Eulers Theorem and the Gibbs-Duhem Relation

Euler s theorem is a mathematical theorem that applies to homogeneous functions. A function of several independent variables, f ni,n2,ti3,. ..,nc), is said to be homogeneous of degree k if 

Let Y stand for any extensive quantity. Since Y is homogeneous of degree 1 in the n s if T and P are constant, Euler s theorem becomes 

Equation (4.6-3) is a remarkable relation that gives the value of an extensive quantity as a weighted sum of partial derivatives. An unbiased newcomer to thermodynamics would likely not believe this equation without its mathematical proof. 

Euler s theorem can also be written in terms of the mean molar quantity, defined 

From Euler s theorem, Eq. (4.6-3), we can write an expression for dY, the differential of an extensive quantity denoted by Y  

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The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

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