Question

Enumerate the postulates of Bohr's model. How does Bohr's model explain the spectrum of hydrogen?

Short answer

The postulates of Bohr's model are: 1) An electron in an atom revolves in certain stable orbits without radiating energy. 2) The energy of an electron in an atom is quantized, meaning it has specific values. 3) An electron can transition between energy states, absorbing or emitting energy in the form of light with a frequency related to the difference in energy. Bohr's model explains the hydrogen spectrum as follows: Each line in the hydrogen spectrum corresponds to an electron transitioning between energy levels, emitting or absorbing light with energy corresponding to the difference between these energy levels, which gives a particular color or wavelength in the hydrogen spectrum.

Step-by-step solution

Postulates of Bohr's Model

Bohr's model of the atom is based on three essential postulates. 1. An electron in an atom revolves in certain stable orbits without the emission of radiant energy, unlike what classical electromagnetic theory suggests. These stable orbits are called stationary states. 2. An electron in a stable orbit has a specific, quantized energy. This means the energy of an electron in an atom is not continuous but has certain fixed, specific values. 3. The electron in the atom can change its state from one stationary state to another, absorbing or emitting energy in the process. This change in energy \( \Delta E \) of the electron is emitted or absorbed in the form of light and is directly proportional to the frequency of the light ( \( \Delta E = h \nu \) ), where \( h \) is Planck’s constant and \( \nu \) is the frequency of light. Therefore, only certain frequencies of light are involved in the process, which results in line spectra, series of discrete lines of colour, corresponding to different energies of light.

Explanation of Spectrum of Hydrogen

According to Bohr's model, an electron can exist only in certain specific energy levels. When it transitions between these levels, it emits or absorbs energy as light. The energy of emitted or absorbed light, and thus its frequency, is determined by the difference in energy levels. For hydrogen, these energy levels are represented by \(E_n = -13.6/n^2 eV\), where \(n\) is the principal quantum number. Each line in the spectrum of hydrogen corresponds to an electron transitioning between two energy levels. For example, when an electron transitions from the \(n=3\) state to the \(n=2\) state, it emits light with frequency given by \(\Delta E = h \nu\), where \(\Delta E\) is the energy difference between those states (-3.4 eV - (-1.51 eV) = -1.89 eV for this transition), \(h\) is Planck's constant, and \(\nu\) is the frequency of the emitted light. This frequency corresponds to a specific color or wavelength of light in the visible spectrum, explaining why the spectrum of hydrogen appears as it does.

Key concepts

Postulates of Bohr's Model

Niels Bohr introduced a remarkable model of the atom early in the 20th century, laying the foundation for how we understand atomic structure and quantum mechanics. One of the key elements he proposed are the postulates that govern an electron's behavior in an atom.

Firstly, Bohr suggested that electrons travel in distinct orbits or shells around the nucleus without losing energy. Contrary to classical theories of the time, these orbits are stable and do not result in the radiation of energy.

Secondly, he introduced the groundbreaking idea of quantization within an atom: the energy levels of electrons are not continuous, but rather exist in fixed, discrete values. This quantization is akin to a ladder where one can only stand on the rungs, not in between.

• An electron on one of these 'rungs' possesses a specific amount of energy, and this position defines its energy state.

Lastly, Bohr proposed that electrons could leap from one energy level to another. These jumps are not random but occur when the electron absorbs or emits a precise amount of energy, which manifests as photons of light. The relationship between the energy change and the light emitted is given by the equation \( \Delta E = h u \), linking the two through Planck’s constant \( h \) and the frequency \( u \). In essence, these energy leaps lead to the distinctive lines seen in emission and absorption spectra.

Quantized Energy Levels

The concept of quantized energy levels is a core tenet of Bohr's atomic theory, and it marks a departure from the classical physics perspectives of the day.

In Bohr's view, the atom is structured like a tiny solar system, with the nucleus at the center and electrons orbiting in shells at specific distances. These distances are not arbitrary; they correspond to certain energy levels that are 'quantized', meaning electrons can only inhabit these particular energy states and no others.

This idea can be visualized using the analogy of a staircase:

• Just like you can only stand on the steps of a staircase and not float in between, electrons can only exist in certain energy levels, not in between them.

The energy associated with each 'step' in the atomic staircase is precisely defined, and when an electron 'jumps' from one step to another, it must absorb or release an exact amount of energy equal to the difference between those steps. These jumps result in the absorption or emission of light with distinct wavelengths, contributing to the unique spectra of each element.

Hydrogen Spectrum Explanation

The hydrogen spectrum is perhaps the most famous application of Bohr's model. It's a testament to the quantized nature of energy in an atom and helps us comprehend why we observe certain lines of color when looking at light emitted by hydrogen.

According to Bohr, when an electron in the hydrogen atom transitions between these fixed energy levels, it either emits or absorbs a photon of light with an energy that corresponds exactly to the difference between these levels.

• For instance, if an electron falls from a higher to a lower orbit, it releases energy in the form of light, producing a line in the emission spectrum of hydrogen.

The energies of the hydrogen electron orbits are given by the formula \( E_n = -13.6/n^2 eV \) where \( n \) represents the principal quantum number or the level number.

Different transitions yield different energy photons, and this is what leads to the characteristic lines seen in the spectrum. The Balmer series, which is visible to the human eye, results from transitions to the \( n=2 \) level, whereas other series like Lyman and Paschen account for ultraviolet and infrared emissions, respectively.

Electron Transitions

Electron transitions are movements of an electron between energy levels within an atom, and they're central to the generation of atomic spectra. These transitions carry significance in both atomic theory and practical applications such as spectroscopy.

When an electron absorbs sufficient energy, it can jump up to a higher energy level, a process called absorption. Conversely, when an electron falls to a lower energy level, it emits energy in the form of light, which is known as emission.

Key Points in Electron Transitions:

• The amount of energy involved in the transition between levels is fixed and is related to the distance between the levels.
• The emitted or absorbed light has a wavelength that corresponds to the energy change, leading to the observable spectral lines.

Therefore, the light emitted by atoms under excitation isn't a continuous spectrum but consists of a series of discrete wavelengths, each marking a specific electron transition. It is this discreteness of electron energy levels and transitions that gives each element its own unique fingerprint in the form of an atomic spectrum.