The Boltzmann formula

Boltzmann made the link between the distribution of molecules over energy levels and the entropy. He proposed that the entropy of a system is given by 

Suppose we raise the temperature just enough for two molecules of a 100-molecule system to be able to leave their lowest energy state and occupy the first excited state. Two possible microstates are 

Suppose that a protein molecule of 100 amino acid residues denatured into a random coil can adopt 1.0 x 10 different conformations of the same energy. We set W= 1.0 X 10 and calculate the entropy as 

---

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. 

Consider continuous radiation with specific intensity I incident normally on a uniform slab with a source function 5 = Bv(Tex) unit volume per unit solid angle to the volume absorption coefficient Kp and is equal to the Planck function Bv of an excitation temperature Tcx obtained by force-fitting the ratio of upper to lower state atomic level populations to the Boltzmann formula, Eq. (3.4). For the interstellar medium at optical and UV wavelengths, effectively S = 0. 

The situation becomes even worse when the Boltzmann formula is used to interpret the absorption of radiant energy by molecules. Electromagnetic radiation considered as a fluctuating electric field interacts with electrons in... 

One can calculate the ratio of populations of spin-up to spin-down electron orientations at room temperature (T = 300 K) from the Boltzmann formula finding that Nl / N is approximately equal to one (0.999), indicating that there is about a 0.1% net excess of spins in the more stable, spin-down orientation at room temperature. Using the same mathematical expression, this difference in populations can be shown to increase as the temperature is lowered. Actually, the EPR signal will be linearly dependent on 1/ T, and this linear dependence is called the Curie law. Because of the excited state population s temperature dependence, most EPR spectra are recorded at temperatures between 4 and 77 K. 

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... 

However, if there is no thermodynamic equilibrium, as is common for interstellar cases, it is convenient to define between two energy levels an excitation temperature Tex which is determined by fitting the observed molecular distribution to the Boltzmann formula, Eq. (13). 

C104 salt did obey KR approximately. We concluded [23] that there was no qualitative change in magnetoresistance even though the (inelastic) mean free path became as low as 1/250 of a lattice parameter in the c direction. This is the experimental basis for the remarks in Section III about the possible applicability of the Boltzmann formulas even when the inelastic mean free path perpendicular to the chains is very small, and the apparent absence of a minimum metallic conductivity in the transverse directions. 

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 ... 

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 ... 

Substituting the above into (D.35), we obtain the Boltzmann formula of the molecular partition function as... 

The kinetics of ion transfer across ITIES is closely linked to the double-layer structure. The concentration of the ion transferred in each phase close to the plane dividing the both phases is related to the concentration outside the space charge region by the Boltzmann formula, for example, for the aqueous phase (cf. [9])... 

---