where Q(e,v) is the number of microstates that have energy, e, and volume, v. The double summation of the right hand side of Eq. (17) can be simplified when the new function of number of microstates, Q.(h,p) is introduced as
h=e+pv where the microscopic enthalpy, h, can be introduced using the microscopic energy and volume as h=e+pv. As seen in Eq. (17), the statistical weight of the microscopic state is determined by the microscopic enthalpy. Therefore it is convenient to introduce the new microscopic variable. Using this new function, Y(T,p) is represented as
The Gibbs energy, G(T,p) and entropy S(h,p) are defined statistical thermo-dynamically as
The probability function f(h,p;T) is defined as expj --f^- |Q(h, p)
The enthalpy function is defined as the average of the enthalpy of microstates as
It is easy to show Eq. (5) from Eqs (20) and (22). From now, the average value using the probability function of the system is represented as < >(T,p). For example, the Eq. (23) can be represented as H(T,p) = <h>(T,p).
By partial derivative to both sides of Eq. (23) with temperature, the following well-known equation can be obtained:
The Eqs (21-24) show that the enthalpy and heat capacity can be calculated by statistical thermodynamics when the entropy is obtained as a function of micro scopic enthalpy.
It should be noted that the entropy previously obtained using Eq. (15) as a function of macroscopic enthalpy, S(H,p), is different from the entropy as a function of the microscopic enthalpy, S(h,p). If She)(H,p) is a good approximation for S(h,p), the heat capacity Cp (T) with Legendre transformation can be calculated from the entropy of Eq. (15). In Fig. 1, the calculated heat capacity is shown. It is largely deviated from the initial heat capacity function, Cp(T), indicating that S(Le)(H,p) is not a good approximation for S(h,p).
On the other hand, the total entropy as the function of microscopic enthalpy can be approximated from the entropy of each thermodynamic state. The entropy functions of N and D sate, SN(T,p) and SD(T,p) are derived from Eqs (6-9). The Legendre transformation is applied to each thermodynamic state such as:
Using the entropy and the enthalpy function of temperature, the entropy can be obtained as a function of the macroscopic enthalpy in the same way as above discussed in the case of S(Le)(H,p). Therefore Sn(Hp) and Sd(H,p) can be obtained. If these entropy functions of the macroscopic enthalpy can approxi mate those of the microscopic enthalpy, these entropy functions are related to the number of states as
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