In the spectrum of hydrogen, there is a line with a wavelength of \(410.3 \mathrm{nm}\). (a) What color is this line? (b) What is its frequency? (c) What is the energy of each of its photons?

The visible light spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. It comprises a range of wavelengths from approximately 380 to 750 nanometers (nm). Each wavelength within this range corresponds to a specific color, starting with violet at the shortest wavelengths and ending with red at the longest wavelengths.

When we look at a rainbow or use a prism, we are actually viewing the visible light spectrum spread out into its component colors. This natural phenomenon happens because different wavelengths of light bend by varying amounts when passing through a medium, separating them into the colors we see. The importance of the visible light spectrum lies in its applications in various fields like optics, astronomy, and even art.

Wavelength to color conversion is the process of correlating the wavelength of light within the visible spectrum to the perceived color. This conversion is based on the way human eyes and brains interpret different wavelengths of light. For example, light that has a wavelength of 410.3 nm is perceived as violet to blue.

However, the relationship between wavelength and color isn't always straightforward due to factors like color mixing and perception variances among individuals. For simplification, specific ranges of wavelengths can be approximately assigned to colors: violet (380-450 nm), blue (450-495 nm), green (495-570 nm), yellow (570-590 nm), orange (590-620 nm), and red (620-750 nm). The light with a wavelength of 410.3 nm, as in our example, would thus be identified as lying in the violet to blue range.

The speed of light calculation is a fundamental concept in physics, crucial for understanding the behavior of light and electromagnetic waves. The speed of light in a vacuum is a constant, denoted by the symbol 'c', and is approximately equal to 299,792,458 meters per second or about 3 x 10^8 m/s in scientific notation.

To calculate the speed of light through various mediums, scientists must consider the refractive index of the medium. However, for most practical and theoretical purposes, such as the analysis of the hydrogen spectrum, we assume that light travels in a vacuum. Employing the speed of light allows us to connect wavelength and frequency, giving us a deeper understanding of the nature of light waves.

Planck's equation, named after the physicist Max Planck, is a pivotal concept in quantum mechanics. It describes the quantized nature of energy carried by photons, the most fundamental particles of light. The equation is expressed as \( E = h \times f \), where \( E \) is the energy of the photon, \( f \) is the frequency of the light, and \( h \) is the Planck constant, a physical constant that has a value of approximately 6.626 x 10^-34 joule seconds (J·s).

The introduction of Planck's constant established the beginning of quantum theory, indicating that energy is not continuous but rather comes in discrete 'packets' or quanta. According to Planck's equation, as the frequency of light increases, so does the energy of its photons, which has far-reaching implications in fields such as spectroscopy, photochemistry, and electronics.