Question

Neutron diffraction is an important technique for determining the structures of molecules. Calculate the velocity of a neutron needed to achieve a wavelength of \(0.955 \AA\). (Refer to the inside cover for the mass of the neutron).

Short answer

To calculate the velocity of a neutron needed to achieve a wavelength of \(0.955\ \textup{Å}\), we first convert the wavelength to meters (\(0.955 \times 10^{-10}\ \textup{m}\)). Using the de Broglie wavelength formula \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34}\ \textup{Js}\)) and \(m_n\) is the mass of a neutron (\(1.675 \times 10^{-27}\ \textup{kg}\)), we get the equation \(v = \frac{h}{m_n\lambda}\). Substituting the values and calculating, we find that the velocity of the neutron is approximately \(4.374 \times 10^5\ \textup{m/s}\).

Step-by-step solution

Convert wavelength to meters

We have been given the wavelength as 0.955 Å. To convert this value to meters, we use the following conversion factor: \(1 \textup{Å} = 10^{-10} \textup{m}\) So, the wavelength in meters is: \(0.955\ Å \times 10^{-10} \frac{\textup{m}}{\textup{Å}} = 0.955 \times 10^{-10}\ \textup{m}\)

Use the de Broglie wavelength formula

The de Broglie wavelength formula relates a particle's wavelength to its momentum: \(\lambda = \frac{h}{p}\) where \(\lambda\) is the wavelength, \(h\) is the Planck's constant (\(h = 6.626 \times 10^{-34}\ \textup{Js}\)) and \(p\) is the momentum. Since momentum is defined as the product of mass and velocity (i.e., \(p = mv\)), we can rewrite the de Broglie formula as: \(\lambda = \frac{h}{mv}\)

Rearrange the formula to solve for velocity

We know the values for \(\lambda\) and \(h\), and we have been told that the mass of the neutron can be found inside the cover of the book. The mass of the neutron is: \(m_n = 1.675 \times 10^{-27}\ \textup{kg}\) Rearranging the de Broglie formula to solve for the velocity, we get: \(v = \frac{h}{m_n\lambda}\)

Calculate the velocity of the neutron

Now, we substitute the values to find the velocity: \(v = \frac{6.626 \times 10^{-34}\ \textup{Js}}{(1.675 \times 10^{-27}\ \textup{kg})(0.955 \times 10^{-10}\ \textup{m})} \) \(v = 4.374 \times 10^{5}\ \textup{m/s}\) Therefore, the velocity of the neutron needed to achieve a wavelength of 0.955 Å is approximately \(4.374 \times 10^5\ \textup{m/s}\).