Entropy microstates

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 some cases, an alternative explanation is possible. It may be assumed that any very complex organic counterion can also interact with the CP matrix with the formation of weak non-ionic bonds, e.g., dipole-dipole bonds or other types of weak interactions. If the energy of these weak additional interactions is on the level of the energy of the thermal motion, a set of microstates appears for counterions and the surrounding CP matrix, which leads to an increase in the entropy of the system. The changes in Gibbs free energy of this interaction may be evaluated in a semiquantitative way [15]. 

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... 

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 ... 

Because the number of microstates available to the system depends only on its current state, not on its past history, W depends only on the current state of the system, and therefore the statistical entropy does, too. 

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. 

In any irreversible change, the overall disorder of the system and its surroundings increases, which means that the number of microstates increases. If W increases, then so does In W, and the statistical entropy increases, too. 

When the temperature of the system increases, more microstates become accessible, and so the statistical entropy increases. 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

Macrostates are collections of microstates [9], which is to say that they are volumes of phase space on which certain phase functions have specified values. The current macrostate of the system gives its structure. Examples are the position or velocity of a Brownian particle, the moments of energy or density, their rates of change, the progress of a chemical reaction, a reaction rate, and so on. Let x label the macrostates of interest, and let x(r) be the associated phase function. The first entropy of the macrostate is... 

The density dependence of the entropy can also be studied by introducing fluctuations in volume rather than particle number. Typically the particle number approach is favored the computational demands of volume scaling moves scale faster with system size than do addition and deletion moves. Nevertheless, the Wang-Landau approach provides a means for studying volume fluctuations as well. In this case, the excess entropy is determined as a function of volume and potential energy for fixed particle number one, therefore, calculates (V, U). Here the microstate probabilities follow ... 

Now suppose that some external constraint on the system, is removed. New microstates, previously inaccessible, become available and transition into these new states may occur. As a consequence the number of microstates among which transitions occur, increases to the maximum permitted by the remaining constraints. This statement is strikingly reminiscent of the entropy postulate of thermodynamics, according to which the entropy increases to the maximum permitted by the imposed constraints. It appears that entropy may be identified with the number of allowed microstates consistent with the imposed macroscopic constraints. 

The identification of entropy with available microstates presents one difficulty Entropy is additive, but the number of microstates is multiplicative3. The answer is to identify entropy with the logarithm of the number of available microstates. Thus... 

The relationship between the statistical expression (equation (5.6)) and the classical expression (equation (5.8)) for determination of the entropy can be explained by the statement that, due to the additional heat taken up, the system acquires more available microstates (Edsall and Gutfreund, 1983). Equation (5.8) introduces a procedure for the direct calorimetric measurement of the entropy change for a specific process such as the reversible formation of a new set of biopolymer interactions. 

C. E. Shannon (1916-2001) developed an information-theoretic definition of entropy that (although not equivalent to the physical quantity) carries similar associations with microstates and probability theory. Shannon recognized that Boolean bit patterns (sequences of l s and 0 s) can be considered the basis of all methods for encoding information. ... 

The quantal energy packets are so small that the total stored molecular energy (the sum of all the molecular excitation quanta) is perceived at the macroscopic level as the continuously variable temperature T rather than a countable microscopic quantity. However, this countable aspect of molecular-level energy excitations underlies proper evaluation of Boltzmann s ft (number of possible molecular microstates consistent with total macrostate energy), and thus the entropy. 

Thus, assuming that the ground level E0 is truly nondegenerate (i.e., there is no alternative of equal energy), the T = 0 microstate is unique, and the corresponding Boltzmann entropy is therefore zero ... 

Similarly, if one is interested in a macroscopic thermodynamic state (i.e., a subset of microstates that corresponds to a macroscopically observable system with bxed mass, volume, and energy), then the corresponding entropy for the thermodynamic state is computed from the number of microstates compatible with the particular macrostate. All of the basic formulae of macroscopic thermodynamics can be obtained from Boltzmann s definition of entropy and a few basic postulates regarding the statistical behavior of ensembles of large numbers of particles. Most notably for our purposes, it is postulated that the probability of a thermodynamic state of a closed isolated system is proportional to 2, the number of associated microstates. As a consequence, closed isolated systems move naturally from thermodynamic states of lower 2 to higher 2. In fact for systems composed of many particles, the likelihood of 2 ever decreasing with time is vanishingly small and the second law of thermodynamics is immediately apparent. 

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