Theorem of Statistical Thermodynamics

In this section, we focus on a relation between entropy and the probability distribution function P,. If P, obeys Boltzmann statistics for the canonical ensemble, then one arrives at a correspondence between entropy S and the partition function Z. Equation (28-26) is interpreted within the context of the first law of thermodynamics in differential form  

Furthermore, the addition of thermal energy affects the distribution of molecules among the available stationary states and allows the system to populate states of higher energy. Hence, 

In summary, p-V work perturbs the energy levels and heat input perturbs the occupational probabilities of the available equilibrium states. One aspect of the second law identifies /T as a factor that makes the heat function an exact differential via the entropy state function 

The parameter as defined in the ergodic problem via equation (28-20), has dimensions of reciprocal energy and is given by l/kT. This claim will be justified, and consistency with classical thermodynamics will be demonstrated in Section 28-5. 

The differential statement of the second law, given by equation (28-30), is manipulated as follows  

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Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

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