Question

If gamma radiation has a wavelength of \(1.00 \times 10^{-12} \mathrm{~m}\), what is the energy of gamma radiation in joules?

Short answer

The energy of gamma radiation with a wavelength of \(1.00 \times 10^{-12} \mathrm{~m}\) is \(1.989 \times 10^{-13} \mathrm{~J}\).

Step-by-step solution

Identify the values of h and c

Planck's constant (h) is \(6.63 \times 10^{-34} \mathrm{~Js}\) and the speed of light (c) is \(3.00 \times 10^{8} \mathrm{~m/s}\).

Substitute the values in the Planck's equation

Substitute the values of h, c, and λ in the Planck's equation \(E = \dfrac{hc}{\lambda}\): \(E = \dfrac{(6.63 \times 10^{-34}\mathrm{~Js})(3.00 \times 10^{8}\mathrm{~m/s})}{1.00 \times 10^{-12}\mathrm{~m}}\)

Calculate the energy

Perform the multiplication and division to find the energy: \(E = \dfrac{(1.989 \times 10^{-25}\mathrm{~Jm})}{1.00 \times 10^{-12}\mathrm{~m}}\) \(E = 1.989 \times 10^{-13} \mathrm{~J}\)

Write the final answer

The energy of gamma radiation in joules is \(1.989 \times 10^{-13} \mathrm{~J}\).

Key concepts

Planck's constant

Understanding Planck's constant is critical when delving into the quantum aspects of physics. Represented by the symbol h, it is a fundamental constant with a value of approximately \(6.62607015 \times 10^{-34} \text{Js}\), where J stands for joules and s for seconds.

It plays a vital role in quantum mechanics, linking the energy carried by a photon to its frequency. Max Planck, a German physicist, first introduced the concept, and it serves as the foundation for describing quantized energy levels in systems such as atoms.

Speed of light

The speed of light in a vacuum, denoted as c, is one of the constants of the universe. Its value is \(3.00 \times 10^{8} \text{m/s}\), roughly 299,792,458 meters per second. Albert Einstein's theory of relativity established that this is the maximum speed at which all conventional matter and information can travel.

It is not just important for theories of relativity but also plays a key role in various fields of physics, including electromagnetism and quantum mechanics. When calculating the energy of electromagnetic waves such as gamma radiation, the speed of light is a necessary variable.

Wavelength

Wavelength is a fundamental concept in the study of waves and is defined as the distance between consecutive crests or troughs in a wave. In symbols, it's represented as \(\lambda\) and typically measured in meters. Wavelength is inversely proportional to frequency -- the higher the frequency of the wave, the shorter its wavelength.

This concept is essential for understanding electromagnetic radiation, including gamma rays which have very short wavelengths and, consequently, high frequencies and energies. The wavelength is a crucial parameter in determining a wave's energy using Planck's equation.

Planck's equation

Planck's equation is a cornerstone of quantum mechanics. It links the energy of a photon (E) to its frequency (f) using Planck's constant (h). The full form of the equation is \(E = hf\), where h is Planck's constant, and f is the frequency of the photon.

However, when the frequency is not known directly, but the wavelength (\(\lambda\)) is, the equation can be rewritten using the speed of light (c) to account for the inverse relationship between frequency and wavelength: \(E = \dfrac{hc}{\lambda}\).

This equation is what we use to calculate the energy of electromagnetic radiation, including gamma rays. It succinctly encapsulates the quantum view that electromagnetic waves can also be described as a stream of particles called photons, each carrying a discrete amount of energy.