Question $5.$

The ideal gas law for a gas at low temperature and pressure is $PV=nRT,$

where $P$ is pressure $($in atm$),V$ is volume $($in L$),T$ is temperature $($in $K),n$ is the number of moles of

the gas, and $R=0\mathrm{.}0820578$ is the molar gas constant. The van der Waals equation

$(P+\frac{n^{2}a}{V^2})(V-nb)=nRT$

covers the nonideal case where these assumptions do not hold. Use the ideal gas law to compute an initial

guess, followed by Newton's Method applied to the van der Waals equation to find the volume of one

mole of oxygen at $320K$ and a pressure of $15$atm. For oxygen, $a=1\mathrm{.}36L^2-$atm $\frac{?}{mol{e}^2}$ and $b=0\mathrm{.}003183$

$\frac{L}{m}$ole.

What is your initial guess for $V?$

How do you convert calculating $V$ in the nonideal case to a root$-$finding problem?

Write a MATLAB code to use Newton's method to calculate $V$ in the nonideal case with three

significant digits. Please output the value of $V$ at each Newton iteration.