Boltzmann formula for the entropy

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... 

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. 

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 ... 

Using this equation in conjunction with the Boltzmann formula for the entropy, eq. (II.ll), we obtain... 

An isolated system is defined to be a system that does not exchange material or energy with its environment. Thus the extensive thermodynamic variables N, V, and E are held fixed. Boltzmann s formula for the entropy of such a system is... 

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 ... 

Applying Boltzmann s formula (Equation 1.4-1) for the entropy of a molecular coil, and in view of Equations 104 and 106, we write... 

Albert Einstein (1879-1955) proposed a formula for the probability of a fluctuation in thermod5mamic quantities by using Boltzmann s idea in reverse whereas Boltzmann used microscopic probability to derive thermod5mamic entropy, Einstein used thermodynamic entropy to obtain the probability of a fluctuation through the following relation ... 

We won t describe how the entropy of a substance is determined, except to note that two approaches are available (1) calculations based on Boltzmann s formula and (2) experimental measurements of heat capacities (Section 8.8) down to very low temperatures. Suffice it to say that standard molar entropies, denoted by S°, are known for many substances. 

Consider the distribution of ideal gas molecules among three bulbs (A, B, and C) of equal volume. For each of the following states, determine the number of ways (W) that the state can be achieved, and use Boltzmann s formula to calculate the entropy of the state ... 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

In order to calculate the entropy, we use its fundamental definition [Eq. (5)]. For distinguishable particles (Boltzmann statistics), we use formula (8) for the number of configurations ... 

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. 

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,... 

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 statistics for the initial conditions, Oy(0), are determined by the equilibrium distribution obtained from the entropy in (A3.2.12) and in accordance with the Einstein-Boltzmann-Planck 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... 

According to the famous formula of Ludwig Boltzmann, entropy S = kg In Q (E), where S is the number of the states available for the system at energy E. The more states there are. the larger the entropy is. 

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

The third law is a consequence of the statistical nature of entropy as reflected in Boltzmann s formula (Equation 8.1), which relates the entropy to W, the number of molecular quantum states (microstates) consistent with the macroscopic conditions. At r = 0, there is no available thamal energy, and the thermodynamically most stable state is the lowest possible energy state (the ground state). In general, this state is unique, so W = 1 for a system at 0 K. From Boltzmann s formula we then have... 

Since viscosity measurements of uncharged, weak and strong PEs [60] were successfully modeled with the Kuhn entropy [80], it should theoretically be possible to describe weak PE solutions with it as well. This assumption is supported by the fact that the Kuhn entropy is based on the Boltzmann entropy formula, and attempts to use this approach become inaccurate to determine the ion distribution around the charged groups of strong PEs at low salt concentrations, but not the PE structure [81]. Since weak PEs are not strongly affected by counter ion condensation, these effects pose no problem for the Kuhn entropy approach. Another reason for the infrequent use of the Kuhn entropy is the fact that the Flory approach is much simpler and better known than the Kuhn approach [49, 80, 82]. 

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