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.