Molecules microstates

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

Entropy is often described as a measure of disorder or randomness. While useful, these terms are subjective and should be used cautiously. It is better to think about entropic changes in terms of the change in the number of microstates of the system. Microstates are different ways in which molecules can be distributed. An increase in the number of possible microstates (i.e., disorder) results in an increase of entropy. Entropy treats tine randomness factor quantitatively. Rudolf Clausius gave it the symbol S for no particular reason. In general, the more random the state, the larger the number of its possible microstates, the more probable the state, thus the greater its entropy. 

In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

In this case, every member of the ensemble is identical, and we can be sure that any selection results in the same microstate, (b) Because each of the four molecules can take two orientations, the total number of ways of arranging them is... 

For a more realistic sample size than that in Example 7.7, one that contains 1.00 mol CO, corresponding to 6.02 x 1023 CO molecules, each of which could be oriented in either of two ways, there are 2602x10 (an astronomically large number) different microstates, and a chance of only 1 in 2< 02x l0" of drawing a given microstate in a blind selection. We can expect the entropy of the solid to be high and calculate that... 

We can show that the thermodynamic and statistical entropies are equivalent by examining the isothermal expansion of an ideal gas. We have seen that the thermodynamic entropy of an ideal gas increases when it expands isothermally (Eq. 3). If we suppose that the number of microstates available to a single molecule is proportional to the volume available to it, we can write W = constant X V. For N molecules, the number of microstates is proportional to the Nth power of the volume ... 

Doubling the number of molecules increases the number of microstates from W to W2, and so the entropy changes from k In W to k In W2, or 2k In W. Therefore, the statistical entropy, like the thermodynamic entropy, is an extensive property. 

Enumerate all the microstates of the molecule. Each microstate of the molecule, having k ligands bound to k specific sites, is characterized by an energy level Ej(k). We usually combine many microstates into one macrostate denoted by a, and write the corresponding canonical PF as... 

The effects of solutes on distributions of microstates are analogous in important ways to the effects of changes in temperature (figure 6.13). A structure-stabilizing osmolyte like TMAO will favor compact, stable microstates. In the presence of a stabilizing cosolvent, the ensemble of configurational states thus includes a relatively small fraction of molecules whose... 

Two short pathways that link the a-helical and /3-hairpin macrostates without making use of microstates with an instantaneous temperature above 488K are shown in Fig.5.1. The path shown in Fig.5.1 (upper) involves the unwinding of both ends of the helix, leaving approximately one turn of helix in the middle of the molecule. This turn then serves as a nucleation point for the formation of the /3-turn, which is stabilized by hydrophobic interactions between the side chains of Y45 and F52. The native hydrogen bonds nearest to the turn then form, after which the remainder of the native hairpin structure forms. This pathway is similar to previously proposed mechanisms for the folding of the G-peptide /3-hairpin from a coil state, which emphasize the formation of hydrophobic contacts before hydrogen bond formation [17,18, 140-143] and the persistence of the /3-turn even in the unfolded state [143]. 

Matters are made up of small particles such as molecules and atoms. Thermodynamic laws have been postulated and inferred without looking into the micro-properties or microstates within the systems. A branch of thermodynamics has evolved, which tries to interpret thermodynamic properties based on the properties of micro constituent of the system. This branch is called the Statistical Thermodynamics. An offshoot is the Nuclear Thermodynamics , where matter is treated as another form of energy and role of atomic and subatomic particle forms are studied in determining thermodynamic properties. 

The plot of the population of microstates against energy levels of the microstates would give an inverted bell shaped plot (normal distribution Fig. 9.1). The difference here is when total number of molecules increase, width of bell becomes thinner and thinner. At extremely large number of molecules (of the order of or 6 10 ) the width becomes so thin that it becomes almost a vertical line. 

Conceiving n and n, is not easy, but it suffices at this point to assume that fV(number of microstate) deals with arrangement of molecules into different microstates. 

In molecular systems, storage of a certain sequence of nucleotides (amount of stored information Imax) depends mainly on the chemical sta bility of the DNA molecule. The information capacity is determined by the number of certain combinations of nucleotides but not by the number of microstates, including accounting the vibrations of aU of the atoms in the DNA chain. The formation of macroinformation is coupled here with the work and energy consumption in the course of biosynthesis of the DNA molecule. Similarly, the information can be implemented by con suming energy for the processes of the information translation and synthe sis of the protein chain. 

For two molecules in the flask, there are four possible microstates ... 

Thus a gas placed in one end of a container will spontaneously expand to fill the entire vessel evenly, because for a large number of gas molecule , there is a huge number of microstates corresponding to equal numbers of molecules in both ends. On the other hand, the opposite process,... 

Positional probability can also be invoked to explain the formation of solutions. The change in positional probability associated with the mixing of two pure substances is expected to be positive. There are many more microstates for the mixed condition than for the separated condition because of the increased volume available to the particles of each component of the mixture. For example, when two liquids are mixed, the molecules of each liquid have more available space and thus more available positions. This will be discussed in detail in Chapter 17. 

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