The amount of energy required to dislodge the extra electron from a negative hydrogen ion is \(1.2 \times 10^{-19} \mathrm{~J}\). (a) The extra electron can be dislodged if the ion absorbs a photon of sufficiently short wavelength. (Recall from Section \(5-5\) that the higher the energy of a photon, the shorter its wavelength.) Find the longest wavelength (in \(\mathrm{nm}\) ) that can accomplish this. (b) In what part of the electromagnetic spectrum does this wavelength lie? (c) Would a photon of visible light be able to dislodge the extra electron? Explain. (d) Explain why the photosphere, which contains negative hydrogen ions, is quite opaque to visible light but is less opaque to light with wavelengths longer than the value you calculated in (a).

The electromagnetic spectrum is the range of all types of electromagnetic radiation, which includes a variety of waves with differing energies, frequencies, and wavelengths. From gamma rays that have the highest energies and shortest wavelengths, to radio waves which exhibit the lowest energies with the longest wavelengths, the spectrum is a vast arena of electromagnetic interaction. Each segment of the spectrum has its unique characteristics and applications.

For instance, X-rays are used in medical imaging, while ultraviolet light can cause substances to fluoresce, a principle used in forensics and art restoration. Visible light is the portion of the spectrum that can be detected by the human eye, and infrared radiation, which lies just beyond visible light, is commonly used for remote controls and thermal imaging. The spectrum is crucial in many fields of science and technology, providing the basis for understanding phenomena such as the absorption and emission of light by atoms and molecules, which leads to the color of objects and various spectroscopic techniques.

A photon is a particle of light that carries energy, and its wavelength is inversely proportional to that energy. According to the equation \(E = h u = \frac{hc}{\lambda}\), where \(E\) is energy, \(h\) is Planck's constant, \(u\) is the frequency, \(c\) is the speed of light, and \(\lambda\) is the wavelength, we understand that longer wavelengths correspond to lower energy photons.

In the context of the hydrogen ion, a photon must have a certain minimum energy — and, consequently, a maximum wavelength — to dislodge an extra electron. This relationship between energy and wavelength is what dictates whether a certain type of electromagnetic radiation can cause a specific atomic or ionic transition, such as the ejection of an electron from a negative hydrogen ion.

Longest Wavelength For Ionization

Based on the provided problem, applying the concept that the longer the wavelength, the less energy a photon has, we can deduce why a photon of visible light does not have enough energy to dislodge the electron in the exercise example.

The photosphere of a star, like our Sun, is the region from which light is emitted and can be observed. Its opacity refers to how transparent it is to various wavelengths of light. Negative hydrogen ions found in the photosphere contribute significantly to its opacity, particularly in the visible light range.

The process of absorption and re-emission of light involves specific energy levels that electrons in atoms can occupy. If a photon has enough energy to move an electron to a higher energy state or remove it completely, as the extra electron of a hydrogen ion, it will be absorbed. However, photons below a certain energy threshold, and thus with wavelengths longer than that threshold will not interact with the electrons in the same way and will pass through more easily.

Infrared Transparency and Visible Absorption

Within the example provided, the photons with wavelengths longer than the calculated 1650 nm (infrared light) don't possess the energy necessary to remove the electron, and therefore, pass through the photosphere more freely, making it less opaque to those wavelengths. On the other hand, the visible light with its higher energy (shorter wavelengths) but not high enough to cause ionization is readily absorbed, thus contributing to the photosphere's opacity in this part of the spectrum.