Question

Briefly describe Bohr's theory of the hydrogen atom and how it explains the appearance of an emission spectrum. How does Bohr's theory differ from concepts of classical physics?

Short answer

Bohr's theory describes quantized energy levels in the hydrogen atom, explaining its emission spectrum with discrete spectral lines, differing from classical physics by introducing stable, non-radiating orbits.

Step-by-step solution

Introduction to Bohr's Theory

Niels Bohr proposed a model for the hydrogen atom in 1913, where the electron orbits the nucleus in distinct circular paths or energy levels without radiating energy. This model introduced the idea of quantized energy levels.

Quantization of Energy Levels

According to Bohr's theory, the electron can only occupy certain allowed energy levels. These levels are defined by an integer known as the principal quantum number, denoted by \( n \). The energy of each level is given by the formula \( E_n = - \frac{k}{n^2} \), where \( k \) is a constant related to the electron and proton interaction.

Emission Spectrum Explanation

Bohr's theory explains the emission spectrum by stating that when an electron falls from a higher energy level \( n_i \) to a lower energy level \( n_f \), it emits energy in the form of light, which corresponds to the difference in energy between these levels: \( E = E_{n_i} - E_{n_f} \). This released energy forms the spectral lines seen in the hydrogen emission spectrum.

Deviations from Classical Physics

Unlike classical physics, which predicts that an electron in orbit should continuously emit radiation and spiral into the nucleus, Bohr introduced the idea of stable orbits with quantized angular momentum \( L = n \hbar \). Classical physics couldn't explain stationary states or the hydrogen's discrete emission spectrum, whereas Bohr's theory successfully did.

Key concepts

hydrogen atom

The hydrogen atom, being the simplest form of an atom, serves as a fundamental building block in quantum physics. In 1913, Niels Bohr introduced a revolutionary model to understand its behavior. He proposed that the hydrogen atom consists of a positively charged nucleus surrounded by an electron which orbits this nucleus in well-defined paths. These paths are called "orbits" or "energy levels." What set Bohr's model apart was the idea that these energy levels were discrete, not continuous.
This concept of quantized orbits was mainly due to Bohr's desire to integrate Planck's quantum theory with the rudimentary understanding of atomic structure at the time. This integration allowed Bohr to explain the stability of the electron, as it doesn’t radiate energy while in these fixed orbits, countering earlier models that couldn't account for the atom's stability.

emission spectrum

An emission spectrum is a series of lines representing light emitted by an element when its electrons transition between different energy levels. For hydrogen, Bohr's model brilliantly elucidated its emission spectrum. Electrons in a hydrogen atom can "jump" from higher to lower energy levels. During this transition, they release energy in the form of light. This light is not a continuous range but rather distinct lines, each corresponding to a specific wavelength.
Bohr's theory ties the emitted light's frequency to the difference in energy between the higher and lower levels. This linkage means that the spectral lines visible in hydrogen's emission spectrum are direct representations of electron transitions, making each line unique to hydrogen.

quantized energy levels

The concept of quantized energy levels is foundational in Bohr's theory. In the Bohr model, electrons exist only in specific orbits or levels, each quantized or fixed at particular energies. These energy levels are depicted by an integer known as the principal quantum number, represented as \( n \).
Quantization is crucial because it signifies that electrons cannot reside in between these levels; they can only "hop" from one level to another. The energy associated with each level depends inversely on the square of \( n \), represented by the equation \( E_n = - \frac{k}{n^2} \). Thus, higher values of \( n \) correspond to less negative energies—closely related to how electrons absorb or emit energy when shifting energy levels.

classical physics

Bohr's theory dramatically deviated from classical physics, especially concerning atomic structure. Classical physics would argue an orbiting electron should constantly emit radiation due to centripetal acceleration, losing energy, and quickly spiraling into the nucleus. However, this did not align with reality, as atoms are stable.
In contrast, Bohr proposed that electrons occupy stable orbits with quantized angular momenta, represented by \( L = n \hbar \), where \( \hbar \) is the reduced Planck's constant. This notion introduces stable energy levels where electrons do not radiate energy. Bohr's theory allowed for the explanation of hydrogen’s distinct emission spectrum, which classical ideas failed to justify, marking a cornerstone in the shift towards modern quantum physics.