Question

In the nuclear fusion reaction: ${}_1^2H + {}_1^3H \to {}_2^4He + n$ given that the repulsive potential energy between the two nuclei is $ \sim 7.7 \times {10^{ - 14}}J$, the temperature to which the gases must be heated to initiate the reaction is nearly ( Boltzmann’s constant, $K = 1.38 \times {10^{ - 23}}J$)
A) ${10^{^7}}K$
B) ${10^5}K$
C) ${10^3}K$
D) ${10^9}K$

Answer 1

Hint: In radioactivity, there are types of reactions occurring. One is nuclear fusion and another one is nuclear fission. These types of reactions occur only when the atom has an unstable nucleus. In this condition, if the smaller nuclei combine to form a larger nucleus is called nuclear fusion. The nuclear fusion reaction needs high temperatures to initiate. There are many examples of nuclear fusion reactions. In stars, nuclear fusion reactions occur naturally. And the man-made hydrogen bombs are also one of the examples of this reaction.

Formula used:
\[K.E = \dfrac{3}{2}KT\]
\[ \Rightarrow T = \dfrac{{2K.E}}{{3K}}\]\[\]
Where, \[K.E\]– Kinetic energy
\[K\]- Boltzmann’s constant which is equal to $K = 1.38 \times {10^{ - 23}}J$.
\[T\]- Temperature required for the process.

Complete step by step solution:
(i) Here we have repulsive potential energy and it equals kinetic energy. Hence, Kinetic energy equals $ \sim 7.7 \times {10^{ - 14}}J$.
(ii) And we have all the values required to find the temperature. Applying all the values in the formula,
\[ \Rightarrow T = \dfrac{{2K.E}}{{3K}}\]
\[T = \dfrac{{2 \times 7.7 \times {{10}^{ - 14}}}}{{3 \times 1.38 \times {{10}^{ - 23}}}}\]
\[\therefore T \approx 3.7 \times {10^9}K.\]
From the options given, we can say that \[{10^9}K\]. As it is close to the value found.

Therefore the correct option is D.

Additional Information:
(i) When the smaller nuclei combine to form the bigger nucleus, the process is called nuclear fusion. And in turn, if the bigger nucleus splits into the number of smaller nuclei is called nuclear fission.
(ii) nuclear fusion is the process that requires a huge amount of energy which is nearly equal to \[200MeV\]. It absorbs this amount of energy to start.
(iii) But nuclear fission does not need temperature. But it releases an enormous amount of energy while the bigger nucleus splits into smaller nuclei.

Note:
The kinetic energy of the nuclei is due to the temperature absorbed. Hence the kinetic energy is directly proportional to the temperature. The temperature cannot be less than \[200MeV\]. Hence the other options given are not suitable for this question. Other options are less than \[200MeV\].