Question

a) Calculate the kinetic energy of a neutron that has a de Broglie's 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 would be relevant for investigating biological samples with neutrons vs. X-rays.

Short answer

Answer: The energy difference between the neutron and the X-ray photon in this example is significant, with the X-ray photon having a much higher energy (1.325 x 10^-15 J) compared to the neutron's kinetic energy (5.775 x 10^-21 J). This difference in energy has implications for studying biological samples. Neutrons can penetrate deeper without causing damage due to their neutral charge, while X-rays may scatter and be absorbed, potentially damaging the sample. Neutrons can provide information about atomic structure, chemical composition, and hydrogen content, while X-rays are better suited for studying electron density. The choice between neutrons and X-rays depends on the goals and requirements of the investigation, as each method has its own advantages and limitations.

Step-by-step solution

Find the kinetic energy of a neutron using de Broglie's equation.

First, let's find the momentum of the neutron using de Broglie's equation: \(p = \dfrac{h}{\lambda}\), where \(p\) is the momentum, \(h\) is Planck's constant, and \(\lambda\) is the wavelength. Substitute the values for \(h = 6.626 \times 10^{-34} \mathrm{Js}\) and \(\lambda = 0.15 \times 10^{-9} \mathrm{m}\) to find the momentum: \(p = \dfrac{6.626 \times 10^{-34} \mathrm{Js}}{0.15 \times 10^{-9} \mathrm{m}} \approx 4.417 \times 10^{-24} \mathrm{kg m/s}\) Now we can calculate the kinetic energy (KE) of the neutron using the momentum: \(KE = \dfrac{p^{2}}{2m}\), where \(m\) is the mass of the neutron. The mass of a neutron is approximately \(1.675 \times 10^{-27} \mathrm{kg}\). \(KE_{neutron} = \dfrac{(4.417 \times 10^{-24} \mathrm{kg m/s})^2}{2(1.675 \times 10^{-27} \mathrm{kg})} \approx 5.775 \times 10^{-21} \mathrm{J}\)

Find the energy of an X-ray photon with the same wavelength.

Now let's find the energy of an X-ray photon with the same wavelength as the neutron. The energy of a photon can be found using the equation: \(E = h\nu\), where \(E\) is the energy, \(\nu\) is the frequency of the photon, and \(h\) is Planck's constant. Let's find the frequency using the speed of light: \(c = \lambda\nu\) => \(\nu = \dfrac{c}{\lambda}\), where \(c\) is the speed of light \((2.998 \times 10^8 \mathrm{m/s})\) and \(\lambda\) is the wavelength of the photon. \(\nu = \dfrac{2.998 \times 10^8 \mathrm{m/s}}{0.15 \times 10^{-9} \mathrm{m}} \approx 1.999 \times 10^{18} \mathrm{Hz}\) Now, we can find the energy of the X-ray photon: \(E_{Xray} = (6.626 \times 10^{-34} \mathrm{Js})(1.999 \times 10^{18} \mathrm{Hz}) \approx 1.325 \times 10^{-15} \mathrm{J}\)

Compare the energies of the neutron and X-ray photon.

Now that we have the kinetic energy of the neutron and the energy of the X-ray photon, we can compare the two values: The kinetic energy of the neutron is \(5.775 \times 10^{-21} \mathrm{J}\), and the energy of the X-ray photon is \(1.325 \times 10^{-15} \mathrm{J}\).

Discuss the relevance for investigating biological samples with neutrons vs. X-rays.

The energy of the X-ray photon is much higher than the kinetic energy of the neutron. This difference in energy has implications for investigating biological samples with neutrons vs. X-rays. Neutrons, being neutral particles, can penetrate deeper into materials without interacting, unlike charged particles such as X-ray photons, which can scatter and be absorbed by the sample. This can potentially cause damage to the sample. Neutrons can reveal information about atomic structure, chemical composition, and hydrogen content in biological samples, which is vital in understanding their functions. In contrast, X-rays are better suited for studying the electron density of a sample, which can provide insights into the bonding and the electrostatic potential of a molecule. In conclusion, the choice between neutrons and X-rays depends on the specific goals and requirements of the investigation, as each method has its advantages and limitations.

Key concepts

Neutron Kinetic Energy Calculation

Understanding the kinetic energy of a neutron is key to grasping the behavior of particles at a quantum level. Neutrons, subatomic particles found in the nucleus of atoms, possess mass but no electric charge. This characteristic allows them to serve as a probe into matter without the complexities of electromagnetic interactions that charged particles experience.

When we want to calculate the kinetic energy of a neutron, we use the de Broglie wavelength, which establishes a relationship between a particle's momentum and its wave-like nature. The equation we use is derived from the de Broglie hypothesis, expressed as \( p = \frac{h}{\lambda} \), where \( p \) is the momentum, \( h \) is Planck's constant, and \( \lambda \) is the wavelength. By rearranging the equation of kinetic energy \( KE = \frac{p^{2}}{2m} \), where \( m \) is the mass, we can find the kinetic energy of a neutron given its wavelength.

For a neutron with a de Broglie wavelength of 0.15 nm, after finding its momentum, the kinetic energy is calculated by squaring the momentum and dividing by twice the mass of the neutron. This provides a direct way to measure the neutron's energy in the context of its motion and is foundational to understanding its interaction with other particles or fields.

X-ray Photon Energy

X-ray photons are a form of electromagnetic radiation, which, unlike neutrons, carry energy in the form of electric and magnetic fields oscillating perpendicularly to each other. The energy of an X-ray photon is directly related to its frequency, which in turn inversely relates to its wavelength. This relationship is given by the equation \( E = hu \) (where \( h \) is Planck's constant and \( u \) is the frequency of the photon).

To calculate the photon's energy with a given wavelength, we use \( u = \frac{c}{\lambda} \) to find its frequency, where \( c \) represents the speed of light in vacuum. Once the frequency is known, we can determine the energy of the X-ray photon. Thus, an X-ray photon and a neutron with the same de Broglie wavelength will have vastly different energies due to the difference in their nature; the X-ray's being derived from its electromagnetic properties, whereas the neutron's is derived from its mass and velocity.

Investigating Biological Samples

Probing biological samples with either neutrons or X-rays is crucial for understanding their structure and function at the molecular level. The choice of particle for investigation depends heavily on the type of information required.

Neutrons, because of their neutrality, are less likely to cause damage by avoiding the electron cloud interactions that charged particles, like X-rays, have with matter. They can therefore offer unique insights, especially in identifying the location of lighter atoms like hydrogen, which is abundant in organic compounds and water. Neutrons can penetrate deep into a sample and give information about the atomic structure and chemical composition, which is essential in biochemistry and materials science.

Conversely, X-rays interact strongly with electrons and can offer detailed maps of electron density. This makes them especially suitable for studying the overall shape and size of molecules, understanding chemical bonding, and the molecular frameworks within cells. However, this interaction with electrons can potentially break chemical bonds or ionize atoms, leading to damage. Thus, when investigating biological samples, scientists weigh these factors to decide the most appropriate method for their specific research objectives.