Bohr's model of the hydrogen atom: quantization of electronic structure

Bohr’s model of the hydrogen atom started from the planetary model, 

but he added one assumption regarding the electrons. 

What if the electronic structure of the atom was quantized? Bohr suggested that the perhaps the electrons could only orbit the nucleus in specific orbits or shells with a fixed radius. Only shells with a radius given by the equation below would be allowed, and the electron could not exist in between these shells. Mathematically, we could write the allowed values of the atomic radius as $\mathrm{ <b>\; r</b>}$(n)=n2⋅r(1)r(n)=n^2\cdot r(1)r(n)=n​2​​⋅r(1)r, left parenthesis, n, right parenthesis, equals, n, start superscript, 2, end superscript, dot, r, left parenthesis, 1, right parenthesis,

where $n$nn is a positive integer, and $r(1)$r(1)r, left parenthesis, 1, right parenthesis is the Bohr radius, the smallest allowed radius for hydrogen.

He found that $r(1)$r(1)r, left parenthesis, 1, right parenthesis has the value

$\text{Bohr radius}=r\left(1\right)=0\mathrm{.}529\times 10^{-10}\text{m}$
$\text{Bohr radius}=r(1)=0.529 \times 10^{-10} \,\text{m}$By keeping the electrons in circular, quantized orbits around the positively-charged nucleus, Bohr was able to calculate the energy of an electron in the $n$nnth energy level of hydrogen: $E(n)=-\dfrac{1}{n^2} \cdot 13.6\,\text{eV}$E(n)=−​n​2​​​​1​​⋅13.6eVE, left parenthesis, n, right parenthesis, equals, minus, start fraction, 1, divided by, n, start superscript, 2, end superscript, end fraction, dot, 13, point, 6, space, e, V, where the lowest possible energy or ground state energy of a hydrogen electron—$E(1)$E(1)E, left parenthesis, 1, right parenthesis—is $-13.6\,\text{eV}$−13.6eVminus, 13, point, 6, space, e, V. Bohr radius=r(1)=0.529×10​−10​​m




Note that the energy is always going to be a negative number, and the ground state, $n=1$n=1n, equals, 1, has the most negative value. This is because the energy of an electron in orbit is relative to the energy of an electron that has been completely separated from its nucleus, $n=\infty$n=∞n, equals, infinity, which is defined to have an energy of $0\,\text{eV}$0eV0, space, e, V. Since an electron in orbit around the nucleus is more stable than an electron that is infinitely far away from its nucleus, the energy of an electron in orbit is always negative.

Atomic line spectra

Atomic line spectra are another example of quantization. When an element or ion is heated by a flame or excited by electric current, the excited atoms emit light of a characteristic color. The emitted light can be refracted by a prism, producing spectra with a distinctive striped appearance due to the emission of certain wavelengths of light.

For the relatively simple case of the hydrogen atom, the wavelengths of some emission lines could even be fitted to mathematical equations. The equations did not explain why the hydrogen atom emitted those particular wavelengths of light, however. Prior to Bohr's model of the hydrogen atom, scientists were unclear of the reason behind the quantization Light emitted or absorbed by single atoms contributes only very little to the colours of our surroundings. Neon signs (or other gas discharge tubes) as used for advertising, sodium or mercury vapour lamps show atomic emission; the colours of fireworks are due to it. The aurora borealis (northern light) is very rare at our latitudes, and to appreciate the colours of cosmic objects, powerful telescopes are necessary. Neon, which gives red colour in a gas discharge, is a colourless gas. If the light of the sun is spread out into different colours by a simple glass prism, the narrow absorption lines cannot be seen.
of atomic emission spectra.

Quantization and photons

By the early 1900s, 

scientists were aware that some phenomena occurred in a discrete, as opposed to continuous, manner. 

Physicists 

                 Max Planck      and 

                                                     Albert Einstein 

had recently theorized that electromagnetic radiation not only behaves like a wave, 

but also sometimes like particles called photons. 

Planck studied the electromagnetic radiation emitted by heated objects, and he proposed that the emitted electromagnetic radiation was "quantized" since the energy of light could only have values given by the following equation $E_{\text{photon}}=nh\nu$ E​photon​​=nhν, where $n$nn is a positive integer, $h$hh is Planck’s constant—$6.626 \times10^{-34}\,\text {J}\cdot \text s$6.626×10​−34​​J⋅s6, point, 626, times, 10, start superscript, minus, 34, end superscript, space, J, dot, s—and $\nu$ν is the frequency of the light, which has units of $\dfrac{1}{\text s}$​s​​1​​start fraction, 1, divided by, s, end fraction.

As a consequence, the emitted electromagnetic radiation must have energies that are multiples of $h\nu$hν. Einstein used Planck's results to explain why a minimum frequency of light was required to eject electrons from a metal surface in the photoelectric effect.

quantized

When something is quantized, it means that only specific values are allowed, such as when playing a piano. Since each key of a piano is tuned to a specific note, only a certain set of notes—which correspond to frequencies of sound waves—can be produced. As long as your piano is properly tuned, you can play an F or F sharp, but you can't play the note that is halfway between an F and F sharp.

 

The planetary model of the atom

The planetary model of the atom

At the beginning of the 20th century, 

a new field of study known as 

                                quantum mechanics 

emerged. One of the founders of this field was Danish physicist Niels Bohr,

                          who was interested in explaining 

                              the discrete line spectrum 

                         observed when light was emitted 

                                by different elements. 

 Bohr was also interested in the structure of the atom, which was a topic of much debate at the time. Numerous models of the atom had been postulated based on experimental results including the discovery of the electron by J. J. Thomson and the discovery of the nucleus by Ernest Rutherford. Bohr supported the planetary model, in which electrons revolved around a positively charged nucleus like the rings around

However, scientists still had many unanswered questions:$$

• Where are the electrons, and what are they doing?
• If the electrons are orbiting the nucleus, why don’t they fall into the nucleus as predicted by classical physics?
  

  According to classical physics, a negatively charged electron moving around in the positive electric field created by the nucleus should emit electromagnetic energy. The electron would continue to lose energy as it orbited the nucleus until it eventually collapsed into the nucleus. Unfortunately, this reasoning would suggest that all atoms are inherently unstable!
• How is the internal structure of the atom related to the discrete emission lines produced by excited elements?
  Bohr addressed these questions using a seemingly simple assumption: what if some aspects of atomic structure, such as electron orbits and energies, could only take on certain values?