Entropy Boltzmann formula

In 1877, the Austrian physicist Ludwig Boltzmann proposed a molecular definition of entropy that enables us to calculate the absolute entropy at any temperature (Fig. 7.6). His formula provided a way of calculating the entropy when measurements could not be made and deepened our insight into the meaning of entropy at the molecular level. The Boltzmann formula for the entropy is... 

SOLUTION (a) Because T = 0, all motion has ceased. We expect the sample to have zero entropy, because there is no disorder in either location or energy. This value is confirmed by the Boltzmann formula because there is only one way of arranging the molecules in the perfect crystal, W = 1. 

EXAMPLE 7.8 Using the Boltzmann formula to interpret a residual entropy... 

STRATEGY The existence of residual entropy at T = 0 suggests that the molecules are disordered. From the shape of the molecule (which can be obtained by using VSEPR theory), we need to determine how many orientations, W, it is likely to be able to adopt in a crystal then we can use the Boltzmann formula to see whether that number of orientations leads to the observed value of S. 

The Boltzmann formula relates the entropy of a substance to the number of arrangements of molecules that result in the same energy when many energy levels are accessible, this number and the corresponding entropy are large. 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

When the length of the box is increased at constant temperature (with T > 0), more energy levels become accessible to the molecules because the levels now lie closer together (Fig. 7.9b). The disorder has increased and we are less sure about which energy level any given molecule occupies. Therefore, the value of W increases as the box is lengthened and, by the Boltzmann formula, the entropy increases, too. The same argument applies to a three-dimensional box as the volume of the box increases, the number of accessible states increases, too. 

We can also use the Boltzmann formula to interpret the increase in entropy of a substance as its temperature is raised (Eq. 2 and Table 7.2). We use the same par-ticle-in-a-box model of a gas, but this reasoning also applies to liquids and solids, even though their energy levels are much more complicated. At low temperatures, the molecules of a gas can occupy only a few of the energy levels so W is small and the entropy is low. As the temperature is raised, the molecules have access to larger numbers of energy levels (Fig. 7.10) so W rises and the entropy increases, too. 

Boltzmann formula (for the entropy) The formula S = k In W, where k is Boltzmann s constant and W is the number of atomic arrangements that correspond to the same energy. 

It is easiest to start with the configurational entropy, Sc- Suppose that the number of defects, which is equal to the number of mobile (localized) holes or electrons, is nd and moreover that only one type of mobile carrier, either holes or electrons, is present. The configurational entropy Sc is given by using the Boltzmann formula ... 

If the number of Schottky defects is ns per unit volume at T K, then there will be ns cation vacancies and anion vacancies in a crystal containing TV possible cation sites and A possible anion sites per unit volume. The Boltzmann formula tells us that the entropy of such a system is given by ... 

EXAMPLE 7.5 Using the Boltzmann formula to calculate the entropy of a substance... 

Let s calculate the entropy of a tiny solid made up of four diatomic molecules of a binary compound such as carbon monoxide, CO. Suppose the four molecules have formed a perfectly ordered crystal in which all molecules are aligned with their C atoms on the left. Because T = 0, all motion has ceased (Fig. 7.5). We expect the sample to have zero entropy, because there is no disorder in either location or energy. This value is confirmed by the Boltzmann formula because there is only one way of arranging the molecules in the perfect crystal, W = l and S = k In 1 =0. Now suppose thar the molecules can lie with their C atoms on either side yet still have the same total energy (Fig. 7.6). The total number of ways of arranging the four molecules is... 

Versions of the Boltzmann formula, Eq. 8, can be used to calculate standard molar entropies of substances that are in very good agreement... 

The unstable degrees of freedom determine the number of various allowed microstates that are responsible for creating the given macrostate. This is namely the number of the microstates, or their thermodynamic probabihty Qj , which determines a total of entropy S of the system. According to the Boltzmann formula,... 

While comparing formula (6.1) to the Boltzmann formula for a physical system with the identical number of microstates Q, one can easily discover a formal relationship between entropy and information ... 

We consider first an isolated system having a fixed internal energy E, volume V, and number of particles N. Let W(E, V, N) be the number of quantum mechanical states of the system characterized by the variables E, V, N. That is the number of eigenstates of the Hamiltonian of the system having the eigenvalue E. We assume for simplicity that we have a finite number of such eigenstates. The first relationship is between the entropy S of the system and the number of states, W (E, V, N). This is the famous Boltzmann formula ... 

The concept of entropy has been widened to take in the general idea of disorder - the higher the entropy, the more disordered the system. For instance, a chemical reaction involving polymerization may well have a decrease in entropy because there is a change to a more ordered system. The thermal definition of entropy is a special case of this idea of disorder -here the entropy measures how the energy transferred is distributed among the particles of matter. See also Boltzmann formula. 

Indeed, we can use Boltzmann formula (7.2), for instance, to estimate the entropic price of building a hmnan body from its parts. There are about 10 cells in a human body, and if we assmne that all of them are different and each must occupy a uniquely defined position, the entropy loss due to their arrangement will be fcBln(l0 ) 10 Similarly, each... 

Even if there were no forces at all acting between the polymer and the solvent molecules, the polymer particles would still try to become scattered as regularly as possible all over the liquid, because the entropy aims at a maximum value. From the Boltzmann formula for the entropy ... 

N is the number of defects in a matrix of No self-assembled moieties or features and AEact is the activation energy of defect formation. The activation energy AEad should not be confused with AHdf- The activation energy represents energy difference between the initial ideally arranged state and a transition state towards the defective structure. Equation 13 can be used to estimate the defect concentration by use of the Boltzmann formula to estimate entropy S... 

The Boltzmann formula gives us the entropy for a system of N particles, the i + and AL of which are indistinguishable from each other ... 

Boltzmann formula A fundamental result in statistical mechanics stating that the entropy S of a system is related to the number W of distinguishable ways in which the system can be realised by the equation S = k InW, where k is the Boltzmann constant. This formula is a quantitative expression of the idea that the entropy of a system is a measure of its disorder. It was discovered by the Austrian physicist Ludwig Boltzmann in the late 19th century in the course of his investigations into the foundations of statistical mechanics. 

The first relationship that we adopt is between the entropy S of the system and the number of states Q. This is the famous Boltzmann formula... 

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