Technetium (Tc; Z=43 ) is a synthetic element used as a...

Technetium (Tc; $Z=43$ ) is a synthetic element used as a radioactive tracer in medical studies. A Tc atom emits a beta particle (electron) with a kinetic energy $\left(E_{\mathrm{k}}=\frac{1}{2} m v^{2}\right)$ of $4.71 \times 10^{-15} \mathrm{~J}$ What is the de Broglie wavelength of this electron?

Key Concepts

Wave-Particle Duality

This concept explains that particles such as electrons can exhibit both wave-like and particle-like properties. It is a fundamental principle of quantum mechanics, highlighting that entities traditionally considered as particles can also display interference and diffraction, phenomena typically associated with waves.

de Broglie Hypothesis

Proposed by Louis de Broglie, this hypothesis states that every particle has an associated wavelength, known as the de Broglie wavelength, which is given by the relation ? = h/p, where h is Planck’s constant and p is the momentum. This idea forms a bridge between classical mechanics and quantum mechanics by linking particle motion to wave characteristics.

Kinetic Energy and Momentum Relationship

For a non-relativistic particle, the kinetic energy (KE) is related to its momentum (p) by the equation KE = p²/(2m), where m is the mass of the particle. This relationship allows one to determine the momentum from the kinetic energy, which is a necessary step in calculating the de Broglie wavelength of the particle.

Planck's Constant

A fundamental constant in quantum mechanics, denoted by h, which appears in various important equations such as the energy of a photon (E = h?) and de Broglie’s relation (? = h/p). It sets the scale for quantum effects and is essential in relating energy, momentum, and wavelength in the quantum realm.

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