Thermodynamic Paradoxes

It was argued that the most celebrated violation of thermodynamics is the fact that CVD (chemical vapor deposition) diamond forms under thermodynamically unstable pressure - temperature conditions. A careful analysis of thermodynamic data led to the conclusion that certain equilibria are taking part in the process, so that the process no longer appears as a thermodynamic paradox [18]. The key idea was to assume that CVD diamond formation is a chemical process consisting in accretion of polymantane macromolecules. Thus, violations of thermodynamic principles could be avoided. 

The thermodynamic paradox might be resolved because (1) the time-symmetric behaviour of the trajectory dynamics contributes nothing more to the global evolution of the statistical mechanical system than the necessary conditions for the existence of such a system and (2) in a large Poincare system trajectories exhibit Brownian motion, and correlation dynamics dominate the macroscopic dynamics. Thermodynamics is then an emergent global phenomenon possessing a temporal direction. [Bishop, 2004, p. 25]... 

Christian Anfmsen s experiments demonstrated that proteins can fold reversibly. A corollary result of Anfmsen s work is that the native structures of at least some globular proteins are thermodynamically stable states. But the matter of how a given protein achieves such a stable state is a complex one. Cyrus Levinthal pointed out in 1968 that so many conformations are possible for a typical protein that the protein does not have sufficient time to reach its most stable conformational state by sampling all the possible conformations. This argument, termed Levinthal s paradox, goes as follows consider a protein of 100 amino acids. Assume that there are only two conformational possibilities per amino acid, or = 1.27 X 10 ° possibilities. Allow 10 sec for... 

The commonest form of phosphorus, and the one which is usually formed by condensation from the gaseous or liquid states, is the waxy, cubic, white form o -P4 (d 1.8232 gcm at 20°C). This, paradoxically, is also the most volatile and reactive solid form and thermodynamically the least stable. It is the slow phosphorescent oxidation of the vapour above these crystals that gives white phosphorus its most characteristic property. Indeed, the emission of yellow-green light from the oxidation of P4 is one of the earliest recorded examples of chemiluminescence, though the details of the reaction... 

The earliest hint that physics and information might be more than just casually related actually dates back at least as far as 1871 and the publication of James Clerk Maxwell s Theory of Heat, in which Maxwell introduced what has become known as the paradox of Maxwell s Demon. Maxwell postulated the existence of a hypothetical demon that positions himself by a hole separating two vessels, say A and B. While the vessels start out being at the same temperature, the demon selectively opens the hole only to either pass faster molecules from A to B or to pass slower molecules from B to A. Since this results in a systematic increase in B s temperature and a lowering of A s, it appears as though Maxwell s demon s actions violate the second law of thermodynamics the total entropy of any physical system can only increase, or, for totally reversible processes, remain the same it can never decrease. Maxwell was thus the first to recognize a connection between the thermodynamical properties of a gas (temperature, entropy, etc.) and the statistical properties of its constituent molecules. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

The overall result is that in the melt the polymer molecules adopt Gaussian configurations and behave as thermodynamically ideal entities. This combination of ideality and chain entanglement has been confirmed by neutron scattering experiments and is well established despite the apparent paradox. 

The evolution of structures and mechanisms in plants to regulate water fluxes down these steep thermodynamic gradients and yet maintain the cellular conditions for biochemical activity was a major factor in the colonisation of the terrestrial habitat. Paradoxically, therefore, some water stress is completely normal , though some plants are better than others at accommodating large deviations. 

While hydrolysis is a thermodynamically favored reaction, the amide and phosphoester bonds of polypeptides and ohgonucleotides are stable in the aqueous environment of the cell. This seemingly paradoxic behavior reflects the fact that the thermodynamics governing the equihbrium of a reaction do not determine the rate at which it will take place. In the cell, protein catalysts called enzymes are used to accelerate the rate... 

The authors discuss Schroeder s paradox, referred to elsewhere in this review, and the fact that liquid water uptake increases but saturated water uptake decreases with temperature. And, at low temperature, the water uptake by membranes in contact with saturated vapor is greater than that by membranes in contact with liquid water, which suggests a fundamental difference in membrane microstructure for the two situations. An energy level diagram of thermodynamic states versus temperature was proposed, based on this Flory—Huggins-based model. 

A point of occasional confusion arises with respect to units. In Eq. (15.22), all portions are unitless except for k T/h, which has units of sec , entirely consistent with the units expected for a unimolecular rate constant. In Eq. (15.23), the same is true with respect to the r.h.s., but a bimolecular rate constant has units of concentration" sec", which seems paradoxical. The point is diat, as with any thermodynamic quantity, one must pay close attention to standard-state conventions. Recall that die magnitude of die iranslational partition function depends on specification of a standard-state volume (or pressure, under ideal gas conditions). Thus, a more complete way to write Eq. (15.23) is... 

However small the effect, it violates the second law of thermodynamics. An array of 1012 diodes would be able to run a flashlight. The answer to this paradox is that one cannot trust the phenomenological equation to the extent that one may use it for deducing a result that is itself of the order of the fluctuations. This example demonstrates the danger of adding a Langevin term to a nonlinear equation. ... 

The diagram above refers to thermodynamic stability. When we discuss addition reactions you will see that the most stable alkene when mixed with an electrophile is the most reactive according to this diagram. This paradox is due to the intermediate, usually a carbocation. Since a tertiary carbocation is more stable, the energy of activation is lowered and a reaction with a tertiary intermediate proceeds more quickly in general, to predict the alkene product, use the above diagram as a reference, but to predict the most reactive alkene to an electrophile, the order is based on cation formation and is nearly reversed. 

J. Kestin, J. R. Rice Paradoxes in the Application of Thermodynamics to Strained Solids. A Critical Review of Thermodynamics, ed by E.B. Stuart (Mono Book Corp., Baltimore, 1970), p.275-280... 

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