Synoptic Exercises

1. You have been provided with an experimentally measured pressure/volume isotherm for \(\ce{SF6}\) at \(298\,\mathrm{K}\) in the form of a csv file.

You can see some familiar isotherms here.

a. Read in the experimental data using NumPy, convert each quantity to SI units, and plot the isotherm using Matplotlib.

b. Assuming ideality, the isotherm should be well modelled by the ideal gas law:

\[p = \frac{RT}{V_\mathrm{m}},\]

where \(p\) is the pressure, \(V_\mathrm{m}\) is the molar volume (\(V / n\)), \(R\) is the gas constant and \(T\) is the temperature.

Write a function to calculate the pressure with the ideal gas law and use this to model the \(\ce{SF6}\) isotherm between \(V_\mathrm{m} = 6.87 \times 10^{-4}\,\mathrm{m}^{3}\,\mathrm{mol}^{-1}\) and \(V_\mathrm{m} = 4.66 \times 10^{-3}\,\mathrm{m}^{3}\,\mathrm{mol}^{-1}\). Plot your modelled isotherm against the experimental data.

c. Non-ideality due to intermolecular interactions and other factors can be accounted for (in part) by the Van der Waals equation of state:

\[p = \frac{RT}{V_\mathrm{m} - b} - \frac{a}{V_\mathrm{m}^{2}},\]

where \(p\) is the pressure, \(R\) is the gas constant, \(V_\mathrm{m}\) is the molar volume and \(a\) and \(b\) are system dependent constants that describe the strength of the interactions and excluded volume effects respectively.

Write a function to calculate the pressure with the Van der Waals equation and use this to model the \(\ce{SF6}\) isotherm using

\[\begin{equation*} a = 7.857 \times 10^{-1}\,\mathrm{m}^6\,\mathrm{Pa\, mol}^{-2} \qquad b = 8.79 \times 10^{-5}\,\mathrm{m}^3\,\mathrm{mol}^{-1} \end{equation*}\]

between \(V_\mathrm{m} = 6.87 \times 10^{-4}\,\mathrm{m}^3\,\mathrm{mol}^{-1}\) and \(V_\mathrm{m} = 4.66 \times 10^{-3}\,\mathrm{m}^3\,\mathrm{mol}^{-1}\).

Plot your modelled isotherm and compare this against the idealised curve and the experimental data.

2. The mean activity coefficient of ions in solution can be measured experimentally. You have been provided with the mean activity coefficient of \(\ce{Na2SO4}\) at \(298\,\mathrm{K}\) for a range of ionic strengths.

a. Use Matplotlib to create a scatter plot of the experimental mean activity coefficient as a function of the ionic strength \(I\).

b. The mean activity coefficient can be predicted by the Debye-Huckel limiting law:

\[\ln\gamma_{\pm} = -|z_{+}z_{-}|A\sqrt{I},\]

where \(z_{+}\) and \(z_{-}\) are cation and anion charges, \(I\) is the ionic strength and \(A\) is a solvent- and temperature-dependent constant.

Write a function to calculate the mean activity coefficient according to the Debye-Huckel limiting law (feel free to reuse your code from previous exercises). Using \(A = 1.179\,\mathrm{M}^{-\frac{1}{2}}\), plot the Debye-Huckel mean activity coefficient between \(I = 0\,\mathrm{M}\) and \(I = 6\,\mathrm{M}\) and compare this with the experimental values.

c. There are several extensions of the Debye-Huckel limiting law that aim to improve its description of the mean activity coefficient outside of the dilute limit. One such extension is:

\[\ln\gamma_{\pm} = -|z_{+}z_{-}|\frac{A\sqrt{I}}{1 + Ba_{0}\sqrt{I}},\]

where \(B\) is another solvent- and temperature-dependent constant and \(a_{0}\) is the distance of closest approach (expected to be proportional to the closest distance between ions in solution).

Write a function to calculate the mean activity coefficient according to the extended Debye-Huckel limiting law. Using

\[\begin{equation*} B = 18.3\, \mathrm{nm^{-1}\, M^{-\frac{1}{2}}} \qquad a_{0} = 0.071\, \mathrm{nm} \end{equation*}\]

plot the extended Debye-Huckel mean activity coefficient between \(I = 0\,\mathrm{M}\) and \(I = 6\,\mathrm{M}\) and compare this with the experimental values and the original Debye-Huckel limiting law.