Entropy, configurational residual

When a crystal of ice is cooled to very low temperatures it is caught in some one of the many possible configurations but it does not assume (in a reasonable period of time) a uniquely determined configuration with no randomness of molecular orientation. It accordingly retains the residual entropy k In IF, in which k is the Boltzmann constant and W is the number of configurations accessible to the crystal. 

The change in configurational entropy, AS g, on forming a loop of n residues from an open random coil has been estimated from simple polymer theory and the probability that the two points where the loop in the random coil is linked lie simultaneously within a defined volume element V. Thus, equations of the form... 

So what drives an amino acid chain to fold The chain contains sufficient hydro-phobic residues to favor minimization of area by folding. However, as there are fewer possible conformational states in the folded state than in the unfolded one, folding is still entropically disfavored, and a positive configurational entropy has to be considered. 

Solution Carbon monoxide has a small electric dipole moment (approx 0.1 Debye), which gives the molecules an energetically preferred orientation as T — 0. However, this dipole moment is so small that the preference is not appreciable until very low temperatures, and the random orientation of the molecules (the dipole has equal probability of pointing in one direction or its opposite) remains as the temperature is lowered. For a mole of CO, each molecule can point in either of two directions and there are 2Na configurations that are about equally probable. This model predicts a residual entropy of... 

At this temperature, the entropy change for dissolution of liquid hydrocarbons in water is zero. However, the entropy of protein denaturation is far from zero at this temperature but amounts to 17.6 J - K l per mole of amino acid residues (Privalov, 1979), a value that corresponds to an 8-fold increase of the number of possible configurations and is close to the value expected for the helix-coil transition of polypeptides (Schellman, 1955). This difference shows that an oil drop is an inadequate model for a globular protein. A more suitable model resembles that of a small crystal with a quite definite positive melting entropy (see also Bellow, 1977, 1978). 

One of the most challenging tasks in the theory of liquids is the evaluation of the excess entropy Sex, which is representative of the number of accessible configurations to a system. It is well known that related entropic quantities play a crucial role, not only in the description of phase transitions, but also in the relation between the thermodynamic properties and dynamics. In this context, the prediction of Sex and related quantities, such as the residual multiparticle entropy in terms of correlation functions, free of any thermodynamic integration (means direct predictive evaluation), is of primary importance. In evaluating entropic properties, the key quantity to be determined is the excess chemical potential (3pex. Calculation of ppex is not straightforward and requires a special analysis. 

According to Orr [210] the residual entropy is negligible as is, according to Temperley [211], the configurational entropy. Sawada [212] described the entropy of copolymerization randomness (unit placement in the chain) by the relation... 

The complexity and diversity of structures in the native proteins eluded any attempt to produce some simple conformation that accounted for their interfacial properties. The study of synthetic polypeptides with non-polar side chains has provided good evidence to support the view that the a-helix can be stable at the air-water interface (5), and it is therefore possible that the interfacial denaturation of proteins is mainly a loss of the tertiary structure (6, 7, 8). Since for a typical protein an a-helix takes up about the same area per residue as the p conformation, it can be accommodated as easily. Moreover, like the p conformation but unlike a more randomly coiled structure, it is linear and therefore compatible with a plane surface without loss of configurational entropy (5). In this respect a plane surface may favor an ordered over a more random structure. The loss of solubility of the spread protein can then be attributed to intermolecular association between hydrophobic side chains exposed as a result of the action of the interface on the polar exterior of the molecules. 

The total number of various proton configurations in ice-like systems is rather large. By analogy with well-known Pauling s formula we obtained the following expression for residual entropy of PWCs ... 

This is equal to the measured experimental value. Let us note that the Pauling s evaluation is a first approximation, as it treats all 0-atoms as being independent and neglects correlations of configuration between neighbour 0-atoms. This correlation enters for a small part only in the value of the residual entropy, but it has up to now not been calculated. The existence of such a residual entropy has no important consequences, except that if it is not properly taken into account, thermodynamical values may differ when measured from different experiments, thus lacking consistency. 

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