Final answer:
To calculate the temperature needed for fusion with a given kinetic energy, one must utilize the relation between kinetic energy and temperature, considering the Boltzmann constant. The formula E_k = 3/2 k_B T allows us to solve for the temperature after rearranging and plugging in the values. The resulting temperature is vital for overcoming the Coulomb barrier in nuclear fusion reactions.
Step-by-step explanation:
To find the temperature necessary for the atoms in a fusion experiment to have average kinetic energies of 6.40×10⁻¹⁴ J, we can use the equipartition theorem, which relates kinetic energy to temperature. The equation for the average kinetic energy (E_k) for a particle in a one-dimensional space is E_k = 1/2 k_B T, where k_B is the Boltzmann constant and T is the temperature in kelvins. Since the energy given is the average kinetic energy per particle in three-dimensional space, we need to consider all three dimensions, so the equation becomes E_k = 3/2 k_B T. Using this equation, we can calculate the temperature T required for the fusion process.
Calculating it step by step:
- Use the formula for average kinetic energy in three dimensions: E_k = 3/2 k_B T.
- Rearrange the formula to solve for temperature: T = (2/3) • (E_k / k_B).
- Insert the given average kinetic energy (E_k = 6.40×10⁻¹⁴ J) and the Boltzmann constant (k_B = 1.38×10⁻²³ J/K).
- Calculate the temperature: T = (2/3) • (6.40×10⁻¹⁴ J / 1.38×10⁻²³ J/K).
- Perform the calculation to find the temperature in kelvins.
This temperature is critical to achieving conditions for nuclear fusion, as high temperatures are necessary to overcome the Coulomb barrier that prevents atomic nuclei from coming close enough to fuse. Fusion reactions require temperatures in the range of tens of millions to hundreds of millions of degrees Kelvin.