Question

What does it mean to say that the hydrogen atom has discrete energy levels? How is this fact reflected in the radiation that excited hydrogen atoms emit?

Short answer

The hydrogen atom has discrete energy levels, meaning its electron can only occupy specific, quantized energy levels around the nucleus. These energy levels can be expressed as \(E_n = -\frac{13.6 eV}{n^2}\). The discrete energy levels are reflected in the radiation emitted by excited hydrogen atoms when the electron transitions between energy levels, releasing energy in the form of a photon. The energy difference between the initial and final energy levels corresponds to the energy of the emitted photon, resulting in distinct wavelengths of light that form the atomic emission spectrum. This spectrum appears as discrete, separate lines, also known as spectral lines, when viewed using a spectrometer.

Step-by-step solution

Understanding discrete energy levels of hydrogen atom

The hydrogen atom consists of a single proton and a single electron. According to the quantum mechanical model, the electron can only occupy specific energy levels around the nucleus. These energy levels are known as discrete or quantized energy levels because there is no continuous transition between them. The energy of an electron in a hydrogen atom can be expressed as: \(E_n = -\frac{13.6 eV}{n^2}\) where n is the principal quantum number (an integer), and E_n is the energy of the electron in the nth energy level.

Electronic transitions and radiation emission

When a hydrogen atom absorbs energy, an electron can be excited to a higher energy level by absorbing a specific amount of energy called a quantum. This excited state is unstable, and the electron tends to return to a lower, more stable energy level. When the electron transitions back to a lower energy level, it releases the excess energy in the form of electromagnetic radiation (a photon). The energy difference between the initial and final energy levels corresponds to the energy of the emitted photon: \(E_{photon} = E_{final} - E_{initial}\)

Discrete energy levels and spectral lines

The discrete energy levels of a hydrogen atom are reflected in the radiation that excited hydrogen atoms emit. The transition of an electron between energy levels results in the emission of photons with specific energies, which in turn correspond to specific wavelengths of light. The collection of these specific wavelengths forms the atomic emission spectrum, which appears as discrete, separate lines when viewed using a spectrometer. This is why the spectrum is also called a line spectrum or spectral lines. For hydrogen, these spectral lines are grouped into several series, such as the Lyman, Balmer, and Paschen series, corresponding to the ultraviolet, visible, and infrared regions of the electromagnetic spectrum, respectively. In summary, the fact that the hydrogen atom has discrete energy levels is reflected in the radiation emitted by excited hydrogen atoms in the form of distinct, separate spectral lines in the atomic emission spectrum. This line spectrum is a direct consequence of the quantization of energy levels and the allowed electron transitions between these quantized levels.