The excess Gibbs energy of a binary liquid mixture at $T$ and $P$ is given by

$\frac{G^E}{R}T=A{x}_1{x}_2$

This is the simplest realistic expression for the excess Gibbs energy, and applies to binary systems

comprised of species that are chemically similar.

$($a$)$ What are the expressions for $ln{}_1$ and $ln{}_2$ that result from this expression?

$($b$)$ For a particular binary system, the following data are known for a single data point:

$t=45C,P=37$kPa,$x_1=0\mathrm{.}398$ and $y_1=0\mathrm{.}428$

In addition, $P_1^{\text{s}\text{a}\text{t}}=27\mathrm{.}78$kPa and $P_2^{\text{s}\text{a}\text{t}}=29\mathrm{.}82$kPa. Determine the value of $A.$