To use the van der Waals equation of state to find $V_{?}$ at a given $T$ and $P,$ it is numerically

convenient to rewrite it as a cubic polynomial in the compressibility factor $Z$ :

$Z^3-(B+1)Z^2+AZ-AB=0,$

where

Determine $Z_L,$ the compressibility factor for liquid water and smallest root of this polyno$-$

mial, and $Z_V,$ the compressibility factor for water vapor and largest root of this polynomial,

at water's normal boiling point $(T=10C,P=1(atm))$ using:

$($a$)$ Fixed$-$point iteration. For only this part, it is recommended to rearrange the polynomial

to solve for the liquid as

$Z_L=B+Z_L^2\frac{B+1-Z_L}{A}$

and the equation to solve for the vapor as

$Z_V=B+1-A\frac{Z_V-B}{Z_V^2}.$

You can implement fixed$-$point iteration in MATLAB ${?}^{}$ using a loop. For example, this

loop iterates $100$ times to solve our example problem from class starting from

$\{$:$x_0=0)$ :

$x=0$;

for $i=1$:$100$

$x=$exp$(-x^{{?}^{{?}^{?}}}2)$;

end

$($b$)$ The built$-$in fzero function in MATLAB$.$ It is recommended to define an anonymous

function for the equation you will solve. For example, to solve $e^{-x^2}-x=0$ with initial

guess $x_0=0,$ you can do:

eqn $=$@$(x)exp(-x^{{?}^{{?}^{?}}}2)-x$;

fzero $(eqn,0\mathrm{.}0)$

$($c$)$ The built$-$in roots function in MATLAB$.$ Read the MATLAB$$ documentation for an

example of how to use this function.

Use your results from #$2($c$)$ to compute the liquid and vapor mass densities, then compare

to the properties of saturated water from NIST $[1].$ If there are discrepancies, comment on

what you think the most likely reason is$.$