The molar Helmholtz free energy $($in $\frac{J}{m}ol)$ of a mixture of two similar polar organic liquids at $30C$ and bar is given by:

$A=90x_1+10x_1{x}_2-70$

where $x_1$ and $x_2$ represent the mole fractions of compounds $1$ and $2.$

a$)$ Derive an equation for partial molar Helmholtz free energy of compound $1,$ and compute its value for $x_1=50\%.$

b$)$ Calculate the pure component molar Helmholtz free energies of both compounds.

c$)$ Calculate the change of mixing of the Helmholtz free energy for $x_1=50\%.$

d$)$ Calculate the Excess Helmholtz free energy for $x_1=50\%.$