. For a given Kinetic Energy, which of the following has smallest De- Broglie wave lengt- electron, proton and a8 – particle?

To determine which particle has the smallest De Broglie wavelength for a given kinetic energy, we need to understand the relationship between kinetic energy, mass, and wavelength. The De Broglie wavelength is given by the formula:

De Broglie Wavelength Formula

The formula for the De Broglie wavelength (λ) is:

λ = h / p

where:

• h is Planck's constant (approximately 6.626 x 10^-34 Js)
• p is the momentum of the particle, which can be expressed as p = mv, where m is mass and v is velocity.

Relating Kinetic Energy to Momentum

The kinetic energy (KE) of a particle is given by the equation:

KE = (1/2)mv²

From this, we can express velocity in terms of kinetic energy:

v = sqrt(2KE/m)

Substituting this expression for velocity into the momentum equation gives us:

p = m * sqrt(2KE/m) = sqrt(2mKE)

Substituting into the De Broglie Equation

Now, substituting this momentum back into the De Broglie wavelength formula:

λ = h / sqrt(2mKE)

From this equation, we can see that the De Broglie wavelength is inversely proportional to the square root of the mass of the particle. This means that for a given kinetic energy, the particle with the largest mass will have the smallest De Broglie wavelength.

Comparing the Masses of the Particles

Now, let’s look at the masses of the particles in question:

• Electron: Approximately 9.11 x 10^-31 kg
• Proton: Approximately 1.67 x 10^-27 kg
• Alpha particle (α-particle): Approximately 6.64 x 10^-27 kg (which is about 4 times the mass of a proton)

Determining the Smallest Wavelength

Given that the De Broglie wavelength is inversely related to the square root of the mass, we can conclude:

• The electron has the smallest mass, leading to the largest De Broglie wavelength.
• The proton has a larger mass than the electron, resulting in a smaller De Broglie wavelength than the electron but larger than the alpha particle.
• The alpha particle has the largest mass, which means it will have the smallest De Broglie wavelength among the three.

Thus, for a given kinetic energy, the particle with the smallest De Broglie wavelength is the alpha particle. This relationship highlights how mass plays a crucial role in wave-particle duality, illustrating the fascinating nature of quantum mechanics.