The partial molar volume of cyclohexane $(1)$ in benzene $(2)$ at $30C$ is found to depende on mole fraction as

$\frac{?}{\text{b}\text{a}\text{r}}(V{)}_1(c\frac{m^3}{m}ol)=112\mathrm{.}24-5\mathrm{.}28x_1+2\mathrm{.}64x_1^2,$

while at the same temperature pure benzene has a molar volume of $89\mathrm{.}96c\frac{m^3}{m}ol.$

$($a$)$ Determine $V_{{?}_{?}}(1)($pure state$)$ and $\frac{?}{\text{b}\text{a}\text{r}}(V{)}_1^{}($extreme dilution$)$ for cyclohexane in benzene.

$($b$)$ Estimate the volume change when mixing $1$ mole of cyclohexane and $2$ moles of benzene.

$($c$)$ Knowing $(\frac{de\frac{l}{b}ar(G{)}_i}{\text{d}\text{e}\text{l}\text{P}}{)}_T\frac{?}{b}=ar(V{)}_i,$ determine cyclohexane's $(\frac{G_{{?}_{?}}(i)}{RT})$ and $(\frac{\frac{?}{\text{b}\text{a}\text{r}}(G{)}_i^{}}{RT})$ when the system pressure increases from $1$atm to $100$atm at $30C.$

$($d$)$ Given that the partial molar volume is in polynomial form, we can conveniently retrieve the corresponding correlations for $\frac{?}{\text{b}\text{a}\text{r}}(V{)}_1$ and for $V.$