When were the Bohr model and the quantum-mechanical model for the atom developed? What purpose do these models serve?

The Bohr model, introduced by Niels Bohr in 1913, revolutionized our understanding of atomic structure. Before Bohr, scientists struggled to explain how electrons orbit the nucleus without spiraling into it and why atoms emitted light at specific wavelengths. Bohr proposed that electrons exist in distinct orbits or energy levels, circling the nucleus much like planets orbit the sun, but with a key difference: they could only occupy certain allowed orbits.

He suggested that electrons could 'jump' from one energy level to another, emitting or absorbing a photon in the process. This was pivotal in explaining why hydrogen's emission spectra showed discrete lines. The model was a significant leap forward, but it had limitations, only accurately describing hydrogen-like, single-electron systems.

Building on the limitations of the Bohr model, the quantum-mechanical model emerged in the mid-1920s to provide a more comprehensive framework for atomic and subatomic processes. In contrast to the Bohr model's defined orbits, the quantum-mechanical model uses complex mathematical functions called wavefunctions to describe the probability of finding an electron in a certain region around the nucleus.

This model, deeply rooted in quantum theory, considers electrons as both particles and waves, embracing the concept of wave-particle duality. It accounts for the behavior of atoms with more than one electron and explains the chemical properties of elements by their electron configurations. This sophisticated approach offers a window into the intricate dance of subatomic particles and is the most accurate model we have to date.

The Schrödinger equation, formulated by Erwin Schrödinger in 1926, is a fundamental theory in quantum physics and the cornerstone of the quantum-mechanical model. It mathematically describes how the quantum state of a physical system changes over time. In its simplest form, the equation reflects the wave nature of matter and is central to predicting the behavior of particles at the atomic and subatomic scale.

Written as \(\hat{H}\psi = E\psi\), it relates the energy of the system (denoted by ''E'') to the wavefunction (denoted by \(\psi\)) through the Hamiltonian operator (denoted by \(\hat{H}\)). The solutions to this equation give us the wavefunctions and corresponding energy levels of electrons in an atom, a profound insight into the probability distributions that govern the natural world.

Emission spectra are like fingerprints for elements, each unique, allowing scientists to identify the atomic constitution of a material based solely on the light it emits. When an atom's electrons are excited to higher energy levels, they eventually 'relax' back to lower levels, releasing energy in the form of light. This light is composed of photons with specific wavelengths, producing a spectrum of distinct lines, known as the emission spectrum of the atom.

These spectra provide critical information in various scientific fields, from determining the composition of distant stars to studying the quantum mechanical properties of materials. The spectrum's discrete lines validate both the Bohr model's postulation of quantized energy levels and the quantum-mechanical model's predictions of electron transitions.