The neutrons produced in fission reactors have a wide range...

The neutrons produced in fission reactors have a wide range of kinetic energies. After the neutrons make several collisions with atoms, they give up their excess kinetic energy and are left with the same average kinetic energy as the atoms, which is $\frac{3}{2} k_{\mathrm{B}} T$. If the temperature of the reactor core is $T=400.0 \mathrm{~K}$, find $(\mathrm{a})$ the average kinetic energy of the thermal neutrons, and (b) the de Broglie wavelength of a neutron with this kinetic energy.

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Key Concepts

Relationship Between Kinetic Energy and Momentum

For a non-relativistic particle, its kinetic energy (E) is related to its momentum (p) by the equation E = p²/(2m), where m is the mass of the particle. This relationship provides a bridge between the energy obtained from the equipartition theorem and the momentum required to compute the de Broglie wavelength, enabling the calculation of quantum mechanical properties from classical thermodynamic quantities.

Equipartition Theorem

This concept from statistical mechanics states that in thermal equilibrium, the average energy associated with each quadratic degree of freedom is (1/2)kT. For a particle moving in three dimensions, this results in an average kinetic energy of (3/2)kT. This is crucial in estimating the kinetic energy of particles, such as neutrons in a reactor core, based solely on the temperature.

De Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics which asserts that every particle exhibits wave-like behavior, with a wavelength given by ? = h/p, where h is Planck's constant and p is the momentum. This allows us to understand and calculate the wave properties of matter, such as neutrons, based on their kinetic energy.

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