$($a$)$ Write down the Gibbs$-$Helmholtz equation with reference to the relative partial molar properties of a solution.

$($b$)$ Write down the general Gibbs$-$Duhem equation for the extensive state property $Q$ in a binary $A-B$ system.

$($c$)$ Show that the chemical potential defined in terms of the Gibbs free energy is identical to the partial molar Gibbs free energy $.$

$($d$)$ Show that the chemical potential defined in terms of the Helmholtz free energy is not identical to the partial molar Helmholtz free energy.

$($e$)$ Show that the two definitions ${}_i^A$ and ${}_i^U$ are equivalent.