Boltzmann s relation

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

By Boltzmann s relation S = k In W, this gives for the ent ropy of mixing... 

Consider the free expansion of 1 mol gas from V/2 to V (see Fig. 13.2). Use Boltzmann s relation to estimate the change in entropy for this process. 

Inserting this expression into Boltzmann s relation gives the entropy change for the free expansion of 1 mol (No atoms) gas from a volume V/2 to V ... 

This example illustrates why Boltzmann s relation must involve the logarithmic function. Because entropy is an extensive variable, its value is proportional to N. But fl depends on N through the power to which cV is raised. Therefore, doubling N doubles S but leads to fl being squared. The only mathematical function that can connect two such quantities is the logarithm. 

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 entropy increases when a solid melts or a liquid vaporizes, and it decreases when the phase transition occurs in the opposite direction. Again, Boltzmann s relation provides the molecular interpretation. When a solid melts or a liquid vaporizes, the number of accessible microstates O increases, and thus the entropy... 

Entropy always increases with increasing temperature. From the kinetic theory of ideal gases in Chapter 9, it is clear that increasing the temperature of the gas increases the magnitude of the average kinetic energy per molecule and, therefore, the range of momenta available to molecules. This, in turn, increases O for the gas and, by Boltzmann s relation, the entropy of the gas. 

To calculate the entropy for our Gaussian chain, we use the Boltzmann s relation from statistical mechanics... 

M makes an angle j8 with the field F, its potential energy is proportional to —(jlF ooa. By Boltzmann s relation (p. 79) the number of molecules at temperature T which possess this potential energy is proportional to e +i BOBplkT trigonometrical factors. If kT... 

Using Boltzmann s relation, the partial molar entropy of excess oxygen in UO2+ ,. is expressed by... 

Since the fourth term is obtained from the combination of chain units and solvent molecules, the first three terms for the whole chain are far less than the fourth one if considering their global contributions in the lattice space. Therefore, only the combinatirm entropy Scombinate is calculated. According to the Boltzmann s relation Scombinate = k aQ, we only need to calculate the total amount of arrangement of molecules in the lattice space. 

At the right-hand side of (10.8), the first five terms come from Floiy s semiflexibility treatment (8.55), the sixth term comes from the mean-field estimation for the pair interactions of parallel bonds (10.7), and the last term comes from the mean-field estimation for the mixing interactions between the chain units and the solvent molecules (8.21). According to the Boltzmann s relation F = —kTlnZ, the free energy of the solution system can be obtained as... 

Equation (24) is called Boltzmann s relation. At absolute zero, the system is in its ground state, and the number of accessible states is unity. Thus, the entropy of a system tends to zero as the temperature goes to zero. This is called the third law of thermodynamics. 

Thus, if we substitute equation [3.120] back into Boltzmann s relation, we find the entropy term of conformation ... 

This may be regarded as an extension to time-dependent systems of Boltzmann s relation between probability and entropy. 

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