Question

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 \(\left.\mathrm{J} \cdot \mathrm{m} / \mathrm{C}^{2}\right]\).

Short answer

Using Coulomb's Law formula, given values for the charges and the helium nucleus's distance, we can calculate the energy required for the fusion of deuterium to make an α particle. After obtaining the energy (E) in Joules, we can estimate the necessary temperature for the fusion using the kinetic energy formula with the Boltzmann constant. The temperature (T) in Kelvin can be calculated with the equation \(T = \frac{2E}{3k}\), where k is the Boltzmann constant \(1.38 \times 10^{-23} \mathrm{J/K}\). Plug in the energy value and the Boltzmann constant to compute the required temperature for the fusion process.

Step-by-step solution

Using the Coulomb's Law formula: \(E = 9.0 \times 10^{9} \left(\frac{Q_1 Q_2}{r}\right)\), we have to find the energy required for the fusion of deuterium to make an α particle. Given, \(Q_1\) and \(Q_2\) are the charges of the helium nucleus, which can be expressed as the charge of a proton \(1.6 \times 10^{-19} \mathrm{C}\), and the distance \(r = 2 \times 10^{-15} \mathrm{m}\). We'll plug these values into the formula to find the energy (E) required. #Step 2: Substitute the given values and calculate the energy#

Now, we will substitute the given values into the formula: \(E = 9.0 \times 10^{9} \left(\frac{(1.6 \times 10^{-19})(1.6 \times 10^{-19})}{2 \times 10^{-15}}\right)\) Now calculate the energy (E) by performing the required operations. #Step 3: Convert the energy to temperature#

Once we have calculated the energy (E) in Joules required for the fusion process, we can estimate the temperature needed for the fusion using the kinetic energy formula and the energy calculated: \(E = \dfrac{3}{2}kT\), where k is the Boltzmann constant \(k = 1.38 \times 10^{-23} \mathrm{J/K}\), and \(T\) is the required temperature in Kelvin. Now solve for the temperature (T). #Step 4: Estimate the temperature needed for the fusion of deuterium to α particles#

With the calculated energy (E) and the Boltzmann constant (k) in hand, we can finally estimate the temperature (T) needed for the fusion of deuterium to form α particles by rearranging the equation: \[T = \frac{2E}{3k}\] Plug in the energy value and the Boltzmann constant, and compute the temperature.