Question

In the following nuclear fusion reaction ${ }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+{ }_{0} \mathrm{n}^{1}$ the repulsive potential energy between the two fusing nuclei is $7.7 \times 10^{-14} \mathrm{~J}$. The Temperature to which the gas must be heated is nearly (Boltzman constant \(\mathrm{K}=1.38 \times 10^{-23} \mathrm{JK}^{-1}\) ) (A) \(10^{3} \mathrm{~K}\) (B) \(10^{5} \mathrm{~K}\) (C) \(10^{7} \mathrm{~K}\) (D) \(10^{9} \mathrm{~K}\)

Short answer

The temperature required to overcome the repulsive potential energy for this nuclear fusion reaction is around \(5.58 \times 10^{9} K\). Therefore, the closest answer is option (D) - \(10^{9} K\).

Step-by-step solution

Understand the problem

The problem is asking us to determine the minimal temperature required to overcome the repulsive potential energy for a nuclear fusion reaction. This is done by rearranging the formula of the most probable energy in a Maxwell-Boltzmann distribution, \(E = kT\) for some constant \(k\), to solve for \(T\) (Temperature).

Rearrange the formula to solve for T

We rearrange the formula \(E = kT\) to solve for \(T\). This gives us \(T = \frac{E}{k}\).

Substitute the values into the formula

We substitute the repulsive potential energy \(E = 7.7 \times 10^{-14} J\) and the Boltzmann constant \(k = 1.38 \times 10^{-23} \: JK^{-1}\) into the formula from step 2. When we do this, we get \(T = \frac{7.7 \times 10^{-14} \: J}{1.38 \times 10^{-23} \: JK^{-1}}\).

Calculate T (Temperature)

Evaluate \(T = \frac{7.7 \times 10^{-14} \: J}{1.38 \times 10^{-23} \: JK^{-1}}\) to find the temperature. A careful calculation yields \(T = 5.58 \times 10^{9} K\).

Make a conclusion

From our calculations, the temperature required to overcome the repulsive potential energy for this nuclear fusion reaction is around \(5.58 \times 10^{9} K\). This temperature is greater than all the options given except (D) - \(10^{9} K\). Hence, option D is the closest and is accepted as the answer.