The interatomic spacing in a crystal of table salt is 0.281 nm. This crystal is being studied in a neutron diffraction experiment, similar to the one that produced the photograph in the figure. How fast must a neutron (mass = 1.67 x 10^-27 kg) be moving to have a de Broglie wavelength of 0.281 nm?

λ = h/p = h/m•v,

v = h/m•λ =
=6.63•10^-34/1.67•10-27•0.281•10^-9 =
= 1420 m/s

To find the velocity at which a neutron must be moving to have a de Broglie wavelength of 0.281 nm, we can use the de Broglie wavelength equation:

λ = h / p

where:
λ = de Broglie wavelength
h = Planck's constant (6.63 x 10^-34 J·s)
p = momentum of the neutron

The momentum of an object with mass m and velocity v is given by:

p = m * v

Substituting this into the de Broglie wavelength equation, we get:

λ = h / (m * v)

Rearranging the equation to solve for v:

v = h / (m * λ)

Now, let's substitute the known values into the equation:

λ = 0.281 nm = 0.281 x 10^-9 m (convert nm to meters)
m = 1.67 x 10^-27 kg
h = 6.63 x 10^-34 J·s

Now we can calculate the velocity v:

v = (6.63 x 10^-34 J·s) / (1.67 x 10^-27 kg * 0.281 x 10^-9 m)
v ≈ 1.98 x 10^6 m/s

Therefore, a neutron must be moving at approximately 1.98 x 10^6 m/s in order to have a de Broglie wavelength of 0.281 nm.

To determine the speed of a neutron with a given de Broglie wavelength, we can use the de Broglie wavelength equation:

λ = h / p

where λ represents the wavelength, h is the Planck's constant (6.626 x 10^-34 J·s), and p is the momentum of the particle.

In this case, we are given the de Broglie wavelength (0.281 nm) and the mass of the neutron (1.67 x 10^-27 kg). We need to find the speed of the neutron, which is related to its momentum.

The momentum of a particle is given by:

p = mv

where m represents the mass of the neutron and v is its velocity.

Substituting the equation for momentum into the de Broglie wavelength equation, we get:

λ = h / (mv)

Solving for v, we rearrange the equation:

v = h / (mλ)

Now we can plug in the given values:

v = (6.626 x 10^-34 J·s) / ((1.67 x 10^-27 kg)(0.281 x 10^-9 m))

Calculating this expression will give us the speed of the neutron required to have a de Broglie wavelength of 0.281 nm.