Question

Determine the wavelength of an X-ray photon that can impart, at most, \(80 \mathrm{keV}\) of kinetic energy to a free electron.

Short answer

The wavelength of the X-ray photon is approximately \(1.55 * 10^{-11} m\) or \(15.5 pm\) (picometers).

Step-by-step solution

Write Down the Given Data

The maximum kinetic energy given to electron by the X-ray photon is 80 KeV. Converting this to Joules, \( 1eV = 1.6*10^{-19} J\), gives \(E_k = 80*10^3 * 1.6 * 10^{-19}\) Joules.

Apply the Photon Energy Formula

The energy of an X-Ray photon is given by \(E = hf\), where \(E\) is the energy, \(h\) is the Planck constant (\(6.63*10^{-34} Js\)) and \(f\) is the frequency.

Apply the Wave Equation

Frequency can be written as \(f = c/λ\), where \(c\) is the speed of light and \(λ\) is the wavelength. Substituting that into the energy equation, we get \(E = hc/λ\). Now, we can solve for the wavelength \(λ\), knowing the energy \(E\).

Solve for the Wavelength

The wavelength \(λ\) is given by \(λ = hc/E\). Substituting the given values, \(λ = (6.63*10^{-34} Js * 3*10^8 m/s) / (80 *10^3 eV * 1.6 * 10^{-19} J/eV)\)

Key concepts

Photon Energy Formula

To understand how an X-ray photon functions, it's crucial to grasp the photon energy formula. This formula is defined as \(E = hf\). Here, \(E\) stands for energy, \(h\) is Planck’s constant, which is approximately \(6.63 \times 10^{-34}\) Joule-seconds, and \(f\) is the frequency of the photon. The photon energy determines how much energy is transferred when photons interact with other particles.

This formula is fundamental in quantum physics and helps explain how light and electromagnetic waves transfer energy. When applied to X-rays, the energy levels involved are quite high, usually measured in electronvolts (eV). For instance, 1 electronvolt is equivalent to \(1.6 \times 10^{-19}\) Joules.

Understanding this concept will help in calculating various physical phenomena, especially those involving high-energy emissions like X-rays.

Wavelength Calculation

To find the wavelength of an X-ray photon, you utilize the relationship between energy, frequency, and wavelength. The wave equation \(f = \frac{c}{\lambda}\) plays a key role here, where \(c\) denotes the speed of light (approximately \(3 \times 10^8\) meters per second).

Once you have the frequency, you can determine the wavelength \(\lambda\) using the formula \(\lambda = \frac{hc}{E}\). Here, \(E\) represents the energy of the photon, measured in Joules. In practical calculations, like determining the wavelength for an X-ray photon providing 80 keV to an electron, you would first convert that energy into Joules and then insert it into the formula. This makes use of constants and given values to solve for the wavelength. Such calculations are essential in fields like medical imaging and crystallography.

Kinetic Energy Conversion

Kinetic energy conversion is a technique that explains how photonic energy can be converted and transferred to particles such as electrons. In the context of X-rays, the photon's energy can transfer substantial kinetic energy to an electron. This concept is essential in understanding processes like Compton scattering, where photons lose energy by transferring it to electrons.

To compute this, start with the kinetic energy formula for the electron, where \( K = \frac{1}{2}mv^2 \), indicating the kinetic energy \(K\), mass \(m\), and velocity \(v\) of the electron. However, when considering X-rays, this kinetic energy is primarily provided by the photons.

It's crucial to convert the given values from electronvolts to Joules to ensure accurate calculations. By understanding how this energy transition occurs, scientists can better grasp high-energy interactions, a principle that supports a myriad of applications in medical and scientific fields.