Problem $2-$ Linearized Model: Cubic Equation of State $(15$ pts$)$

The van der Waals $($vdW$)$ cubic equation of state is a model that describes the thermody$-$

namic properties of a non$-$ideal gas using two parameters a and $b$ :

$P=\frac{RT}{V_m-b}-\frac{a}{V_m^2}$

where $P$ is the pressure, $R$ is the gas constant, $T$ is the temperature, and $V_m$ is the molar vol$-$

ume.

We would like to use linear regression to solve for the parameters of the model in the case of

methane gas. Experimental data for the pressure as a function of molar volume is provided in

the "eos$-$data.dat" file. All data was collected at $T=300K.$ However, since the equation of

state is not linear with respect to the parameters, a linearized version should be derived first.

$($a$)$ Derive a linearized form of the vdW equation of state shown above. There are several

ways to linearize the equation of state. However, some methods are better than

others; aim for three unknowns where one is technically a function of the other two. Fi$-$

nally, rewrite into general matrix form $(Ax=b)$ where $x$ is the vector of unknowns.

$($b$)$ Using Python and the matrix method for linear least squares regression, solve and re$-$

port the model parameters for a and $b.$ Sanity check that your third unknown is close

to its expected value based on the fit values of a and $b.$

$($c$)$ Code a function in Python that computes an array of pressures given an array of molar

volumes and the model parameters a and $b.$ Use this function to plot the model predic$-$

tions against the experimental data for the pressure of methane.

$($d$)$ Finally, discuss if other direct solvers, e$.$g$.,$ Gaussian Elimination, and iterative solvers,

e$.$g$.,$ Gauss$-$Seidel, would be appropriate for this problem. Explain your reasoning with

evidence.

"eos$-$data.dat" file content:

# Vm$($L$/$mol$)$ P$($bar$)$

$0\mathrm{.}100207\mathrm{.}621$

$0\mathrm{.}127154\mathrm{.}076$

$0\mathrm{.}162121\mathrm{.}974$

$0\mathrm{.}20798\mathrm{.}645$

$0\mathrm{.}26480\mathrm{.}085$

$0\mathrm{.}33664\mathrm{.}848$

$0\mathrm{.}42852\mathrm{.}274$

$0\mathrm{.}54641\mathrm{.}941$

$0\mathrm{.}69533\mathrm{.}509$

$0\mathrm{.}88626\mathrm{.}677$

$1\mathrm{.}12921\mathrm{.}175$

$1\mathrm{.}43816\mathrm{.}769$

$1\mathrm{.}83313\mathrm{.}254$

$2\mathrm{.}33610\mathrm{.}460$

$2\mathrm{.}9768\mathrm{.}245$

$3\mathrm{.}7936\mathrm{.}493$

$4\mathrm{.}8335\mathrm{.}109$

$6\mathrm{.}1584\mathrm{.}018$

$7\mathrm{.}8483\mathrm{.}159$

$10\mathrm{.}0002\mathrm{.}482$Problem $2-$ Linearized Model: Cubic Equation of State $(15$ pts$)$

The van der Waals $($vdW$)$ cubic equation of state is a model that describes the thermody$-$

namic properties of a non$-$ideal gas using two parameters a and $b$ :

$P=\frac{RT}{V_m-b}-\frac{a}{V_m^2}$

where $P$ is the pressure, $R$ is the gas constant, $T$ is the temperature, and $V_m$ is the molar vol$-$

ume.

We would like to use linear regression to solve for the parameters of the model in the case of

methane gas. Experimental data for the pressure as a function of molar volume is provided in

the "eos$-$data.dat" file. All data was collected at $T=300K.$ However, since the equation of

state is not linear with respect to the parameters, a linearized version should be derived first.

$($a$)$ Derive a linearized form of the vdW equation of state shown above.

Let's define a new variable, $(z=\frac{V}{V-b}).$ So$,$

the equation becomes:

$P=\frac{RT}{z-1}-\frac{a}{(z-1{)}^2}$

$(=\frac{a}{R^2{T}^2})$ and $,(=\frac{b}{RT}).$

The equation now reads:

$P=\frac{RT}{z-1}-\frac{a}{(z-1{)}^2}=\frac{RT}{z}-\frac{a}{z^2}$

Create new variables:

$(x_1=\frac{T}{z}),(x_2=z),$ and $,(x_3=\frac{1}{z})$

The equation can be rewritten as follows:

$P=R{x}_1+x_2+x_3$

This equation is now linear in relation to the

parameters alpha, beta, and $R.$

Finally, rewrite this equation in the general

matrix form $(Ax=b),$

Start Answering from here using the equation above

$($b$)$ Using Python and the matrix method for linear least squares regression, solve and re$-$

port the model parameters for a and $b.$ Sanity check that your third unknown is close

to