niels bohr hydrogen spectrum experiment

Development of Current Atomic Theory

Emission Spectrum of Hydrogen

When an electric current is passed through a glass tube that contains hydrogen gas at low pressure the tube gives off blue light. When this light is passed through a prism (as shown in the figure below), four narrow bands of bright light are observed against a black background.

These narrow bands have the characteristic wavelengths and colors shown in the table below.

Four more series of lines were discovered in the emission spectrum of hydrogen by searching the infrared spectrum at longer wave-lengths and the ultraviolet spectrum at shorter wavelengths. Each of these lines fits the same general equation, where n 1 and n 2 are integers and R H is 1.09678 x 10 -2 nm -1 .

Explanation of the Emission Spectrum

Max Planck presented a theoretical explanation of the spectrum of radiation emitted by an object that glows when heated. He argued that the walls of a glowing solid could be imagined to contain a series of resonators that oscillated at different frequencies. These resonators gain energy in the form of heat from the walls of the object and lose energy in the form of electromagnetic radiation. The energy of these resonators at any moment is proportional to the frequency with which they oscillate.

To fit the observed spectrum, Planck had to assume that the energy of these oscillators could take on only a limited number of values. In other words, the spectrum of energies for these oscillators was no longer continuous. Because the number of values of the energy of these oscillators is limited, they are theoretically "countable." The energy of the oscillators in this system is therefore said to be quantized . Planck introduced the notion of quantization to explain how light was emitted.

Albert Einstein extended Planck's work to the light that had been emitted. At a time when everyone agreed that light was a wave (and therefore continuous), Einstein suggested that it behaved as if it was a stream of small bundles, or packets, of energy. In other words, light was also quantized. Einstein's model was based on two assumptions. First, he assumed that light was composed of photons , which are small, discrete bundles of energy. Second, he assumed that the energy of a photon is proportional to its frequency.

In this equation, h is a constant known as Planck's constant, which is equal to 6.626 x 10 -34 J-s.

Example: Let's calculate the energy of a single photon of red light with a wavelength of 700.0 nm and the energy of a mole of these photons.

Red light with a wavelength of 700.0 nm has a frequency of 4.283 x 10 14 s -1 . Substituting this frequency into the Planck-Einstein equation gives the following result.

A single photon of red light carries an insignificant amount of energy. But a mole of these photons carries about 171,000 joules of energy, or 171 kJ/mol.

Absorption of a mole of photons of red light would therefore provide enough energy to raise the temperature of a liter of water by more than 40 o C.

The fact that hydrogen atoms emit or absorb radiation at a limited number of frequencies implies that these atoms can only absorb radiation with certain energies. This suggests that there are only a limited number of energy levels within the hydrogen atom. These energy levels are countable. The energy levels of the hydrogen atom are quantized.

The Bohr Model of the Atom

The electron in a hydrogen atom travels around the nucleus in a circular orbit.
The energy of the electron in an orbit is proportional to its distance from the nucleus. The further the electron is from the nucleus, the more energy it has.
Only a limited number of orbits with certain energies are allowed. In other words, the orbits are quantized.
The only orbits that are allowed are those for which the angular momentum of the electron is an integral multiple of Planck's constant divided by 2p.
Light is absorbed when an electron jumps to a higher energy orbit and emitted when an electron falls into a lower energy orbit.
The energy of the light emitted or absorbed is exactly equal to the difference between the energies of the orbits.

Some of the key elements of this hypothesis are illustrated in the figure below.

Three points deserve particular attention. First, Bohr recognized that his first assumption violates the principles of classical mechanics. But he knew that it was impossible to explain the spectrum of the hydrogen atom within the limits of classical physics. He was therefore willing to assume that one or more of the principles from classical physics might not be valid on the atomic scale.

Second, he assumed there are only a limited number of orbits in which the electron can reside. He based this assumption on the fact that there are only a limited number of lines in the spectrum of the hydrogen atom and his belief that these lines were the result of light being emitted or absorbed as an electron moved from one orbit to another in the atom.

Finally, Bohr restricted the number of orbits on the hydrogen atom by limiting the allowed values of the angular momentum of the electron. Any object moving along a straight line has a momentum equal to the product of its mass ( m ) times the velocity ( v ) with which it moves. An object moving in a circular orbit has an angular momentum equal to its mass ( m ) times the velocity ( v ) times the radius of the orbit ( r ). Bohr assumed that the angular momentum of the electron can take on only certain values, equal to an integer times Planck's constant divided by 2p.

Bohr then used classical physics to show that the energy of an electron in any one of these orbits is inversely proportional to the square of the integer n .

The difference between the energies of any two orbits is therefore given by the following equation.

In this equation, n 1 and n 2 are both integers and R H is the proportionality constant known as the Rydberg constant.

Planck's equation states that the energy of a photon is proportional to its frequency.

Substituting the relationship between the frequency, wavelength, and the speed of light into this equation suggests that the energy of a photon is inversely proportional to its wavelength. The inverse of the wavelength of electromagnetic radiation is therefore directly proportional to the energy of this radiation.

By properly defining the units of the constant, R H , Bohr was able to show that the wavelengths of the light given off or absorbed by a hydrogen atom should be given by the following equation.

Thus, once he introduced his basic assumptions, Bohr was able to derive an equation that matched the relationship obtained from the analysis of the spectrum of the hydrogen atom.

According to the Bohr model, the wavelength of the light emitted by a hydrogen atom when the electron falls from a high energy ( n = 4) orbit into a lower energy ( n = 2) orbit.

Substituting the appropriate values of R H , n 1 , and n 2 into the equation shown above gives the following result.

Solving for the wavelength of this light gives a value of 486.3 nm, which agrees with the experimental value of 486.1 nm for the blue line in the visible spectrum of the hydrogen atom.

Wave-Particle Duality

The theory of wave-particle duality developed by Louis-Victor de Broglie eventually explained why the Bohr model was successful with atoms or ions that contained one electron. It also provided a basis for understanding why this model failed for more complex systems. Light acts as both a particle and a wave. In many ways light acts as a wave, with a characteristic frequency, wavelength, and amplitude. Light carries energy as if it contains discrete photons or packets of energy.

When an object behaves as a particle in motion, it has an energy proportional to its mass ( m ) and the speed with which it moves through space ( s ).

When it behaves as a wave, however, it has an energy that is proportional to its frequency:

By simultaneously assuming that an object can be both a particle and a wave, de Broglie set up the following equation.

By rearranging this equation, he derived a relationship between one of the wave-like properties of matter and one of its properties as a particle.

As noted in the previous section, the product of the mass of an object times the speed with which it moves is the momentum ( p ) of the particle. Thus, the de Broglie equation suggests that the wavelength ( l ) of any object in motion is inversely proportional to its momentum.

De Broglie concluded that most particles are too heavy to observe their wave properties. When the mass of an object is very small, however, the wave properties can be detected experimentally. De Broglie predicted that the mass of an electron was small enough to exhibit the properties of both particles and waves. In 1927 this prediction was confirmed when the diffraction of electrons was observed experimentally by C. J. Davisson.

De Broglie applied his theory of wave-particle duality to the Bohr model to explain why only certain orbits are allowed for the electron. He argued that only certain orbits allow the electron to satisfy both its particle and wave properties at the same time because only certain orbits have a circumference that is an integral multiple of the wavelength of the electron, as shown in the figure below.

The Bohr Model vs. Reality

At first glance, the Bohr model looks like a two-dimensional model of the atom because it restricts the motion of the electron to a circular orbit in a two-dimensional plane. In reality the Bohr model is a one-dimensional model, because a circle can be defined by specifying only one dimension: its radius, r . As a result, only one coordinate ( n ) is needed to describe the orbits in the Bohr model.

Unfortunately, electrons aren't particles that can be restricted to a one-dimensional circular orbit. They act to some extent as waves and therefore exist in three-dimensional space. The Bohr model works for one-electron atoms or ions only because certain factors present in more complex atoms are not present in these atoms or ions. To construct a model that describes the distribution of electrons in atoms that contain more than one electron we have to allow the electrons to occupy three-dimensional space. We therefore need a model that uses three coordinates to describe the distribution of electrons in these atoms.

Wave Functions and Orbitals

We still talk about the Bohr model of the atom even if the only thing this model can do is explain the spectrum of the hydrogen atom because it was the last model of the atom for which a simple physical picture can be constructed. It is easy to imagine an atom that consists of solid electrons revolving around the nucleus in circular orbits.

Erwin Schrödinger combined the equations for the behavior of waves with the de Broglie equation to generate a mathematical model for the distribution of electrons in an atom. The advantage of this model is that it consists of mathematical equations known as wave functions that satisfy the requirements placed on the behavior of electrons. The disadvantage is that it is difficult to imagine a physical model of electrons as waves.

The Schrödinger model assumes that the electron is a wave and tries to describe the regions in space, or orbitals , where electrons are most likely to be found. Instead of trying to tell us where the electron is at any time, the Schrödinger model describes the probability that an electron can be found in a given region of space at a given time. This model no longer tells us where the electron is; it only tells us where it might be.

Chapter 30 Atomic Physics

30.3 Bohr’s Theory of the Hydrogen Atom

Learning objectives.

Describe the mysteries of atomic spectra.
Explain Bohr’s theory of the hydrogen atom.
Explain Bohr’s planetary model of the atom.
Illustrate energy state using the energy-level diagram.
Describe the triumphs and limits of Bohr’s theory.

The great Danish physicist Niels Bohr (1885–1962) made immediate use of Rutherford’s planetary model of the atom. ( Figure 1 ). Bohr became convinced of its validity and spent part of 1912 at Rutherford’s laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the planetary model of the atom. For decades, many questions had been asked about atomic characteristics. From their sizes to their spectra, much was known about atoms, but little had been explained in terms of the laws of physics. Bohr’s theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics.

Mysteries of Atomic Spectra

As noted in Chapter 29.1 Quantization of Energy , the energies of some small systems are quantized. Atomic and molecular emission and absorption spectra have been known for over a century to be discrete (or quantized). (See Figure 2 .) Maxwell and others had realized that there must be a connection between the spectrum of an atom and its structure, something like the resonant frequencies of musical instruments. But, in spite of years of efforts by many great minds, no one had a workable theory. (It was a running joke that any theory of atomic and molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra.) Following Einstein’s proposal of photons with quantized energies directly proportional to their wavelengths, it became even more evident that electrons in atoms can exist only in discrete orbits.

This figure has two parts. Part a shows a discharge tube at the extreme left. Light from the discharge tube passes through a rectangular slit and a grating, going from left to right. From the grating, light of different colors falls on a photographic film. Part b of the figure shows the emission line spectrum for iron.

In some cases, it had been possible to devise formulas that described the emission spectra. As you might expect, the simplest atom—hydrogen, with its single electron—has a relatively simple spectrum. The hydrogen spectrum had been observed in the infrared (IR), visible, and ultraviolet (UV), and several series of spectral lines had been observed. (See Figure 3 .) These series are named after early researchers who studied them in particular depth.

The observed hydrogen-spectrum wavelengths can be calculated using the following formula:

$\boldsymbol{\frac{1}{\lambda}}$

Calculating Wave Interference of a Hydrogen Line

$\boldsymbol{15 ^{\circ}}$

Strategy and Concept

For an Integrated Concept problem, we must first identify the physical principles involved. In this example, we need to know (a) the wavelength of light as well as (b) conditions for an interference maximum for the pattern from a double slit. Part (a) deals with a topic of the present chapter, while part (b) considers the wave interference material of Chapter 27 Wave Optics .

Solution for (a)

$\boldsymbol{n_{\textbf{i}}=3}$

Discussion for (a)

This is indeed the experimentally observed wavelength, corresponding to the second (blue-green) line in the Balmer series. More impressive is the fact that the same simple recipe predicts all of the hydrogen spectrum lines, including new ones observed in subsequent experiments. What is nature telling us?

Solution for (b)

Double-slit interference ( Chapter 27 Wave Optics ). To obtain constructive interference for a double slit, the path length difference from two slits must be an integral multiple of the wavelength. This condition was expressed by the equation

$\boldsymbol{d \textbf{sin} \theta = m \lambda}$

Discussion for (b)

This number is similar to those used in the interference examples of Chapter 29 Introduction to Quantum Physics (and is close to the spacing between slits in commonly used diffraction glasses).

Bohr’s Solution for Hydrogen

Bohr was able to derive the formula for the hydrogen spectrum using basic physics, the planetary model of the atom, and some very important new proposals. His first proposal is that only certain orbits are allowed: we say that the orbits of electrons in atoms are quantized . Each orbit has a different energy, and electrons can move to a higher orbit by absorbing energy and drop to a lower orbit by emitting energy. If the orbits are quantized, the amount of energy absorbed or emitted is also quantized, producing discrete spectra. Photon absorption and emission are among the primary methods of transferring energy into and out of atoms. The energies of the photons are quantized, and their energy is explained as being equal to the change in energy of the electron when it moves from one orbit to another. In equation form, this is

$\boldsymbol{\Delta E = hf = E_i - E_f}$

Figure 5 shows an energy-level diagram , a convenient way to display energy states. In the present discussion, we take these to be the allowed energy levels of the electron. Energy is plotted vertically with the lowest or ground state at the bottom and with excited states above. Given the energies of the lines in an atomic spectrum, it is possible (although sometimes very difficult) to determine the energy levels of an atom. Energy-level diagrams are used for many systems, including molecules and nuclei. A theory of the atom or any other system must predict its energies based on the physics of the system.

The energy level diagram is shown. A number of horizontal lines are shown. The lines are labeled from bottom to top as n is equal to one, n is equal to two and so on up to n equals infinity; the energy levels increase from bottom to top. The distance between the lines decreases from the bottom line to the top line. A vertical arrow shows an electron transitioning from n equals four to n equals two.

To get the electron orbital energies, we start by noting that the electron energy is the sum of its kinetic and potential energy:

$\boldsymbol{E_n = \textbf{KE} + \textbf{PE}}$

Thus, for hydrogen,

$\boldsymbol{\frac{13.6 \;\textbf{eV}}{n^2}}$

Figure 7 shows an energy-level diagram for hydrogen that also illustrates how the various spectral series for hydrogen are related to transitions between energy levels.

An energy level diagram is shown. At the left, there is a vertical arrow showing the energy levels increasing from bottom to top. At the bottom, there is a horizontal line showing the energy levels of Lyman series, n is one. The energy is marked as negative thirteen point six electron volt. Then, in the upper half of the figure, another horizontal line showing Balmer series is shown when the value of n is two. The energy level is labeled as negative three point four zero electron volt. Above it there is another horizontal line showing Paschen series. The energy level is marked as negative one point five one electron volt. Above this line, some more lines are shown in a small area to show energy levels of other values of n.

Finally, let us consider the energy of a photon emitted in a downward transition, given by the equation to be

$\boldsymbol{E_n = (-13.6 \;\textbf{eV/n}^2)}$

It can be shown that

$\boldsymbol{(\frac{13.6 \;\textbf{eV}}{hc})}$

is the Rydberg constant . Thus, we have used Bohr’s assumptions to derive the formula first proposed by Balmer years earlier as a recipe to fit experimental data.

$\boldsymbol{\frac{1}{n_i^2})}$

Triumphs and Limits of the Bohr Theory

Bohr did what no one had been able to do before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from basic physics. Some of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and molecules. Angular momentum is quantized. The electrons do not spiral into the nucleus, as expected classically (accelerated charges radiate, so that the electron orbits classically would decay quickly, and the electrons would sit on the nucleus—matter would collapse). These are major triumphs.

But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

PhET Explorations: Models of the Hydrogen Atom

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

Section Summary

$\boldsymbol{(\frac{1}{n_f^2}}$

The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Conceptual Questions

1: How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

2: Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.

3: What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?

Problems & Exercises

1: By calculating its wavelength, show that the first line in the Lyman series is UV radiation.

2: Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.

$\boldsymbol{a_{\textbf{B}} = \frac{h^2}{4 \pi ^2m_ekq_e^2}}$

9: What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

10: Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.

11: Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

12: (a) Which line in the Balmer series is the first one in the UV part of the spectrum?

(b) How many Balmer series lines are in the visible part of the spectrum?

$\boldsymbol{n_f = 5}$

(b) How much energy in eV is needed to ionize the ion from this excited state?

$\boldsymbol{\textbf{C}^{+5}}$

(a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen?

(b) What is the wavelength of the first line in this ion’s Paschen series?

$\boldsymbol{r_n = \frac{n^2}{Z}a_{\textbf{B}}}$

5: 0.850 eV

$\boldsymbol{2.12 \times 10^{-10} \;\textbf{m}}$

It is in the ultraviolet.

11: No overlap

(b) 54.4 eV

$\boldsymbol{\frac{kZq_e^2}{r_n^2} = \frac{m_eV^2}{r_n}}$

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COMMENTS

30.3: Bohr’s Theory of the Hydrogen Atom - Physics LibreTexts
Bohr’s theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics. Figure $\PageIndex{1}$. Niels Bohr, Danish physicist, used the planetary model of the atom to explain the atomic spectrum and size of the hydrogen atom.
Emission Spectrum of Hydrogen - Division of Chemical ...
Niels Bohr proposed a model for the hydrogen atom that explained the spectrum of the hydrogen atom. The Bohr model was based on the following assumptions. The electron in a hydrogen atom travels around the nucleus in a circular orbit.
Bohr model | Description, Hydrogen, Development, & Facts ...
Bohr model, description of the structure of atoms proposed in 1913 by the Danish physicist Niels Bohr. The Bohr model of the atom, a radical departure from earlier, classical descriptions, was the first that incorporated quantum theory and was the predecessor of wholly quantum-mechanical models.
6.5: Bohr’s Model of the Hydrogen Atom - Physics LibreTexts
Bohr’s model of the hydrogen atom, proposed by Niels Bohr in 1913, was the first quantum model that correctly explained the hydrogen emission spectrum. Bohr’s model combines the classical mechanics of planetary motion with the quantum concept of photons.
6.3: Line Spectra and the Bohr Model - Chemistry LibreTexts
In 1913, a Danish physicist, Niels Bohr (1885–1962; Nobel Prize in Physics, 1922), proposed a theoretical model for the hydrogen atom that explained its emission spectrum. Bohr’s model required only one assumption: The electron moves around the nucleus in circular orbits that can have only certain allowed radii.
Experiment 9: The Spectrum of the Hydrogen Atom
Niels Bohr suggested an alternative theory for atoms, suggesting that the electrons can only exist in certain energy states and can only jump between these discreet states discontinuously.
Experiment 7: Spectrum of the Hydrogen Atom - Columbia University
In 1913 Neils Bohr proposed a physical model to describe the spectrum of the hydrogen atom. It was the birth of Quantum Mechanics! As a consequence, the energy of the electron around the nucleus is also quantized! In a real life experiment we can never measure the energy of the electron. We can however measure differences in its energy.
12.7: Bohr’s Theory of the Hydrogen Atom - Physics LibreTexts
Explain Bohr’s theory of the hydrogen atom. Distinguish between correct and incorrect features of the Bohr model, in light of modern quantum mechanics. The great Danish physicist Niels Bohr (1885–1962) made immediate use of Rutherford’s planetary model of the atom.
30.3 Bohr’s Theory of the Hydrogen Atom – College Physics
Bohr’s theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics. Figure 1. Niels Bohr, Danish physicist, used the planetary model of the atom to explain the atomic spectrum and size of the hydrogen atom.
7.3.2: The Bohr Theory of the Hydrogen Atom - Chemistry ...
Ernest Rutherford had proposed a model of atoms based on the α -particle scattering experiments of Hans Geiger and Ernest Marsden. In these experiments helium nuclei (α -particles) were shot at thin gold metal foils. Most of the particles were not scattered; they passed unchanged through the thin metal foil.