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BIASED THERMAL MOTION AND THE SECOND LAW OF THERMODYNAMICS Pentcho Valev
Abstract
Thermal motion in an ideal gas system is isotropic. This implies that, in order to counteract the respective force (gas pressure) and return the system to its initial state after the gas has done work, one needs to apply a greater or at least equal force and waste work in the process. For that reason, if one is preoccupied with gas systems, one sees no reason to doubt the second law of thermodynamics. However this could be an illusion due to the fact that the thermal motion is isotropic. Structuralized systems can generate anisotropic (biased) thermal motion. In this case the respective macroscopic force can be counteracted without wasting work and the second law is not so incontestable. By an appropriate phenomenological analysis, the problem can be reduced to the following question: Are the two partial derivatives in a Maxwell relation equal or not? Since these partial derivatives have physical meanings and are measurable, the second law turns out to be an experimentally falsifiable hypothesis. If experiments show that the two partial derivatives are equal, the second law is confirmed. If the partial derivatives differ, the second law is wrong.
There is a strange confusion which, amazingly, resides in elementary electrostatics. Consider a constant-charge parallel-plate capacitor with a polarized solid dielectric between the plates:
Since the molecules of the dielectric material are polarized, they generate an electric field which counteracts the original field and so reduces the voltage between the plates. The question is: How about the attraction between the plates? Does the presence of the dielectric increase or decrease it?
The curious thing is that the attraction actually increases (the polarization obviously reinforces the original attraction) whereas textbooks either beg the question or imply that the attraction decreases! If a student looks for a qualitative solution to the problem (increase or decrease?) he/she may consider the plates as two opposite charges separated by a dielectric and see what textbooks say about the force of attraction between such charges. Sooner or later he/she will come across an expression for Coulomb’s law where a factor 1/k accounts for the influence of the dielectric. The dielectric constant k, as placed in the denominator, indicates a decrease in the force of attraction in the presence of a dielectric!
In fact, the confusion is much greater than that. If the dielectric is liquid and the plates are totally immersed in it, the force of attraction does indeed decrease and the factor 1/k looks relevant. However its physical meaning is obscure so calling k "dielectric constant" is unjustified. The following quotation is from perhaps the only source where the problem is mentioned: "This means that if a system maintained at constant charge is totally surrounded by a dielectric liquid all mechanical forces will drop in the ratio 1/k. A factor 1/k is frequently included in the expression for Coulomb’s law to indicate this decrease in force. The physical significance of this reduction of force, which is required by energy considerations, is often somewhat mysterious. It is difficult to see on the basis of a field theory why the interaction between two charges should be dependent upon the nature or condition of the intervening material, and therefore the inclusion of an extra factor 1/k in Coulomb’s law lacks a physical explanation" ([1], p. 114).
Let us reformulate the problem. In order to be able to quantify the observed decrease in voltage between two opposite charges separated by a dielectric, one introduces the factor 1/k where k is called the dielectric constant and is defined as the ratio of the capacitance in the presence of the dielectric and the capacitance in a vacuum. Then, encouraged by the success of this initial step, one includes the same factor 1/k in Coulomb’s law but immediately gets into trouble. When the dielectric is solid (or liquid but confined within a solid box), the inclusion of 1/k makes Coulomb’s law predict the opposite of what really happens. The solid dielectric reinforces the force of attraction between the plates whereas Coulomb’s law, with the factor 1/k, predicts a reduction of the force.
On the other hand, when the dielectric is liquid and totally surrounds the plates, the inclusion of 1/k leads to a correct qualitative prediction but the physical meaning of 1/k is mysterious and it is not at all clear why this k should be quantitatively identical to the k which accounts for the decrease in voltage. The fact that both quantities have been given the same symbol and name may turn out to be a costly mistake.
Let us try to disentangle the conundrum. When two opposite charges (or capacitor plates) are immersed in a liquid dielectric, e.g. water, some additional pressure between them emerges, pushes them apart and so counteracts their electrostatic attraction ([1], pp. 111-116). If the plates are vertical and only partially immersed, the self-same pressure forces the liquid between the plates to rise above the surface of the water pool (see fig. 6-7 on p. 112 in [1]):
On examining the picture, the first impression may be that the effect (lifting of water between the plates) is due to capillary action. One may imagine that the plate "pulls" molecules upwards, they in turn "pull" their neighbors etc. However it is not difficult to see that, if "pulling" were the only cause for lifting, the original electrostatic attraction between the plates will be reinforced by the "pulling" force so that the resultant attraction between the plates will be greater in water than in a vacuum. Yet Panofsky and Phillips ([1], pp. 111-116) speak of an additional pressure generated in the bulk which pushes the plates apart so that the resultant attraction is substantially weaker than in a vacuum.
What mechanism is responsible for the additional pressure? Let us return to the above picture. If it were not for the stigmatized dipole, other dipoles on the picture are perfectly polarized as if there were no thermal motion. Of course, this is an oversimplification – the thermal motion is a factor which constantly disturbs the polarization order. However the crucial point is that, as can be inferred from the picture, any thermal disturbance contributes to the creation of a pressure between the plates. Consider the stigmatized dipole. It has just received a strong thermal stroke and has rotated. As a result, it rejects adjacent dipoles electrostatically and pushes them towards the plates. Macroscopically, the sum of all such disturbances is expressed as a pressure exerted on the plates. One can also say that the stigmatized dipole has absorbed heat and now, by rejecting adjacent dipoles, is trying to convert it into work. When the distance between the plates is fixed, a horizontal displacement of the liquid is impossible and the work consists in pushing liquid upwards, against gravity. In the process, the heat absorbed is converted into potential energy of the lifted water.
What if one punches a small hole in one of the plates, just above the surface of the pool? Will the lifted water leak through the hole and fall, thereby dissipating its potential energy as heat? If this happens, one will have a perpetual macroscopic cycle capable of producing work. Needless to say, such a cycle contradicts the second law of thermodynamics. No matter how weak the waterfall is, theoretically it can rotate a waterwheel…
Let us repeat the main arguments. If lifting is due to capillary action, i.e. if the pulling upward force exerted by the plate is the only one responsible for lifting, water will refuse to leak through the hole and the second law will be saved. However in this case the substantial reduction of the attraction between the plates in water will remain mysterious since capillary action reinforces rather than weakens the original attraction. If lifting is due to an additional pressure generated within the bulk, as assumed by Panofsky and Phillips, then water will leak through the hole and the second law will be violated. On the other hand, this latter interpretation makes the decrease in the original attraction understandable: the pressure exerted on the plates counteracts the original attraction and so weakens it.
A digression. There is something very strange in the above story. The problem is fundamental so far as a textbook concept – the dielectric constant – is concerned. And the alarm is set off in an authoritative textbook - Panofsky and Phillips use expressions such as "mysterious", "lacks a physical explanation" etc. Yet nobody seems to have discussed the issue since 1962, as if some prohibition existed. Even Panofsky and Phillips beg the question to some extent and avoid calling k "dielectric constant" when discussing the mysterious pressure in dielectric liquids. Where do the fear and confusion come from?
The following conjecture deserves attention. The electrostatic theory is built on the explicit assumption that all the forces the theory deals with are conservative. "Conservative" implies that, as one does work against such a force, isothermally, the supplied energy is stored in the system and not ejected as heat. If the energy is ejected as heat, the work is done against a non-conservative force (e.g. gas pressure). Is the mysterious pressure non-conservative? Panofsky and Phillips claim that "the decrease in force…..cannot be explained by electrical forces alone" (p. 115). If the pressure is non-conservative, the electrostatic theory is just as wrong as any other theory built on a wrong assumption. Such a conclusion would be too painful and physicists prefer to avoid it.
It makes sense to recast the problem in thermodynamic (phenomenological) terms. Let us assume that a closed system in equilibrium is so designed that (at least) two constraints can be removed and two specific isothermal work productions initiated. The total work done by the system can be expressed as
dW = dW1 + dW2 (1)
where W1 and W2 are the two specific works involved. Since the system is initially in equilibrium, the work-producing process can be reduced to a succession of equilibrium states. This will be achieved if the counteracting external force (against which the work is done) is kept almost equal to the work-producing force generated by the system. In this case eq. (1) takes the form
dW = F1dX1 + F2dX2 (2)
where F is a work-producing force generated by the system and X is the respective displacement.
Under the conditions specified (succession of equilibrium states and constant temperature), the variables X1 and X2 determine the state of the system - once their values are fixed, the system remains in a unique equilibrium state. On the other hand, W obviously depends on X1 and X2. But this still does not imply that W is a function of X1 and X2. Let us imagine a process in which X1 and X2 slowly change and return to their initial values so that the difference D W=Wfinal-Winitial proves positive in the end. If such a cycle is possible, W is not a function of X1 and X2. In order for W to be a function of X1 and X2, i.e. for one to be allowed to write W(X1, X2), D W must be zero at the end of the reversible isothermal cycle. Since X1 and X2 determine the state of the system (under the conditions specified), W(X1, X2) can also be defined as a state function.
Now if W is a state function, the second law is obeyed - there is no way to extract work from the cycle. If, however, W is not a state function, the system is able, cyclically, to absorb heat from the surroundings and convert it into work. D W is exactly the maximum work extractable from the isothermal cycle. If, for a given cycle, D W is negative, it can be made positive by carrying out the cycle in the opposite direction.
So far I have shown that, if the second law is correct, the implication is that W is a function of X1 and X2. A next implication is provided by the following mathematical theorem:
If W is a function of X1 and X2, the mixed partial derivatives ¶ 2 W/¶ X1¶ X2 and ¶2 W/¶ X2¶ X1 are equal.
The proof of the theorem is given in mathematical textbooks and will not be discussed here.
Since F1 and F2 are the first partial derivatives, the theorem can be expressed in the form of the Maxwell relation
¶ F1/¶ X2 = ¶ F2/¶ X1 (3)
(It is not quite clear why equations of the type (3) are called Maxwell relations but we shall use this name since it is popular in physical chemistry texts.) The two sides of (3) have independent physical meanings and, even more importantly, are measurable. This leads to the following conclusion. If experiments unambiguously show that the values of the two partial derivatives in (3) differ - e.g. one is positive and the other negative, the second law will have to be rejected. The reason is that, in accordance with the above argumentation, the truth of the Maxwell relation (3) is a necessary condition for W to be a state function.
Let us return to our initial example where a constant-charge capacitor with vertical plates is partially immersed in a pool of water. This system (capacitor plus pool in a gravitational field) is able to produce two types of work. On one hand, one can draw the plates together or apart - the work the system does is positive in the former and negative in the latter case. On the other hand, one can let the capacitor down or lift it - the work the system does is again positive in the former and negative in the latter case. Accordingly, eq. (2) takes the form
dW = -Fdl - Gdh (4)
where F is the magnitude of the force of attraction (which resists drawing the plates apart), G is the force that pulls the capacitor downwards, l is the distance between the plates and h is the height of the capacitor with respect to some point of reference with a zero height. The respective Maxwell relation is
¶ F/¶ h = ¶ G/¶ l (5)
The partial derivative on the left, ¶ F/¶ h, is obviously positive - as the capacitor is gradually immersed (dh<0), the force of attraction between the plates decreases (dF<0). As for the partial derivative on the right, ¶ G/¶ l, there are theoretical arguments suggesting that it is zero or almost zero. But I am not going to present those arguments since my intention is to put the accent on the experimental approach. Since both partial derivatives have physical meanings and are measurable, we must first of all enjoy the availability of a method for experimental verification of the second law of thermodynamics. If an experiment (consisting in just weighing the capacitor for different values of l) shows that ¶ G/¶ l is zero, the unavoidable conclusion is that the second law is violated.
Let us pay some more attention to the molecular mechanism. Our stigmatized dipole underwent rotation but, since the polarization closely packs the dipoles in the horizontal direction, it would have been difficult for the thermal motion do move the stigmatized dipole towards the plates. Rotation was difficult, horizontal displacement was even more difficult. If so, we may say that the thermal motion was biased - some components of the motion were "easier" than others. Another essential feature is that the "easier" components contribute to the building of a macroscopic force which can do work.
The issue is extremely important so let us consider a more spectacular example. There are biological nanomachines - protein motors - which are traditionally regarded as Brownian machines [2]. This image implies that the thermal motion involved is originally isotropic; only secondarily can it be biased by an anisotropic medium involving some gradient or a chemical reaction far from equilibrium. The implication is both wrong and misleading. Structural features of proteins can primarily bias thermal motion so that the latter occurs in the form of large, low-frequency peptide librations [3] [4].
A nice macroscopic model is a toy children in poor families easily assemble. A thread is passed through two of the holes of a button and then the ends are tied. Eventually, the child extends the loop between his/her thumbs, with the button suspended halfway between the thumbs. After some initial winding, the button starts rotating, now clockwise and then anticlockwise. The child reinforces the rotation by forcibly stretching the loop at specific stages of the process.
What amuses the child is that the rotation makes the thread elastic - the elastic force draws the thumbs together. So it takes work to stretch the loop with the thumbs. However, as soon as somebody’s hair gets intertwined with the thread, the elastic force suddenly disappears and then it takes no work to stretch the loop.
The motion of the button is obviously "biased" - it is not isotropic but rather occurs in a plane perpendicular to the line connecting the thumbs. The essential point is that the elastic force the motion generates acts along this line. Another essential point is that biasing allows one to counteract the elastic force without spending work. "To put a spoke in one’s wheel" reflects the same comfortable opportunity.
The only essential difference between the motion of the button and peptide librations is that the latter, being a variety of biased thermal motion, do not need any external reinforcement. Once the impediment to the motion is removed ("the spoke is taken out of the wheel"), librations resume all on their own. In biological systems, the role of impediment can be played by clusters of structured water around ionized weak acid groups or, in a less lucid way, configurations resulting from the interaction between the molecular motor and ATP.
Let us now consider a system where work is done by "chemical springs" - macroscopic contractile polymers driven by biased peptide librations. There are two types of polymers which on acidification (decrease of the pH in the system) contract and can lift a weight. The reader is referred to fig. 16A in [4] for visualization. Here is the essence of the picture:
Although both types contract on acidification, they differ in that polymers designed by Urry (UP) absorb protons on stretching (as the length of the polymer increases), whereas those designed by Katchalsky (KP) release protons on stretching (see discussion on p. 11020 in [4]).
Let us assume that two macroscopic polymers, one of each type (one UP and one KP) are suspended in the same system:
At constant temperature, the maximum work done by the system is given by
dWmax = -FUPdLUP - FKPdLKP (6)
where F is the force of contraction and L is the length of the polymer. The respective Maxwell relation is
¶ FUP/¶ LKP = ¶ FKP/¶ LUP (7)
Although experiments involving this particular setup (one UP and one KP in the same solution) have not been done, the signs of the partial derivatives in eq. (7) can be assessed by using experimental results reported on p. 11020 in [4]. As KP is being stretched (LKP increases), the polymer releases protons, the pH decreases and, accordingly, FUP must increase. Therefore, the left partial derivative in eq. (7) is positive. In contrast, as UP is being stretched (LUP increases), it absorbs protons, the pH increases and FUP must decrease. Therefore, the right partial derivative in eq. (7) is negative. If the estimated difference between the two derivatives does not result from experimental mistakes, the second law should be rejected. Since W is not a state function, the net work extractable from some isothermal reversible cycle is positive.
The usefulness of Maxwell relations is obvious. One does not need to try to build a perpetual motion machine. Rather, one just needs to assess, experimentally, the two partial derivatives in eq. (7). If the experiment unequivocally shows that their values differ, the second law is disproved.
Here it is pertinent to digress and explain the phenomenology of the two-work approach. Consider an oversimplified machine that has an executive unit at one end, e.g. a lever that can lift a weight, and a switch at the other. The machine exchanges work with the surroundings through both the lever and the switch and the two work productions interact with one another - as we press the switch, the executive work-producing force increases. Naturally, we prefer the work exchanged through the switch to be small and negligible. However our theoretical curiosity leads us to the suspicion that this quality of the switch makes the machine too anthropocentric. So we consider the general case where the second work exchange is not negligible. We replace the switch with another lever but continue to assume that the two work productions interact with one another. The interaction can be presented in the following way:
The lever on the left is lifting a weight whereas the lever on the right is fixed. However the interaction between the levers is such that the performance of work on the left increases the upward force on the right.
Then we consider the symmetrical process:
Now the right-hand lever is lifting a weight whereas the left-hand one is fixed. However in this case, in contrast to the previous one, the upward force on the left decreases in the process.
We may not be able to explain the internal mechanism of the interaction between the two work productions - the machine may be a "black box" for us. Still the phenomenological picture is complete: the case is analogous to the chemical-spring case and the respective partial derivatives differ in sign (one is positive and the other negative). We can even imagine a four-step isothermal cycle which produces work in violation of the second law:
Clearly, each step modifies the next one so that, in the next step, either more work is gained or less work is spent. For instance, since in Step 2 the upward force on the left decreases, letting down the left-hand lever in Step 3 consumes less work than it would if Step 2 had not occurred.
Cyclical conversion of heat into work, in violation of the second law of thermodynamics, seems to be possible whenever structural features allow a system to generate biased thermal motion. Then could the second law be an illusion? Remember the popular argument according to which, if the second law were not correct, a ship would extract heat from the ocean (single heat reservoir), convert this heat into work and so travel without any energy problems. Is this a scientific argument? What if the ship cannot use environmental heat for reasons that have nothing to do with the second law - e.g. extreme slowness of isothermal heat engines? The answers to these and many other questions concerning the second law would be easier if one constantly takes into account the history of thermodynamics. This is a long and extremely interesting story where the issues are often ethical rather than scientific. The story deserves a separate discussion but still it is pertinent to mention a few episodes here, as a conclusion to the present paper and an introduction to further publications.
First of all we should bear in mind that, in the period 1780 - 1790, Lavoisier, Berthollet and other chemists place heat among the chemical elements and give it the name calorique. In this capacity heat is material and can be neither created nor transformed. Just as other elements, calorique participates in chemical reactions in fixed proportions. As bodies undergo friction, they get hot because they lose calorique.
The last explanation arouses objections since friction seems to be an inexhaustible source of heat. In the years 1790 - 1850 those objections will produce alternative interpretations which all turn round the statement: "Conversion of work into heat is possible". So the scientific thought is preparing itself, a bit indecisively, to accept one half of the first law of thermodynamics. The other half which implies "Conversion of heat into work is possible" has almost no expression in that period. When Joule publishes, in 1845, the results of his precise experiments, he does not even hint that the heat which increases as a weight falls can, under different circumstances, be transformed into lifting of a weight [5]. The idea of the conversion of heat into work is totally alien to the scientific authorities of that time. The future baron Kelvin is still on the wrong track even in 1848:
"In the present state of science no operation is known by which heat can be absorbed, without either elevating the temperature of matter, or becoming latent and producing some alteration in the physical condition of the body into which it is absorbed; and the conversion of heat (or caloric) into mechanical effect is probably impossible, certainly undiscovered. In actual engines for obtaining mechanical effect through the agency of heat, we must consequently look for the source of power, not in any absorption and conversion, but merely in the transition of heat" [6].
Six years earlier, in 1842, Julius Robert Mayer has published the following:
"Just as heat appears as an effect of the diminution of bulk and of cessation of motion, so also does heat disappear as a cause when its effects are produced in the shape of motion, expansion, or raising of weight…the steam-engine serves to decompose heat again into motion or the raising of weights. A locomotive engine with its train may be compared to a distilling apparatus; the heat applied under the boiler passes off as motion, and this is deposited again as heat at the axles of the wheels."[7]
Here both halves of the first law are formulated with a clarity that is rare even today. The authorities will need a few years to understand Mayer’s discovery and by 1850 they are ready. The discovery turns out to be made by them whereas Mayer attempts suicide and is sent to a mental institution.
The second law is a product of this initial confusion concerning the possible/impossible conversion of heat into work. Then the confusion developed so that at present the second law is a mythological rather than scientific principle. In fact, there are many second laws (nobody knows their number) which often have nothing to do with one another [8]. Yet, just in case, the authorities have coined the expression Perpetuum mobile of the second kind which implies that those who try to test the second law are just as mad as those who try to extract energy out of nothing. What a curse! The situation looks serious but I hope it is not desperate. Or perhaps it is desperate but not serious. Who knows.
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