Planck Boltzmann entropy equation

The value for In W from equation (24.10) may be substituted in the Boltzmann-Planck equation, so that the general expression for the entropy of an ideal gaseous system is found to be... 

It is seen from the foregoing results, c.g., equations (24.11) and (24.12), that by combining statistical mechanics with the Boltzmann-Planck equation it is possible to derive a relationship between the molar entropy of any gas, assuming it to behave ideally, and the partition function of the given species. Since the partition function and its temperature coefficient may be regarded as known, from the discussion in Chapter VI, the problem of calculating entropies may be regarded as solved, in principle. In order to illustrate the procedure a number of cases will be considered. 

In relation to the introductory thermodynamic equations of the Stefan-Boltzmann law we note that Max Planck had been a professor of physics since 1889 specializing in thermodynamics. There is a very interesting history of Planck s discovery on the Internet at http //www.daviddarling. info/encyclopedia/Q/quantum theory origins.html. In fact Planck s interest was initially related to an equation he had tried to find relating Boltzmann s entropy to Wein s law. Wein s law was simply that the color of a hot object shifts with temperature and Planck developed the quantized equation to explain Wein s law. Wein s law is just an empirical observation that Planck tried to put on a firm foundation, although Planck approached the problem from a thermodynamic approach. Wien s Law relates the wavelength of the spectral maximum to temperature as [7]... 

In the first and second equation, E is the energy of activation. In the first equation A is the so-called frequency factor. In the second equation AS is the entropy of activation, the interatomic distance between diffusion sites, k Boltzmann s constant, and h Planck s constant. In the second equation the frequency factor A is expressed by means of the universal constants X2 and the temperature independent factor eAS /R. For our purposes AS determines which fraction of ions or atoms with a definite energy pass over the energy barrier for reaction. 

Strictly speaking, the equation K =S is an extension of Boltzmann s theory, in so far as we have ascribed a definite value to the entropy constant. According to Boltzmann, the probabihty contains an undetermined factor, which cannot be evaluated without the introduction of new hypotheses. Boltzmann and Clausius suppose that the entropy may assume any positive or negative value, and that the change in entropy alone can be determined by experiment. Of late, however, Planck, in connection with Nemst s heat theorem, has stated the hypothesis that the entropy has always a finite positive value, which is characteristic of the chemical behaviour of the substance. The probabihty must then always be greater than unity, since its logarithm is a positive quantity. The thermodynamical probabihty is therefore proportional to, but not identical with, the mathematical probabihty, which is always a proper fraction. The definition of the quantity w on p. 15 satisfies these conditions, but so far it has not been shown that this definition is sufficient under all circumstances to enable us to calculate the entropy. 

The second statement of the third law (which bears Planck s name) is that as the temperature goes to zero, AS goes to zero for any process for which a reversible path could be imagined, provided the reactants and products are perfect crystals. Here, perfect crystals are defined as those which are non-degenerate, that is, they have only a single quantum state in which they can exist at absolute zero. This statement follows rigorously from Boltzmann s equation for entropy,... 

Sackur-Tetrode equation - An equation for the molar entropy of an ideal monatomic gas = Mn(e V/N A ), where R is the molar gas constant, V is the volume, and A/ is Avogadro s number. The constant A is given hy A = h/(2nmkT), where h is Planck s constant, m the atomic mass, k the Boltzmann constant, and T the temperature. 

By applying transition-state theory, we can calculate the activation entropy AS of this Diels-Alder reaction from Eq. (4), where R is the molar gas constant, A is the preexponential factor in the Arrhenius equation, T is the absolute temperature, kt is the Boltzmann constant, and h is Planck s constant. The values of and AS are presented in Table 19.2. 

S= rtfiln(e VlnN/ A ), where A= HI(2iankT) where n is the amount of the gas, ills the gas constant, e is the base of natural logarithms, V is the volume of the system, Af is the Avogadto constant, h is the Planck constant, m is the mass of each atom, fcis the Boltzmann constant, and Tis the thermodynamic temperature. To calculate the molar entropy of the gas both sides are divided by n. The Sackur-Tetrode equation can be used to show that the entropy change AS, when a perfect gas expands isothermally from Vi to Vt, is given by ... 

The free energy defined in Eqn (15.35) G(A) = —T In f Qxp[—V x,X)/T]dx. Along a single stochastic trajectory, the usual thermodynamic relations are valid for ensemble averages in equilibrium (Schmiedl and Seifert, 2007). Using the Fokker-Planck equation andD = Tp, with the Boltzmann constant assumed as unity, the rate of change of the trajectory-dependent total entropy of the system becomes... 

According to the transition state theory (Eyring equation) enthalpy (AH ), entropy (AS ) and free energy (AG ) of activation can be ealculated from the Arrhenius parameters Ea and A obtained experimentally. The activation energy Ea is related to the enthalpy of activation through Eq. 34.1. On the other hand, the pre-exponential factor A can be related to the entropy of activation through Eq. 34.2. Once both AH and AS are known, it is easy to calculate AG by means of Eq. 34.3 ( b and h are the Boltzmann and Planck constants respectively, T the absolute temperature... 

The rate constant is defined by equation (2), according to the Theory of Absolute Reaction Rates (67,83). In equation (2), k refers to the specific reaction rate, the equilibrium between the normal and activated states of the reactants, AF the free energy, AH the heat, AJB the eneigy, AT the volume change, and AS the entropy, all of activation, p the hydrostatic pressure, T the absolute temperature, and R the gas constant. The expression K kT/h) is the universal frequency for the decomposition of the activated complex in all chemical reactions. In this, k is the transmission coefficient, usually equal to 1, IT the absolute temperature, Jb the Boltzmann constant, and h Planck s constant. 

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