They are defined as the difference between the value of the property in a real mixture and the value that would exist in an ideal solution under the same conditions.
The excess volume (VE), internal energy (UE), and enthalpy (HE) are identical to the corresponding mixing properties; that is, These relationships hold because the volume, internal energy, and enthalpy changes of mixing are zero for an ideal solution.
term originates from the entropy of mixing of an ideal mixture.
The excess partial molar Gibbs free energy is used to define the activity coefficient, By way of Maxwell reciprocity; that is, because the excess molar volume of component
This formula can be used to compute the excess volume from a pressure-explicit activity coefficient model.
Similarly, the excess enthalpy is related to derivatives of the activity coefficients via By taking the derivative of the volume with respect to temperature, the thermal expansion coefficients of the components in a mixture can be related to the thermal expansion coefficient of the mixture: Equivalently: Substituting the temperature derivative of the excess partial molar volume, one can relate the thermal expansion coefficients to the derivatives of the activity coefficients.
This quantity can be related to derivatives of the excess molar volume, and thus the activity coefficients: Elliott, J. Richard; Lira, Carl T. (2012).
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