Deriving thermodynamic potentials from equations 3-41 to 3-49

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swmmr1928
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Textbook page 44, Thermodynamic Properties from Volumetric Data, Chapter 3, showing equations 3-40 to 3-48 for internal energy U, enthalpy H, Helmholtz energy A, Gibbs energy G, chemical potential, and integral expressions for U and H.

Printed textbook page 45, section 3.4, showing equations 3-49 to 3-54 for thermodynamic properties with independent variables V and T, including integral expressions for S, A, G, mu_i and RT ln phi_i.

I don't understand how some terms are derived.
How did the last term of 3-47 originate?
How did 3-49 get so many terms from just one term in 3-41?
Why integrate from V to infinite? That is not intuitive.
Thas a functions are unusual because the absolute values of U,H,S cannot be computed. However, the analytical functions are needed to derive the expression for chemical potential?
 
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swmmr1928 said:
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I don't understand how some terms are derived.
How did the last term of 3-47 originate?
How did 3-49 get so many terms from just one term in 3-41?
All the terms you are referring to are the properties of the mixture in the ideal gas state at the given temperature and composition.
Why integrate from V to infinite? That is not intuitive.
Infinite specific volume refers to the ideal gas limit. You need to do the integration to get the correction for the property at finite specific volume.
Thas a functions are unusual because the absolute values of U,H,S cannot be computed. However, the analytical functions are needed to derive the expression for chemical potential?
The terms involving the chemical potentials are for the pure substances at temperature T relative to a specified reference state.

Chet