Example 3.

In this example, we are going to be working with a system of charged particles whose interactions are defined solely by a Coulomb interaction potential (see Figure 1) given by: $$E_{i,j}=\sum_{i}^{N}\sum_{i\neq j}\frac{q_iq_j}{4\pi\epsilon_0r_{ij}}$$ where \(N\) is the number of particles, \(q_i\) and \(q_j\) are the charges for particles i and j, \(ε_o\) is the vacuum permittivity constant, and \(r_{ij}\) is the interatomic distance. This example's main focus is to find a unit to measure current and therefore \(q\) and \(ε_o\).

The approach to define the units in this example is no different from Example 2. We are going to assume straight away that the units used to measure distance, mass, and time will be the angstrom \(Å\), the Dalton \(Da\), and the picosecond \(ps\) respectively. \(ε_o\) is equal to $$ε_o=8.854×10^{-12} \frac{Farad}{m}$$ in SI units. A \(\frac{Farad}{m}\) is derived from fundamental quantities and is equivalent to: $$1\frac{Farad}{m}=1\frac{s^2C^2}{m^3Kg}$$ Notice how we have to convert \(m→Å\), \(Kg→Da\), \(s→ps\), and also Coulombs (\(C\)), the SI unit to measure charge, into a unit of our own. In Chemistry, it is usual to represent electric charge as a multiple of Elementary charges (\(e\)). For example, Sodium has a formal charge of \(+1e\) and Chlorine has a formal charge of \(-1e\) in an NaCl crystal. 1 \(e\) is equivalent to \(1.602×10^{-19} C\) in SI units. When converted to \(Å\), \(Da\), \(ps\), and \(e\), \(ε_o\) becomes $$ε_o = 5.7289×10^{-7} \frac{ps^2e^2}{Å^3Da}$$ For example, the energy between two particles i and j of equal but opposite charges \(q_i = -q_j = 1 e\) distanced 1 \(Å\) apart would be equal to $$E_{i,j}=\frac{q_iq_j}{4\pi\epsilon_0r_{ij}}=\frac{1}{4\pi\epsilon_0}\approx1.389×{10}^5 \frac{DaÅ^2}{ps^2}$$ which is equivalent to \(2.306×10^{-18} J\) (in SI Units). See Figure 2 and Figure 3 to see the results of such a simulation.