Entropy and the Number of Microstates

CHAPTER 20 Thermodynomics Entropy, Free Energy, and the Direction of Chemical Reactions 

In 1877, the Austrian mathematician and physicist Ludwig Boltzmann defined the entropy (S) of a system in terms of W  

Dissolving of salt crystalline solid + liquid — ions in solution 

Chemical change crystalline solids -- gases + ions in solution 

Changes in Entropy If the number of microstates increases during a physical or chemical change, there are more ways for the energy of the system to be dispersed among them. Thus, the entropy increases  

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The Second Law of Thermodynamics Predicting Spontaneous Change Lrnitations of the First Law The Sign of AH and Spontaneous Change Freedom of Motion and Dispersal of Energy Entropy and the Number of Microstates Entropy and the Second Law Standard Molar Entropies and the Third Law... 

The Austrian physicist Ludwig Boltzmann proposed the following relationship between entropy and the number of microstates ... 

FIGURE 13.4 The fundamental relation between entropy (S) and the number of microstates (1/1/) was derived by Ludwig Boltzmann in 1868. On his tombstone in Vienna is carved the equation he obtained, S = k log W. We would write "In" instead of "log" for the natural logarithm. 

As discussed in Section 3.10, Boltzmann recognized that there was a close link between the entropy of a system and the number of microstates that comprise it which he expressed... 

Second, suppose we keep the volume fixed but increase the temperature. How does this change affect the entropy of the system Recall the distribution of molecular speeds presented in Figure 10.17(a). An increase in temperature increases the most probable speed of the molecules and also broadens the distribution of speeds. Hence, the molecules have a greater number of possible kinetic energies, and the number of microstates increases. Thus, the entropy of the sj stem increases with increasing temperature. 

Einally, it is appropriate to consider the third law of thermodynamics briefly in connection with the determination of entropy values. So far we have related entropy to microstates—the greater the number of microstates a system possesses, the larger is the entropy of the system. Consider a perfect crystalline substance at absolute zero (0 K). Under these conditions, molecular motions are kept at a minimum and the number of microstates (W) is one (there is only one way to arrange the atoms or molecules to form a perfect crystal). From Equation (18.1) we write... 

According to the third law of thermodynamics, the entropy of a perfect crystalline substance is zero at the absolute zero of temperature. As the temperature increases, the freedom of motion increases and hence also the nmnber of microstates. Thus, the entropy of any substance at a temperature above 0 K is greater than zero. Note also that if the crystal is impure or if it has defects, then its entropy is greater than zero even at 0 K because it would not be perfectly ordered and the number of microstates would be greater than one. 

It is therefore plausible to associate the entropy to the number of microstates that are accessible to the system. A microstate is a snapshof of the individual classical or quantum-mechanical states of the individual particles in the whole system. Such an individual state is in atomic systems the quantum state of the nth particle a > on larger scales, it can be sufficient to characterize it classically by the mechanical particle state given by position and momentum, s = (x , p ). The microstates of an A7-particle system are then given by the state vectors a1a2. .. a . .. a ) or (si,S2,...,s ,...,sjn), respectively. Typically, an equilibrium ensemble of microstates dominates at a given temperature and represents the macrostate. 

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. 

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. 

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. 

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. 

From Equation 13.9 we see that the entropy of a gas increases during an isothermal expansion (V2 > Vi) and decreases during a compression (V2 < Vt). Boltzmann s relation (see Eq. 13.1) provides the molecular interpretation of these results. The number of microstates available to the system, H, increases as the volume of the system increases and decreases as volume decreases, and the entropy of the system increases or decreases accordingly. 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

In the molecular statistical analysis, Boltzmann defined the entropy S in any thermodynamic state as S = In fl, where O is the number of microstates available to the system in that same thermodynamic state. This equation is used for qualitative interpretations of entropy changes. It shows that any process that increases D will increase S, and any process that decreases O will decrease S. 

It is clear from the derivation presented above that the phase space average, Equation (38), is exactly equal to the desired ensemble average. That is, all phase points with energy E are included with equal probability. Consider the phase space volume, f2 NVE), the number of states with energy E given physical volume, V, and N particles. As the phase space volume increases, obviously, the number of microstates increases and the entropy should increase. This suggest that we postulate that S NVE) = F f2 NVE)) where Q NVE) is now referred to as the microcanonical partion function and F must be a monotonically increasing, function to be determined. 

The quantitative basis for the linkage between macro- and microstates is provided by the prescription for computing the entropy S(E,V, N) in terms of the number of microstates, namely. 

Here, S is entropy, kB is the Boltzmann constant (1.38066 x 10-23), and fl is the number of microstates that the molecule can assume per macrostate. If we consider entropy as the inability of a system s energy to do work, we can see that entropy will decrease as the chain is stretched [6], Likewise, as the chain is relaxed, entropy will increase in an endothermic process. 

The total kinetic energy of a system is the sum of the translational, rotational, and vibrational energies of its particles, each of which is quantized. A microstate of fhe system is any specific combination of the quantized energy states of all the particles. The entropy of a system is directly related to the number of microstates over which the system disperses its energy, which is closely associated with the freedom of motion of the particles. A substance has more entropy in its gaseous state than in its liquid state, and more in its liquid state than in its solid state. 

This is the same result we obtained by the statistical approach. That approach helps us visualize entropy changes in terms of the number of microstates over which the energy is dispersed, but the calculations are limited to simple systems like ideal gases. This approach, which involves incremental heat changes, is less easy to visualize but can be applied to liquids, solids, and solutions, as well as gases. 

Understand the meaning of entropy (5) in terms of the number of microstates over which a system s energy is dispersed describe how the second law provides the criterion for spontaneity, how the third law allows us to find absolute values of standard molar entropies (5°), and how conditions and properties of substances influence 5° ( 20.1) (SP 20.1) (EPs 20.4-20.7, 20.10-20.23)... 

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