(a) For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.40 × 10–10 m?

To find the kinetic energy of a neutron that corresponds to a de Broglie wavelength of 1.40 × 10^-10 m, we can use the de Broglie wavelength formula and the relationship between kinetic energy and momentum. Let's break this down step by step.

Understanding de Broglie Wavelength

The de Broglie wavelength (λ) is given by the formula:

λ = h / p

where:

• h is Planck's constant (approximately 6.626 × 10^-34 Js),
• p is the momentum of the particle.

Relating Momentum to Kinetic Energy

For a particle like a neutron, the momentum (p) can be expressed in terms of its mass (m) and velocity (v):

p = mv

The kinetic energy (KE) of the neutron is given by:

KE = (1/2)mv²

We can also relate kinetic energy to momentum through the equation:

KE = p² / (2m)

Calculating the Kinetic Energy

First, we need to find the momentum using the de Broglie wavelength formula. Rearranging the formula gives us:

p = h / λ

Substituting the values:

p = (6.626 × 10^-34 Js) / (1.40 × 10^-10 m)

Calculating this yields:

p ≈ 4.73 × 10^-24 kg·m/s

Finding Kinetic Energy from Momentum

Now, we can substitute the momentum back into the kinetic energy formula:

KE = p² / (2m)

The mass of a neutron (m) is approximately 1.675 × 10^-27 kg. Plugging in the values:

KE = (4.73 × 10^-24 kg·m/s)² / (2 × 1.675 × 10^-27 kg)

Calculating this gives:

KE ≈ 6.67 × 10^-20 J

Converting to Electronvolts

Often, kinetic energy is expressed in electronvolts (eV) for subatomic particles. To convert joules to electronvolts, we use the conversion factor:

1 eV = 1.602 × 10^-19 J

Thus, converting our kinetic energy:

KE ≈ (6.67 × 10^-20 J) / (1.602 × 10^-19 J/eV) ≈ 0.416 eV

Final Result

In summary, the kinetic energy of a neutron that corresponds to a de Broglie wavelength of 1.40 × 10^-10 m is approximately:

6.67 × 10^-20 J or 0.416 eV