Estimate the temperature needed to achieve the fusion of deuterium to make an \(\alpha\) particle. The energy required can be estimated from Coulomb's law [use the form \(E=9.0 \times 10^{9}\) \(\left(Q_{1} Q_{2} / r\right),\) using \(Q=1.6 \times 10^{-19} \mathrm{C}\) for a proton, and \(r=2 \times\) \(10^{-15} \mathrm{m}\) for the helium nucleus; the unit for the proportionality constant in Coloumb's law is J \(\cdot \mathrm{m} / \mathrm{C}^{2} ]\)

We are given the form of Coulomb's law as: \[ E = 9.0 \times 10^9 \frac{Q_1 Q_2}{r} \] where \(Q_1\) and \(Q_2\) are the charges of the two particles (in this case, deuterium nuclei) with unit charge, i.e., the charge of a proton, \(Q = 1.6 \times 10^{-19} \mathrm{C}\), \(r = 2 \times 10^{-15} \mathrm{m}\) is the separation between the protons in the helium nucleus.

Using the given values, we can calculate the energy required for the fusion process: \[ E = 9.0 \times 10^9 \frac{(1.6 \times 10^{-19})^2}{2 \times 10^{-15}} \] Calculate this expression to find the energy value: \[ E \approx 7.68 \times 10^{-13} \mathrm{J} \]

Now, we need to estimate the temperature needed for the fusion process. We can relate energy and temperature using the equipartition theorem, which states that the average kinetic energy per particle is given by: \[ \frac{3}{2} k_B T = E \] where \(k_B = 1.38 \times 10^{-23} \mathrm{J/K}\) is the Boltzmann constant and \(T\) is the temperature in Kelvin. Plugging in the energy value we found in step 2, we can solve for temperature: \[ T = \frac{2E}{3k_B} = \frac{2 (7.68 \times 10^{-13} \mathrm{J})}{3 (1.38 \times 10^{-23} \mathrm{J/K})} \] Calculate this expression to find the temperature value: \[ T \approx 3.71 \times 10^7 \mathrm{K} \] So, the estimated temperature needed to achieve the fusion of deuterium to make an α particle is approximately \(3.71\times 10^7 \mathrm{K}\).