Legendre transformation in thermodynamics is known as useful tool to convert the thermodynamic variables. One of the most useful transformations is
where G is Gibbs energy, H is enthalpy and S is entropy. It should be remarked that entropy is well defined in statistical thermodynamics as a function of internal energy and volume, while Gibbs energy is defined as a function of temperature and pressure, and enthalpy is defined as an average around the system under fixed temperature and pressure. These functions are approximately related in statistical thermodynamics. In thermodynamics, entropy is converted to a function of temperature and pressure using the Legendre transformation, Eq. (1).
In statistical thermodynamics, the relation of Legendre transformation such as Eq. (1) will be guaranteed only if the property of thermodynamic functions satisfies some conditions. Concerning to the transformation of Eq. (1), the necessary condition is that the probability function of enthalpy under fixed temperature and pressure has only one maximum.
In the thermal transition of proteins, the thermodynamic functions and partition functions accompanied with thermal transition have been precisely determined with calo rimetry [1, 2]. These thermodynamic functions are useful to discuss the mechanism for protein folding and the stability. Using these functions, it has been shown that the probability function has several maximums around
D. Lorinczy (ed.), The Nature of Biological Systems as Revealed by Thermal Methods, 333-341. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
the transition temperature . This indicates apparently that the necessary condition for the Legendre transformation is not satisfied.
In this report, the microscopic enthalpy is introduced to clarify the two kinds of entropy functions. One entropy function is a function of macroscopic enthalpy and another is of microscopic enthalpy. When the total entropy of macroscopic enthalpy was evaluated without deconvolution method, it does not coincide to the total entropy of microscopic enthalpy apparently.
On the other hand, the methods to determine the thermodynamic functions of each thermodynamic state have been developed [1, 2]. They are know as 'deconvolution' method. From the heat capacity of the whole system, the thermody-namic functions of several thermodynamic states can be determined. In the present study, the entropy functions of each deconvoluted thermodynamic states were found to become good approximation of the entropy functions of microscopic enthalpy. This indicates that we should deconvolute the thermodynamic states where the Legendre transformation becomes a good approximation in order to discuss the system properly in statistical mechanics.
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