Question

What evidence do we have that energy levels in an atom are quantized? State and explain the evidence.

Short answer

The evidence for the quantization of energy levels in atoms comes mainly from Bohr's Atomic Model, which proposed that electrons revolve around the nucleus in fixed orbits and that their energy can only change in discrete "chunks" called quanta. This idea is supported by the hydrogen emission spectrum, which displays distinct colored lines corresponding to the specific wavelengths of light emitted by energized hydrogen atoms. Furthermore, the Rydberg formula can accurately predict the wavelengths of these emitted lines, providing additional evidence for the quantization of energy levels in atoms.

Step-by-step solution

Bohr's Atomic Model

The Bohr Atomic Model, proposed by Niels Bohr in 1913, is an early theory that presents energy levels within atoms as quantized. In this model, electrons revolve around the nucleus in fixed orbits, and their energy can only change in discrete "chunks" called quanta. This model formed the basis for the modern understanding of atomic structure and the quantization of energy levels.

Hydrogen Emission Spectrum

The presence of quantized energy levels in atoms can also be observed experimentally through the hydrogen emission spectrum. When energizing hydrogen atoms by applying heat or electricity, these atoms emit light. After passing the emitted light through a prism, distinct colored lines can be observed, known as the hydrogen emission spectrum. This pattern of lines demonstrates that the energy levels in a hydrogen atom are quantized, as only specific wavelengths of light are emitted.

Rydberg Formula

The Rydberg formula further supports the evidence for quantized energy levels in atoms. This formula predicts the wavelengths of light emitted or absorbed by hydrogen atoms using the following equation: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\] Where \( \lambda \) is the wavelength of the emitted or absorbed light, \(R_H\) is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 m^-1), and \(n_1\) and \(n_2\) are two positive integers representing the initial and final energy levels of the electron transition, respectively. The Rydberg formula successfully matches the line spectra observed in the hydrogen emission spectrum, providing further evidence for the quantization of energy levels in atoms.

Key concepts

Bohr's Atomic Model

The Bohr Model is a seminal theory in atomic physics that gives us a fundamental explanation for the quantization of energy levels in atoms. Developed by Niels Bohr in 1913, this model postulates that electrons are found in distinct orbits around an atom's nucleus, much like planets orbiting the sun. In this scenario, each orbit corresponds to a certain energy level.

According to Bohr, electrons can only inhabit these specific orbits and can move between them by absorbing or emitting a discrete amount of energy, known as a quantum. This quantum leap from one energy level to another is what comprises the quantized nature of the atom's energy levels. The remarkable aspect of the Bohr Model is that it explains why electrons do not spiral into the nucleus, as they can only exist in these “allowed” states without losing energy.

Despite its success in explaining many aspects of atomic behavior, the Bohr Model has its limitations and was eventually superseded by quantum mechanics. However, it laid the groundwork for the complex and more accurate description of atomic structure that we use today.

Hydrogen Emission Spectrum

The hydrogen emission spectrum serves as one of the most direct pieces of evidence for the existence of quantized energy levels in atoms. When a hydrogen atom is excited, for instance, by applying heat or an electrical current, it enters a higher energy state. As it returns to a lower energy state, it releases energy in the form of light.

The key observation is that this light, when split by a prism, doesn't form a continuous rainbow of colors. Instead, it creates a series of discrete, bright lines against a dark background, each corresponding to a specific wavelength. This pattern, known as an emission line spectrum, signifies that the light produced by the hydrogen atom includes only certain wavelengths.

These spectral lines are the fingerprints of the energy differences between quantized levels within the atom. Each line represents a photon emitted as an electron transitions from a higher to a lower energy orbit, with the color or wavelength of the light being related to the energy difference between these levels. It is a visual manifestation of energy quantization, and each element has its unique emission spectrum, further supporting the idea that atomic energy levels are quantized.

Rydberg Formula

The Rydberg Formula is an essential mathematical expression that backs up the concept of quantized energy levels in atoms. It was formulated by the Swedish physicist Johannes Rydberg and relates to the emission or absorption spectra of various elements, most notably hydrogen. Here's the formula again for reference:

\[\begin{equation}\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\end{equation}\]
Where \( \lambda \) is the wavelength of the emitted or absorbed light, \(R_H\) is the Rydberg constant specific to hydrogen, and \(n_1\) and \(n_2\) represent the initial and final energy states of the electron, respectively. These states are given as integers reflecting the quantized nature of energy levels within the atom.

The precise prediction of the wavelengths in the spectral lines seen in the hydrogen emission spectrum by the Rydberg Formula provides compelling evidence for the quantized model of the atom. Notably, it predicts wavelengths within a spectrum that corresponds to different possible transitions between energy levels, showcasing the discrete nature of these levels. This strong correlation between the formula's predictions and the observed spectral lines is a clear indication that atomic energy is indeed quantized.