Final answer:
The de Broglie wavelength of a proton will be shorter than that of an alpha particle when both have the same kinetic energy, because wavelength is inversely proportional to momentum, which is dependent on the particle's mass.
Step-by-step explanation:
When a proton and an alpha particle have the same kinetic energy, their de Broglie wavelengths can be compared using de Broglie's hypothesis.
According to this hypothesis, the wavelength λ for any particle can be calculated using the equation λ = h/p, where h is the Planck constant and p is the momentum of the particle.
Since kinetic energy (KE) is given by (1/2)mv2, and momentum (p) is given by mv, where m is the mass and v is the velocity of the particle, we can relate momentum to kinetic energy by the relation p = √(2mKE). For particles with the same kinetic energy, the momentum will be inversely proportional to the square root of their masses.
Considering that the proton has a much smaller mass than the alpha particle (which contains two protons and two neutrons), the proton will have a higher momentum for the same kinetic energy.
Consequently, the de Broglie wavelength of the proton will be shorter than that of the alpha particle. To summarize, de Broglie wavelength is inversely proportional to the momentum, and for the same kinetic energy, the proton's wavelength will be shorter than that of the alpha particle.