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In the book Problem solving in chemical and biochemical engineering with POLYMATH, Excel and Matlab by Cutlip and Shacham there is a problem (7.1) where you want to plot the compressibility factor for CO2 over a range of 0.1≤Pr<=10. To do this, we need to know how the volume varies with Pr for a constant Tr=1.1 using the van der Waal equation of state.

The van der Waal equation of state is:

\(P=\frac{RT}{V−b}−\frac{a}{V^2}\)

We define the reduced pressure as Pr=P/Pc, and the reduced temperature as Tr=T/Tc.

The approach we consider is to derive an expression for \(dV/dPr\). We proceed as follows:

\(\frac{dV}{dPr} = \frac{dV}{dP} \frac{dP}{dPr}\).

The first term on the right is also \((\frac{dP}{dV})^{-1}\) which we can derive from the van der Waal equation. The second term on the right is just a constant from the definition of the reduced pressure.

With some work you can derive this differential equation:

\(\frac{dP}{dV} = -\frac{RT}{(V-b)^2} + \frac{2a}{V^3}\)

Combine this information to derive the required ODE \(\frac{dV}{dPr}\) and express it in one or more functions in Python.

We are given an initial condition: at Pr=0.1, Tr=1.1, V=3.676.

Use this information to calculate V as a function of Pr by integrating the differential equation you derived in the previous step. Here is some additional information you need.

R = 0.08206   # gas constant
Pc = 72.9     # critical pressure
Tc = 304.2    # critical temperature

a = 27 * R**2 * Tc**2 / (Pc * 64)
b = R * Tc / (8 * Pc)

Tr = 1.1

Finally, the compressibility is defined as \(Z = \frac{P V}{R T}\). Plot the compressibility as a function of Pr.

Your solution should look like

this.

When you are done, download a PDF and turn it in on Canvas. Make sure to save your notebook, then run this cell and click on the download link.

%run ~/f23-06623/f23.py
%pdf