Gibbs-Duhem integration , equilibrium

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

The measurement of polymer solutions with lower polymer concentrations requires very precise pressure instruments, beeause the difference in the pure solvent vapor pressure becomes very small with deereasing amount of polymer. At least, no one can really answer the question if real thermodynamie equilibrium is obtained or only a frozen non-equilibrium state. Non-equilibrium data ean be deteeted from unusual shifts of the %-function with some experience. Also, some kind of hysteresis in experimental data seems to point to non-equilibrium results. A eommon eonsisteney test on the basis of the integrated Gibbs-Duhem equation does not work for vapor pressure data of binary polymer solutions because the vapor phase is pure solvent vapor. Thus, absolute vapor pressure measurements need very careful handling, plenty of time, and an experienced experimentator. They are not the method of choiee for high-viseous polymer solutions. 

If the logarithms of the activity coefficients of a binary system are plotted against molar composition, Eq. (53) relates the slopes at any value of the x s. This is useful for testing the consistency of data. Such testing becomes easier if we have a systematic way to correlate and smooth data and extend them over the entire range of composition. Integrated forms of the Gibbs-Duhem relation allow us to do this. Theoretically, one reliable measurement of vapor-liquid equilibrium at any point can then be used to characterize a system. 

The thermodynamic equilibrium constant IC is then obtained by integrating the appropriate form of the Gibbs-Duhem equation to give as corresponding... 

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