Question

Assume that two deuterium nuclei in the core of fusion reactor at temperature T are moving towards each other, each with kinetic energy 1.5kT, when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature T required for them to reach a separation of $4\times {{10}^{-15}}m$ isin the range
$\text{A}\text{. }1\times {{10}^{9}}K < T < 2\times {{10}^{9}}K$
$\text{B}\text{. 2}\times {{10}^{9}}K < T < 3\times {{10}^{9}}K$
$\text{C}\text{. 3}\times {{10}^{9}}K < T < 4\times {{10}^{9}}K$
$\text{D}\text{. 4}\times {{10}^{9}}K < T < 5\times {{10}^{9}}K$

Answer 1

Hint: Use the law of conservation of conservation of energy which says that the mechanical energy of a system is constant. Mechanical energy of a system is equal to the sum of the kinetic energy and the potential energy of the system.
Formula used:
${{K}_{i}}+{{U}_{i}}={{K}_{f}}+{{U}_{f}}$
$U=\dfrac{K{{q}_{1}}{{q}_{2}}}{d}$

Complete answer:
It is given that two deuterium nuclei are very far from each other than the electrostatic potential energy (Coulomb potential energy) can be neglected. It is said that both the particles are moving towards each other. Each of the particles has an initial kinetic equal to 1.5kT, where k is the Boltzmann constant and T is the temperature of the reactor.
When these two nuclei come closer, they will feel electrostatic repulsion. As a result, both the particles will gradually slow down and stop at a separation d. After this, they will move in the opposite direction.
To find the minimum separation between the two nuclei, we will use the law of conservation of energy. According to this law, the sum of the kinetic energy and potential energy of the system (called the mechanical energy) is constant. This means that initial mechanical energy and the final mechanical energy are equal.
Therefore,
${{K}_{i}}+{{U}_{i}}={{K}_{f}}+{{U}_{f}}$ ….. (i).
Here, ${{K}_{i}}=1.5kT$, ${{U}_{i}}=0$ and ${{K}_{f}}=0$.
The potential energy between two charges separated by distance d is given as $U=\dfrac{K{{q}_{1}}{{q}_{2}}}{d}$, where K is a proportionality constant.
In this case, ${{q}_{1}}={{q}_{2}}=e$.
This means that ${{U}_{f}}=\dfrac{K{{e}^{2}}}{d}$.
Substitute the values in (i).
$\Rightarrow 1.5kT+0=0+\dfrac{K{{e}^{2}}}{d}$
$\Rightarrow T=\dfrac{K{{e}^{2}}}{d\times 1.5k}$.
Substitute $K=9\times {{10}^{9}}N{{m}^{2}}{{C}^{-2}}$, $e=1.6\times {{10}^{-19}}C$, $d=4\times {{10}^{-15}}m$ and $k=1.4\times {{10}^{-23}}J{{K}^{-1}}$.
$\Rightarrow T=\dfrac{9\times {{10}^{9}}\times {{\left( 1.6\times {{10}^{-19}} \right)}^{2}}}{4\times {{10}^{-15}}\times 1.5\times 1.4\times {{10}^{-23}}}=1.4\times {{10}^{9}}K$.

So, the correct answer is “Option A”.

Note:
The law of conservation of energy can also be stated as the change in kinetic energy of a system is equal to the negative of the change in potential energy of the system. This means that the loss in kinetic energy is equal to the gain in the potential energy and vice versa.