Free energy of an ideal chain

The entropy S is the product of the Boltzmann constant k and the logarithm of the number of states iV. 

Denote fl N, R) as the number of conformations of a freely jointed chain of N monomers with end-to-end vector R. The entropy is then a function 

The probability distribution function is the fraction of all conformations that actually have an end-to-end vector R between R and R + dR  

The entropy of an ideal chain with JVmonomers and end-to-end vector R is thus related to the probability distribution function  

Equation (2.85) for the probability distribution function determines the entropy  

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The free energy of an ideal chain is purely entropic and changes quad-ratically with the end-to-end vector ... 

The energy of an ideal chain U N,R) is independent of the end-to-end vector R, since the monomers of the ideal chain have no interaction energy. The free energy can be written as... 

The Flory estimate of the entropic contribution to the free energy of a real chain is the energy required to stretch an ideal chain to end-to-end distance 7 [Eq. (2.101)] ------------------------------------------------------... 

Note that this entropic free energy alone has a minimum at i = Nb, which is the conformation of an ideal chain. 

Finally, recalling eqns [68]-]70] we obtain the free energy of an inhomogeneous system of long ideal chains (with c(r) 0 for... 

The first self-consistent PRISM studies by Schweizer et al. considered only the HNC-style solvation potential and were based on an optimized perturbative, not variational, determination of the ideal reference system effective bending energy. The starting point is a simple functional expansion of the true single-chain free energy about an ideal reference system" ... 

The free energy of confining a real linear chain in a good solvent either into a slit of spacing Z) or to a cylindrical pore of diameter D is larger than for an ideal chain because the real chain has repulsive interactions ... 

Depending on the Flory parameter x, there is a particular special temperature T = 6 at X = 1/2, which corresponds to an exact cancellation between steric repulsion and van der Waals attraction between monomers, and thus the chains are nearly ideal. This temperature is known as the collapse temperature or theta temperature [15]. Equation (15.10) for the free energy of mixing is an expression that finds wide use in physical chemistry. A quantitative understanding of hydrophobie eollapse is required to understand the initial stage of protein folding, as proteins are often a finite chain consisting of a 50-300 amino acid residue linear chain, which in many aspects resembles a heteropolymer. 

In Equation 10.8, AG is the folding free energy of the protein, is the first-order rate constant of the slow hydrogen exchange reaction at the C-2 position in the imidazole side chain of an unprotected histidine, m is 54G, / 5[denaturant], T is the temperature in Kelvin, R is the ideal gas constant, and [P] is the protein concentration expressed in n-ma- equivalents. Equation 10.8 can be daived from Equation 10.9, which is commonly used in the linear extrapolation method (LEM) to analyze denaturant-induced equilibrium unfolding curves [34] ... 

A key difference between elecfrostatic and steric stabilization is the significant influence that the solvent quality and the temperature can have in steric stabilization. As we saw above, interpenetration of the polymer chains gives rise to a mixing effect. At a certain temperature, referred to as the (theta) temperature, the interpenetration of the polymer chains does not lead to a change in the free energy of mixing (AGmix = 0), and a system of the polymer dissolved in the solvent behaves like an ideal solution. The solvent is referred to as a 0 solvent. 

If the periodicity is denoted by L, all blocks are stretched to an end-to-end point distance D=L/4, with a corresponding ideal chain elastic free energy of Fei,biock=3D /Na feT. The free energies pa unit volume are... 

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