# Why should we equate the formula for translational kinetic energy of monoatomic molecule with $P$ to find $T$?

In the nuclear fusion reaction,

$$H_1^2 + H_1^3 \rightarrow He_2^4 + n_0^1$$

suppose I have been given the repulsive potential energy (say $P$) between the two nuclei. I need to find the temperature (say $T$) at which the gases must be heated to initiate the reaction.

My textbook says that we can find $T$ using the equation

$$\frac{3}{2}kT=P$$

However, I could not understand why we should equate the formula for translational kinetic energy of monoatomic molecule with $P$ to find $T$. Can anyone explain why this equation will give correct value for $T$ ?

Energy is conserved, so the sum of the potential and kinetic energy is constant. As the incoming particles approach each other their potential energy increases due to the electrostatic repulsion, so the kinetic energy decreases. The maximum increase in the potential energy that is possible, $\Delta U_{max}$, is when the kinetic energy has decreased to zero, so if $T$ is the initial kinetic energy:

$$\Delta U_{max} = T$$

In this case we know the height of the potential energy barrier is $P$, so for the particles to get through the barrier we must have $\Delta U_{max} \ge P$, and therefore $T \ge P$.