The relationship between the wavelength of a spectral line and its energy can be described by the fundamental equation of wave-particle duality in quantum mechanics. This relationship is expressed by the following equation:
E = hc/λ
where: E is the energy of the spectral line, h is Planck's constant (approximately 6.626 x 10^-34 joule-seconds), c is the speed of light in a vacuum (approximately 3 x 10^8 meters per second), and λ is the wavelength of the spectral line.
According to this equation, the energy (E) of a spectral line is inversely proportional to its wavelength (λ). In other words, as the wavelength of a spectral line increases, its energy decreases, and vice versa. This relationship is known as the inverse relationship between energy and wavelength.
This equation is derived from the wave-particle duality concept, which states that light and other electromagnetic radiation can exhibit both wave-like and particle-like properties. In the case of spectral lines, they are associated with the emission or absorption of photons, which are quantized packets of energy. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength, as given by the equation E = hf, where f is the frequency. Since speed of light is constant, the frequency and wavelength of light are inversely related, resulting in the equation E = hc/λ.
This relationship between the wavelength and energy of spectral lines has important implications in various scientific fields, including spectroscopy, astrophysics, and quantum mechanics. By analyzing the wavelengths and corresponding energies of spectral lines, scientists can gain valuable insights into the atomic and molecular structure, energy levels, and interactions of matter.