Synoptic Exercises
1. Define your own functions (with docstrings) to calculate the following quantities.
Use scipy.constants to access values of any physical constants, rather than typing them manually.
Print your result to a reasonable number of decimal places!
a. The de Broglie wavelength
\[\lambda = \frac{h}{p},\]
where \(h\) is Planck’s constant and \(p\) is the momentum.
Test your function for \(p = 1.01 \times 10^{-31}\,\mathrm{kg\ m\ s}^{-1}\).
b. The change in the Gibbs free energy
\[\Delta G = -RT\ln K,\]
where \(R\) is the gas constant, \(T\) is the temperature and \(K\) is the equilibrium constant.
Test your function for \(T = 500\, \mathrm{K}\) and \(K = 1.15\).
c. The mean activity according to the Debye-Huckel limiting law
\[\gamma_{\pm} = \exp\left[-|z_{+}z_{-}|A\sqrt{I}\right],\]
where \(z_{+}\) and \(z_{-}\) are the charges on the cations and anions respectively, \(I\) is the ionic strength and \(A\) is a constant that depends upon the solvent and the temperature.
Test your function with values of
\[ z_{+} = 1\ \ \ \ \ z_{-} = 2 \]
\[ A = 1.179\, \mathrm{M}^{\frac{1}{2}} \ \ \ \ \ I = 0.018\, \mathrm{M} \]
\[\lambda = 6.56 \times 10^{-3}\,\mathrm{m}\]
\[\Delta G = -5.81 \times 10^{-1}\,\mathrm{kJ\ mol}^{-1}\]
\[\gamma_{\pm} = 0.729 \]
2a. The thermal de Broglie wavelength \(\Lambda\) describes the average de Broglie wavelength of particles in an ideal gas at a given temperature:
\[\Lambda = \frac{h}{\sqrt{2\pi mk_{\mathrm{B}}T}},\]
where \(h\) is Planck’s constant, \(m\) is the mass of each particle, \(k_\mathrm{B}\) is Boltzmann’s constant and \(T\) is the temperature.
Write a function to calculate \(\Lambda\).
2b. The total entropy of an ideal gas can be calculated with the Sackur-Tetrode equation:
\[\frac{S}{k_{\mathrm{B}}N} = \ln\left(\frac{V}{N\Lambda^{3}}\right) + \frac{5}{2},\]
where \(S\) is the entropy, \(N\) is the number of particles, \(V\) is the volume, \(k_{\mathrm{B}}\) is Boltzmann’s constant and \(\Lambda\) is the thermal de Broglie wavelength.
Write a function to calculate the entropy of an ideal gas with the Sackur-Tetrode equation.
Using your two functions, calculate the entropy of an ideal gas with
\[ m = 6.63 \times 10^{-26}\,\mathrm{kg} \ \ \ \ \ T = 500\,\mathrm{K} \] \[ N = 1.92 \times 10^{23} \ \ \ \ \ V = 5.00 \times 10^{-3}\,\mathrm{m}^{3} \]
\[S = 50.2\,\mathrm{J\ K}^{-1}\]