If two nuclei are to fuse in a nuclear reaction, they must be moving fast enough so that the repulsive Coulomb force between them does not prevent them for getting within \(R \approx 10^{-14} \mathrm{m}\) of one another. At this distance or nearer, the attractive nuclear force can overcome the Coulomb force, and the nuclei are able to fuse. (a) Find a simple formula that can be used to estimate the minimum kinetic energy the nuclei must have if they are to fuse. To keep the calculation simple, assume the two nuclei are identical and moving toward one another with the same speed \(v\). (b) Use this minimum kinetic energy to estimate the minimum temperature a gas of the nuclei must have before a significant number of them will undergo fusion. Calculate this minimum temperature first for hydrogen and then for helium. (Hint: For fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance \(R\) apart.)

We are given the hint that for fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance \(R\) apart. The Coulomb potential energy is given by: \(U(r) = \frac{k * q_1 * q_2}{r}\), where \(q_1\) and \(q_2\) are the charges of the two nuclei, \(r\) is the distance between them, and \(k\) is the Coulomb constant (\(k = 8.9875 * 10^9 \frac{\mathrm{N m^2}}{\mathrm{C^2}}\)). If the two nuclei are identical and moving towards each other with the same speed \(v\), their combined kinetic energy is given by: \(K.E. = 2 * \frac{1}{2} m v^2 = m v^2\), where \(m\) is the mass of a single nucleus. We know that the minimum kinetic energy required for fusion is equal to the Coulomb potential energy when the nuclei are at a distance of \(R = 10^{-14} \mathrm{m}\). Therefore, we have: \(m v^2 = \frac{k * q_1^2}{R}\). Now, we need to isolate \(v^2\) to find a simple formula for estimating the minimum kinetic energy: \(v^2 = \frac{k * q_1^2}{m * R}\).

Using the equation \(v^2 = \frac{k * q_1^2}{m * R}\), we can calculate the minimum temperature for gas to undergo fusion: \(T_{min} = \frac{m * v^2}{3k_B}\), where \(k_B\) is the Boltzmann constant (1.381 × 10^-23 J/K). For hydrogen nuclei, we have one electron charge (\(q_1 = 1.6 * 10^{-19} \mathrm{C}\)) and mass (\(m = 1.67 * 10^{-27} \mathrm{kg}\)). Substituting these values and the value of \(R\) into the equation for \(v^2\), we get: \(v^2 = \frac{(8.9875 * 10^9)(1.6 * 10^{-19})^2}{(1.67 * 10^{-27})(10^{-14})} \) Now we can calculate the minimum temperature for hydrogen gas: \(T_{min(H)} = \frac{(1.67 * 10^{-27}) * [(8.9875 * 10^9)(1.6 * 10^{-19})^2 / ((1.67 * 10^{-27})(10^{-14}))]}{3 * (1.381 * 10^{-23})} \) For helium nuclei, we have two electron charges (\(q_1 = 3.2 * 10^{-19} \mathrm{C}\)) and 4 times the mass of a hydrogen nucleus (\(m = 4 * 1.67 * 10^{-27} \mathrm{kg}\)). Substituting these values and the value of \(R\) into the equation for \(v^2\), we get: \(v^2 = \frac{(8.9875 * 10^9)(3.2 * 10^{-19})^2}{(4 * 1.67 * 10^{-27})(10^{-14})} \) Now we can calculate the minimum temperature for helium gas: \(T_{min(He)} = \frac{(4 * 1.67 * 10^{-27}) * [(8.9875 * 10^9)(3.2 * 10^{-19})^2 / ((4 * 1.67 * 10^{-27})(10^{-14}))]}{3 * (1.381 * 10^{-23})} \) With these formulas, we can estimate the minimum temperature at which a significant number of hydrogen and helium nuclei will undergo fusion.