The sodium flame test has a characteristic yellow colour due to emissions of wavelength \(589 \mathrm{~nm}\). What is the mass equivalence of one photon of this wavelength?

Understanding the energy-wavelength relation is pivotal when studying the properties of photons, which are particles of light. In essence, this relation describes how the energy of a photon is inversely proportional to its wavelength. This principle can be expressed mathematically by the equation:

\[\begin{equation}E = \frac{hc}{\lambda}\end{equation}\]

where:

• \(E\) signifies the energy of a single photon,
• \(h\) stands for Planck's constant (\(6.626 \times 10^{-34} \) Joule seconds),
• \(c\) represents the speed of light in a vacuum (\(3.00 \times 10^8 \) meters per second), and
• \(\lambda\) is the wavelength of the photon.

The shorter the wavelength, the higher the energy of the photon, and vice versa. This means photons with higher energy have a greater capacity to affect matter, which is why different wavelengths of light cause varied chemical and physical responses. In our specific example, the sodium flame test emits yellow light with a wavelength of 589 nm, which can be plugged into our equation to determine the energy of a photon. This is the first step in understanding the mass equivalence of a photon, as energy and mass are directly interconnected through Einstein's famous equation, which also implies that light, although typically massless, has an equivalent mass when considering its energy.

Planck's constant (\(h\)) is a fundamental quantity in quantum mechanics. It bridges the gap between the macroscopic classical physics and the microscopic quantum world. Planck's constant has a value of approximately \(6.626 \times 10^{-34} \) Joule seconds and plays a critical role in the quantization of energy.

It is the proportionality constant that relates the energy (\(E\)) of a photon to the frequency (\(v\)) of its associated electromagnetic wave:

• \[\begin{equation}E = hv\end{equation}\]

This quantum of action quantifies the smallest possible discrete unit of any physical property that is involved in an interaction. One of the landmark implications of Planck's constant is in the energy-wavelength relation of photons, where it helps ascertain the energy of a photon from its wavelength — a central element when examining phenomena such as the photoelectric effect, blackbody radiation, and in our sodium flame test example, the emission of colored light.

Einstein's mass-energy equivalence is one of the most iconic concepts in physics, embodied by the equation:\[\begin{equation}E = mc^2\end{equation}\]
Here, \(E\) is the energy, \(m\) is the mass, and \(c\) is the speed of light in a vacuum. This revolutionary idea posits that mass and energy are two sides of the same coin, and they can be converted into each other. Essentially, a tiny amount of mass can be converted into a substantial amount of energy, and vice versa.

This equation is particularly relevant when discussing photons which are generally considered to be massless particles. However, they do possess energy and therefore have a 'mass equivalence'—a notional mass that one can calculate by rearranging Einstein's equation to:

• \[\begin{equation}m = \frac{E}{c^2}\end{equation}\]

When you calculate the energy of a photon using the energy-wavelength relation and Planck's constant, you can then derive its mass equivalence through Einstein's formula. This doesn't mean that photons have actual rest mass, but rather an equivalent mass that could theoretically be assigned due to their energy when interacting with other particles. In educational contexts, exercises like the sodium flame test help illustrate this deep and abstract concept with a concrete example, enhancing the student’s understanding of these fundamental principles.