting that mass disappears as it is converted into energy! But today, little passion is aroused and many or most students get the answer right. The mass of the calorimeter and contents is unchanged by the explosion. According to Einstein’s famous equation E = mc 2, mass and energy are equivalent. It is quite true that the rest mass of the contents of the calorimeter after the explosion do not add up to the rest masses before the explosion. Part of the rest mass of the explosive is converted to radiation and to the kinetic energy of the atoms in the calorimeter, but these alternative manifestations of energy possess the weighable mass, E/c 2. It is true that mass is transformed from one form to another and energy is transformed from one form (latent) to another so that the total weight is not changed by the process. This attribution of mass to radiation within the calorimeter does not contradict the current custom of calling photons “massless”. They have no rest mass (and are no longer photons when at rest), but when reflecting from the walls of a container trapping them or when absorbed by an absorber, they impart weighable mass to the object. Recall that photons are attracted by a gravitational field and hence exert a greater radiation pressure on the bottom of a container than on the top. 2. A violation of the conservation of energy? Where the remainder of the energy goes is in work done on the moving object by the radiation pressure of the photons acting over a distance. By applying the first law of thermodynamics it is quite easy to show that this pressure is just the energy density E/V of the radiation. An interesting application is to operate a Carnot engine using a “photon gas” to exert pressure on the piston. From the efficiency of this Carnot engine comes the Stefan–Boltzmann radiation law. It is only a small skip from the concept of pressure exerted by radiation to illustrate Einstein’s favorite way of demonstrating that E = mc 2. 3. A valid perpetual motion machine? What makes the perpetual motion inoperable under equilibrium conditions is the sedimentation equilibrium 1068
law well known to biologists who centrifuge solutions. The individual solute “particles” in solution have an effective “density” (molecular weight/partial molar volume) greater than that of water. Therefore, the gravitational field makes the salt concentration increase with increasing depth, and the increase is just fast enough to keep the hydrostatic pressure difference from exceeding the osmotic pressure, however deeply the tube is inserted. The simplest derivation of the sedimentation equilibrium law, a law analogous to the isothermal barometric law except for the buoyancy effect of the medium, is to calculate the concentration gradient required to prevent the perpetual motion machine from working. This sedimentation law is what allowed Perrin to determine Avogadro’s number in 1909 in his famous experiments with the sedimentation of the colloid gamboge (4 ). Why would the perpetual motion machine probably work in practice? Because the ocean is far from equilibrium. Swimming creatures and, more importantly, ocean currents driven by the sun and tidal forces due to the moon stir up the salt, making it more uniform in concentration than at equilibrium. So the machine would probably work (inefficiently) for some of the same reasons that windmills work.
4. Can heat be converted completely into work? One way for heat to be converted completely into work is in an isothermal, reversible expansion of an ideal gas. This is not a practical way to generate work, however, because the system cannot be returned to its original state without converting some work into heat. All effective heat engines are cyclic. 5. Is entropy of mixing a valid concept? Gibbs showed how to carry out mixing and unmixing of gases reversibly, in principle, with the aid of two pistons that are semipermeable membranes, each passing one of the gases but not the other. Therefore, a reversible mixing path can be constructed and the reversible heat and entropy change can be determined (2). 6. Must entropy increase when gases are spontaneously mixed? If two identical gases such as a given isotopic species of nitrogen are mixed, no one doubts that an irreversible diffusion of molecules takes place. Yet the state of the system hasn’t changed in any demonstrable way, and the Gibbs stratagem of reversible mixing–unmixing is no longer conceivable, even theoretically. In quantum mechanics, identical molecules are truly indistinguishable, and quantum statistics denies any meaning to the changing of a state by physically shuffling indistinguishable particles (2). 7. A violation of the third law? Attempting to calculate the third law entropy of a crystal by evaluating the integral ∫0T1[Cp(T )/T ]dT with the classical Cp(T ) = 3R leads, of course, to an infinite result. For the third law to work, Cp(T ) must vanish sufficiently rapidly as absolute zero is approached for the integral to be well behaved. In real (quantum) systems with quantized energy levels, the energy gap between the ground state and next higher quantum state prevents a free absorption of heat as the bath temperature is raised until kT becomes comparable to the energy gap.
Journal of Chemical Education • Vol. 78 No. 8 August 2001 • JChemEd.chem.wisc.edu
In the Classroom
Therefore, Cp(T ) starts at zero and remains low until that temperature is attained.
radius, the enhancement of vapor pressure and its dependence on drop size is given by the “Kelvin equation”.
8. A violation of Henry’s law for hydrogen dissolved in Pd, or HCl dissolved in water? When gaseous hydrogen is dissolved in Pd or gaseous HCl is dissolved in water, the individual gas molecules dissociate into two atoms or ions. Mass action accounts for the dependency of solute concentration on the square root of the gas pressure.
10. Are common-sense expectations really valid? The answers are no. As explained in some detail in ref 5, it is not impossible to freeze a liquid by adding thermal energy. A seemingly anomalous, counterintuitive example is that of the 3He system, whose properties are conspicuously shaped by the quantum behavior of very light particles. In a system with liquid 3He in equilibrium with its crystalline phase, when the temperature is under 0.3 K, the liquid is frozen by adding heat. Clearly, then, the solid has a higher entropy than the liquid with which it is in equilibrium. As far as I am aware, 3He is the only one-component system possessing such a property. This example may not be very important in ordinary thermodynamic considerations, but it is curious. And it should warn students of the dangers of overgeneralizing.
9. Is the Gibbs–Poynting relation wrong? The Gibbs–Poynting relation is correct. The reason for the discrepancy between theory and experiment, of course, has to do with the effect of nonideality. A liquid under pressure because of compressed gas above it is in contact with what can be considered a solvent. Gas under pressure is not inert. In essence, it is able to dissolve a higher concentration of the pressurized liquid or solid than would be possible in the absence of the “inert” gas. How can the pressure be applied to a liquid in such a way as to satisfy the Gibbs–Poynting requirements in a thermodynamically rigorous way? One way would be to compress the liquid with a Gibbs semipermeable membrane that passes gas but not liquid. Another is for the pressure to be the Laplace pressure exerted on small droplets by surface tension. When a drop is compressed by its own Laplace pressure, a pressure inversely proportional to the drop
Literature Cited 1. Levenspeil, O. Sci. Am. 1971, 225 (6), 100. 2. Pauli, W. Thermodynamics and the Kinetic Theory of Gases; MIT Press: Cambridge, MA, 1973; pp 45–48. 3. Webster, J. T. Discuss. Faraday Soc. 1953, (15), 243. 4. Perrin, J. C. R. Acad. Sci. 1908, 147, 475–479. 5. Bartell, L. S. J. Chem Educ. 2001, 78, 1059–1067.
JChemEd.chem.wisc.edu • Vol. 78 No. 8 August 2001 • Journal of Chemical Education
1069
