Question

What is the frequency (in \(\mathrm{Hz}\) ) of \(\mathrm{X}\) rays whose energy is \(6.82 \mathrm{keV}\) ?

Short answer

The frequency is approximately \(1.65 \times 10^{18} \, \mathrm{Hz}\).

Step-by-step solution

Convert Energy from keV to Joules

The energy given is in kiloelectronvolts (keV), but we need it in Joules to find the frequency. The conversion factor is \(1 \, \mathrm{eV} = 1.602 \times 10^{-19} \, \mathrm{J}\). Therefore, \(1 \, \mathrm{keV} = 1000 \, \mathrm{eV}\). The energy in Joules is calculated as follows:\[E = 6.82 \, \mathrm{keV} \times 1000 \, \mathrm{eV/keV} \times 1.602 \times 10^{-19} \, \mathrm{J/eV} = 1.092 \times 10^{-15} \, \mathrm{J}\]

Use the Energy-Frequency Relation

We use the Planck's equation \(E = h u\) to find the frequency, where \(E\) is the energy, \(h\) is Planck's constant \(6.626 \times 10^{-34} \, \mathrm{Js}\), and \(u\) is the frequency. Solving for \(u\):\[u = \frac{E}{h} = \frac{1.092 \times 10^{-15} \, \mathrm{J}}{6.626 \times 10^{-34} \, \mathrm{Js}} = 1.65 \times 10^{18} \, \mathrm{Hz}\]

Key concepts

Energy Conversion

When dealing with X-ray energy measurements, it's common for energy to be expressed in kiloelectronvolts (keV). However, many scientific calculations, such as those involving frequency, require energy in Joules. This is where energy conversion becomes essential.
To convert energy from kiloelectronvolts to Joules, follow these steps:

• Recognize that 1 electronvolt (eV) is equivalent to \(1.602 \times 10^{-19}\) Joules.
• Since 1 keV equals 1000 eV, just multiply the energy in keV by 1000 to convert to eV.
• Finally, multiply by \(1.602 \times 10^{-19}\) to convert from eV to Joules.

Using these steps, the energy of 6.82 keV, when converted to Joules, results in \(1.092 \times 10^{-15}\) Joules. This conversion is crucial for using this energy in other equations, including those involving frequency.

Planck's Equation

Planck's equation is a fundamental principle in quantum mechanics used to relate the energy of a photon to its frequency. This equation is expressed as \(E = hu\), where:

• \(E\) stands for the energy of the photon in Joules.
• \(h\) is Planck's constant, a universal value of \(6.626 \times 10^{-34}\) Joule-seconds (Js).
• \(u\) is the frequency of the photon in hertz (Hz).

To find the frequency when the energy is given, rearrange the equation to solve for \(u\): \(u = \frac{E}{h}\).
In this way, Planck's equation provides a direct link between the energy and frequency of a photon, enabling us to uncover one quantity, provided we know the other. This reveals the wave-particle duality of light, demonstrating that energy can be observed as both a wave (frequency) and a particle (energy).

Kiloelectronvolt to Joules

Understanding the conversion from kiloelectronvolts to Joules is key to solving problems related to frequency and energy calculations. The process is straightforward but often misunderstood.
Firstly, recall that a single electronvolt (eV) is the energy gained by an electron when it's accelerated through an electric potential of one volt. This energy measurement is small, so to simplify calculations, it's usually scaled up to kiloelectronvolts (1 keV = 1000 eV).
To convert keV to Joules, you need to multiply by the eV to Joules conversion factor \(1.602 \times 10^{-19}\) since this factor is the established conversion from electronvolts to Joules. Step-by-step, \(6.82\) keV becomes \(1.092 \times 10^{-15}\) Joules.
This step is crucial when using energy values in a variety of scientific contexts, ensuring that the units align with conventional physical constants.

Energy-Frequency Relation

The energy-frequency relation is a concept bridging energy and electromagnetic wave phenomena. According to this principle, a direct relationship exists between the energy of a photon and its frequency.Utilizing Planck's equation, \(E = hu\), we see that as energy increases, the frequency of the associated wave also increases proportionally.
In the context of X-rays, this relation allows us to determine how energetic the waves are. Since X-rays are highly energetic, they correspond to high frequencies.
By rearranging the equation to \(u = \frac{E}{h}\), you can calculate the frequency if you know the energy. Using the example of \(6.82\) keV (1.092 x 10^{-15} J), the equation yields a frequency of \(1.65 \times 10^{18}\) Hz. Thus, the energy-frequency relation is a powerful tool for estimating the properties of electromagnetic waves.