$5\mathrm{.}78.$ The van der Waals equation of state $($Equation $5\mathrm{.}3-7)$ is to be used to estimate the specific molar volume hat$(V)(\frac{L}{m}ol)$ of air at specified values of $T(K)$ and $P(atm).$ The van der Waals constants for air are $a=1\mathrm{.}33$atm$*\frac{L^2}{m}o{l}^2$ and $b=0\mathrm{.}0366\frac{L}{m}ol.$

$($a$)$ Show why the van der Waals equation is classified as a cubic equation of state by expressing it in the form

$f(hat(V))=c_3$hat$(V{)}^3+c_2$hat$(V{)}^2+c_1$hat$(V)+c_0=0$

where the coefficients $c_3,c_2,c_1,$ and $c_0$ involve $P,R,T,a,$ and $b.$ Calculate the values of these coefficients for air at $223K$ and $50\mathrm{.}0$atm. $($Include the units when giving the values.$)$