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

Discrete energy levels in a hydrogen atom refer to the specific, quantized energy states that an electron can occupy around the nucleus. When a hydrogen atom is excited, the electrons can transition between these energy levels, resulting in the emission of electromagnetic radiation. The energy difference between the initial and final energy levels during a transition determines the energy of the emitted radiation, according to the Planck-Einstein relation \(E = h\nu\). Thus, the discrete energy levels of a hydrogen atom lead to specific frequencies of emitted radiation, which make up the hydrogen atomic spectrum, featuring well-known line series such as Balmer (visible light) and Lyman (UV light) series.

Step-by-step solution

Discrete energy levels in the hydrogen atom

Discrete energy levels in a hydrogen atom refer to the specific energy states that the electron can occupy around the nucleus. These levels are quantized, meaning that energy states are separated by specific, fixed differences. The electron in the hydrogen atom cannot occupy any other energy levels besides these quantized states. This discreteness results from the quantized angular momentum of the electron, which is a consequence of the wave-like nature of electrons and their satisfying the wave-equation formulated by Schrödinger.

Electron transitions between energy levels

When a hydrogen atom is excited, such as by absorbing energy from an external source, the electron can move from a lower energy level to a higher energy level, known as an upward transition. Conversely, when the electron returns to a lower energy level, it releases energy in the form of electromagnetic radiation – this is called a downward transition.

Radiation emitted during electron transitions

The energy difference between the initial and final energy levels during a transition relates to the energy of the emitted radiation. According to the Planck-Einstein relation, the energy of photons (the quantum of electromagnetic radiation) is given by \(E = h\nu\), where \(E\) is the energy, \(h\) is the Planck constant, and \(\nu\) is the frequency of the radiation.

The link between discrete energy levels and emitted radiation

As the energy levels in a hydrogen atom are discrete, the energy differences between these levels are also discrete. Due to the Planck-Einstein relation, the emitted radiation frequency is directly proportional to the energy difference. Therefore, when excited hydrogen atoms emit radiation, the energy differences between energy levels result in specific frequencies of emitted radiation. These specific frequency values make up the hydrogen atomic spectrum, which consists of several line series, including the well-known Balmer series (visible light) and Lyman series (UV light). Each spectral line corresponds to a specific electron transition between quantized energy levels.

Key concepts

Quantized Energy States

When discussing atoms and their electrons, the concept of quantized energy states is fundamental. Electrons orbiting an atom's nucleus exist at certain fixed energy levels, not just any arbitrary level. Think of this like stairs in a staircase: an electron can only stand on the steps (specific energy levels), not between them. This idea was pivotal in the development of quantum mechanics, contrasting the previously held belief that electrons could have any energy. These steps or energy states are where the electrons 'live' when they are not excited by external energy.

The concept of quantized energy states explains many peculiarities in atomic behavior, especially the distinct lines observed in emission and absorption spectra of gases, reaffirming the fact that electrons inhabit specific orbits around the nucleus, and transition between these orbits, rather than moving freely in any pattern or path.

Planck-Einstein Relation

The Planck-Einstein relation represents a cornerstone in the understanding of quantum mechanics and electromagnetic radiation. It's a simple equation that reveals the energy of a photon is proportional to its frequency: \(E = hu\), with \(E\) being the energy, \(h\) standing for Planck's constant, and \(u\) for frequency.

This relation means that the higher the frequency of light, the higher its energy. Thus, when an electron transitions between energy levels, the frequency of the emitted or absorbed photon is a direct indicator of the energy difference between those levels. This fundamental relationship gives rise to the concept of photons as 'energy packets' and is what enables us to calculate the energies involved in transitions within the atom's energy states.

Hydrogen Atomic Spectrum

The hydrogen atomic spectrum is like the atom's unique fingerprint, characterized by specific lines of light, each corresponding to electron transitions between quantized energy levels of the hydrogen atom. Each of these spectral lines represents a photon emitted when an electron falls from a higher energy state to a lower one.

Notably, when studying these spectral lines—like those in the Balmer or Lyman series—we're seeing the direct impact of the discrete energy levels of hydrogen. The colors you see in the spectrum (reds, violets, etc.) in the case of the visible Balmer series, correspond to specific frequencies of light, reinforcing the idea that only certain energy transitions are allowed.

Schrödinger Wave Equation

The Schrödinger wave equation is akin to a map that illustrates how electrons behave at the quantum level. It provides a mathematical formula to describe the wave-like behavior of particles, notably electrons within an atom.

This wave equation is a cornerstone of quantum mechanics because it allows us to predict where an electron is likely to be at any given time. It essentially provides the shapes and orientations of what we know as 'orbitals' - the regions in space where there is a high probability of finding an electron. Moreover, it's the solutions to this equation that tell us the discrete energy levels that electrons can occupy, reflecting how the wave-like nature of electrons restricts them to specific paths dictated by their wave properties.

Electron Transitions

Electron transitions are the movements of an electron between different energy levels within an atom. Think of it as an electron gaining energy to jump up a step in a staircase (absorption), or losing energy to hop down (emission). These changes are not continuous but happen step-wise, reflecting the quantized nature of atomic energy levels.

When these transitions occur, photons are either absorbed—lifting the electron to a higher level—or emitted, as the electron falls back to a lower energy state. The specific colors of light observed in spectra are tied directly to these transitions. Understanding electron transitions helps in deciphering the language of chemistry, from figuring out how atoms interact to predicting chemical reactions.