Question

a) Calculate the kinetic energy of a neutron that has a de Broglie wavelength of \(0.15 \mathrm{nm}\). Compare this with the energy of an X-ray photon that has the same wavelength. b) Comment on how this energy difference is relevant to using neutrons or X-rays for investigating biological samples.

Short answer

The kinetic energy of a neutron with a de Broglie wavelength of 0.15 nm is approximately 2.88 x 10^-15 J, while the energy of an equivalent X-ray photon is 1.33 x 10^-15 J. Since the energy of the neutron is more than double that of the X-ray photon, it may not be as suitable for investigating fragile biological samples as X-rays, which have lower energy and lower potential to cause damage. However, neutrons can provide complementary information due to their sensitivity to certain elements, like hydrogen. Therefore, using both neutron and X-ray methods can give a more comprehensive view of biological samples.

Step-by-step solution

Determine the neutron's momentum using the de Broglie wavelength

To find the neutron's momentum, we first need to rearrange the de Broglie wavelength formula: \(p = \dfrac{h}{\lambda}\) Given the de Broglie wavelength \(\lambda = 0.15 \,nm\), we can calculate the momentum: \(p = \dfrac{6.626 \times 10^{-34} \, J\cdot s}{0.15 \times 10^{-9} \, m}\) \(p \approx 4.4 \times 10^{-24} \, kg \cdot m/s\) ##Step 2: Calculate the kinetic energy of the neutron##

Determine the kinetic energy using the momentum and mass of the neutron

Now, we can calculate the kinetic energy of the neutron using the formula: \(KE = \dfrac{p^2}{2m}\) The mass of a neutron is approximately \(1.675 \times 10^{-27} \, kg\). \(KE = \dfrac{(4.4 \times 10^{-24} \, kg \cdot m/s)^2}{2(1.675 \times 10^{-27} \, kg)}\) \(KE \approx 2.88 \times 10^{-15} \, J\) ##Step 3: Calculate the energy of the X-ray photon##

Determine the energy of the X-ray photon using its wavelength

We can find the energy of the X-ray photon using the formula: \(E = h\cdot c /\lambda\) Where \(c\) is the speed of light (\(\approx 3 \times 10^8 \, m/s\)). \(E = \dfrac{6.626 \times 10^{-34} \, Js \cdot 3 \times 10^8 \, m/s}{0.15 \times 10^{-9} \, m}\) \(E \approx 1.33 \times 10^{-15} \, J\) ##Step 4: Compare the energies and discuss their relevance##

Compare the calculated energies and discuss the significance

The kinetic energy of the neutron is approximately \(2.88 \times 10^{-15} \, J\), while the energy of the X-ray photon is \(1.33 \times 10^{-15} \, J\). The neutron's kinetic energy is over twice the energy of the X-ray photon. When investigating biological samples, it's essential to use energy forms that can penetrate the sample without causing damage. If the energy is too high, it can break chemical bonds or ionize atoms, causing irreversible changes to the sample. Since the kinetic energy of the neutron is higher than that of the X-ray photon, using X-rays may generally be a safer option for investigating delicate biological samples, as they have lower energy and a lower likelihood of causing such damage. However, neutrons can provide valuable complementary information, as they are more sensitive to certain elements, like hydrogen, which is essential in biological materials. So, a combination of both methods could provide the most comprehensive view.