To determine the temperature at which the nuclear fusion reaction occurs, we need to consider the relationship between kinetic energy and potential energy in the context of particle interactions. The equation you mentioned, \( \frac{3}{2} k T = P.E. \), is indeed related to kinetic energy, but it helps us understand the conditions necessary for overcoming the repulsive forces between the nuclei. Let's break this down step by step.
Understanding the Fusion Reaction
The fusion reaction you provided involves two isotopes of hydrogen: deuterium (\( ^2H_1 \)) and tritium (\( ^3H_1 \)). When these two nuclei collide with sufficient energy, they can overcome their electrostatic repulsion and fuse to form helium-4 (\( ^4He_2 \)) and a neutron (\( n \)). The repulsive potential energy between the two positively charged nuclei is given as \( 7.7 \times 10^{-14} \) J.
The Role of Temperature
Temperature is a measure of the average kinetic energy of particles in a system. In the context of nuclear fusion, we need the particles to have enough kinetic energy to overcome the repulsive potential energy barrier. This is where the equation \( \frac{3}{2} k T = P.E. \) comes into play. Here, \( k \) is Boltzmann's constant, and \( T \) is the temperature in Kelvin.
Why Use Kinetic Energy?
The equation \( \frac{3}{2} k T \) represents the average kinetic energy per particle in a monatomic ideal gas. While it might seem that we are discussing kinetic energy when we are interested in potential energy, the key point is that the kinetic energy of the particles must be sufficient to equal or exceed the potential energy barrier for the fusion reaction to occur. In other words, we need to find the temperature at which the average kinetic energy of the particles is equal to the repulsive potential energy.
Calculating the Required Temperature
To find the temperature, we can rearrange the equation to solve for \( T \):
- Start with the equation: \( \frac{3}{2} k T = P.E. \)
- Rearranging gives: \( T = \frac{2 P.E.}{3 k} \)
Now, substituting the values:
- Potential Energy, \( P.E. = 7.7 \times 10^{-14} \) J
- Boltzmann's constant, \( k = 1.38 \times 10^{-23} \) J/K
Plugging in these values:
\( T = \frac{2 \times 7.7 \times 10^{-14}}{3 \times 1.38 \times 10^{-23}} \)
Calculating this step-by-step:
- Numerator: \( 2 \times 7.7 \times 10^{-14} = 1.54 \times 10^{-13} \) J
- Denominator: \( 3 \times 1.38 \times 10^{-23} = 4.14 \times 10^{-23} \) J/K
Now, dividing the two results:
\( T = \frac{1.54 \times 10^{-13}}{4.14 \times 10^{-23}} \approx 3.72 \times 10^{9} \) K
Final Thoughts
This temperature, approximately \( 3.72 \times 10^{9} \) K, is extraordinarily high and reflects the extreme conditions necessary for nuclear fusion to occur. In practical terms, achieving such temperatures is a significant challenge, which is why fusion reactions are typically studied in controlled environments like stars or experimental reactors. Understanding the relationship between kinetic and potential energy is crucial in this context, as it highlights the energy barriers that must be overcome for fusion to take place.
