Atomic Physics
107 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 states using an energy-level diagram.
- Describe the major successes and limitations of Bohr's theory.
Rutherford's discovery of the atomic nucleus solved one of the greatest mysteries of physics, but it immediately raised another. If negatively charged electrons orbit a positively charged nucleus, why don't they simply spiral inward and collapse into the atom? Classical physics could not answer this question.
The Danish physicist Niels Bohr (1885–1962) proposed the first successful model that explained the structure and behavior of the simplest atom: hydrogen. Building on Rutherford's nuclear model, Bohr introduced the revolutionary idea that electrons can occupy only certain allowed energy states. His theory successfully explained the hydrogen emission spectrum and marked one of the first major steps toward the development of modern quantum mechanics.
Although Bohr's model was later replaced by a more complete quantum mechanical description of the atom, many of its ideas remain fundamental. Concepts such as quantized energy levels, photon emission and absorption, and electronic transitions continue to be used throughout physics, chemistry, biology, and medicine.
Mysteries of Atomic Spectra
One of the strongest clues that atoms possess an internal structure came from the light they emit. When atoms are energized—for example, by heating a gas or passing an electric current through it—they do not emit every possible wavelength of light. Instead, they produce a series of bright, sharply defined spectral lines.
This phenomenon was puzzling because classical physics predicted that electrons could possess any energy and therefore should emit a continuous range of wavelengths. Instead, every element produced its own unique pattern of spectral lines, suggesting that electrons could occupy only specific energy states.
Scientists recognized that these spectra were like fingerprints for the elements. Long before anyone understood their origin, emission spectra had already become powerful analytical tools for identifying unknown substances in laboratories, astronomy, and later in medicine.
Healthcare Connection: Spectroscopy in Medicine
Atomic spectra are more than a laboratory curiosity. Modern medical technologies rely on spectroscopy to identify chemical elements and molecules, measure blood oxygen levels using pulse oximetry, analyze biological samples, detect trace metals, and characterize tissues using fluorescence. The quantum behavior of electrons discussed in this chapter underlies many important diagnostic techniques.
Hydrogen, the simplest atom, contains only one electron and therefore has the simplest atomic spectrum. Its spectral lines have been observed in the ultraviolet (UV), visible, and infrared (IR) regions of the electromagnetic spectrum. Groups of these spectral lines form several distinct series, each corresponding to electrons making transitions to the same final energy level.
Long before the underlying physics was understood, experimentalists discovered that every wavelength in the hydrogen spectrum could be described by a remarkably simple mathematical relationship:
In this equation, λ is the wavelength of the emitted light, R is the Rydberg constant, and ni and nf are positive integers that identify the electron's initial and final energy states.
The value of nf determines which spectral series is produced:
- Lyman series: [latex]n_f=1[/latex] (ultraviolet)
- Balmer series: [latex]n_f=2[/latex] (mostly visible light)
- Paschen series: [latex]n_f=3[/latex] (infrared)
For every transition, the electron must begin in a higher energy state, so
For example, the Balmer series consists of transitions from
to the common final state
Initially, the Rydberg equation was simply an empirical formula—it reproduced experimental measurements with extraordinary accuracy but offered no explanation for why it worked. Johannes Balmer first derived the relationship for only one series of hydrogen lines, and it was later generalized to all of the observed spectral series.
Bohr recognized that this mathematical pattern reflected a deeper physical principle: electrons can occupy only discrete energy levels. His theory transformed the Rydberg equation from an experimental recipe into a consequence of the quantum structure of the atom.
Example 107.1: Calculating the Wavelength of a Hydrogen Spectral Line and Applying Wave Interference
Problem
What is the spacing between the slits of a diffraction grating that produces a first-order maximum for the second Balmer line at an angle of [latex]15^\circ[/latex]?
Strategy
This problem combines concepts from two different areas of physics:
- Determine the wavelength of the second Balmer line using the Rydberg equation.
- Use the condition for constructive interference from a diffraction grating (or double slit) to determine the slit spacing.
This illustrates how ideas developed in different chapters often work together to solve real physical problems.
Part (a): Find the Wavelength of the Hydrogen Line
The Balmer series consists of transitions that end at
The first Balmer line corresponds to the transition
so the second Balmer line corresponds to
Substituting these values into the Rydberg equation gives
Taking the reciprocal gives the wavelength:
Discussion
The calculated wavelength of 486 nm matches the experimentally observed blue-green line in the Balmer series. Even more remarkable is that the same equation accurately predicts every hydrogen spectral line. Bohr's theory later showed that this relationship arises because electrons occupy only discrete energy levels.
Part (b): Determine the Slit Spacing
For a first-order interference maximum, the diffraction condition is
where
- [latex]d[/latex] is the slit spacing,
- [latex]\theta[/latex] is the diffraction angle,
- [latex]m[/latex] is the interference order, and
- [latex]\lambda[/latex] is the wavelength.
Since this is a first-order maximum,
Solving for the slit spacing gives
Substituting the known values,
Final Answer
Discussion
This slit spacing is typical of diffraction gratings used in spectroscopy. Instruments based on this principle separate light into its component wavelengths, allowing scientists to identify elements by their characteristic emission spectra. Similar optical techniques are widely used in chemistry, astronomy, environmental monitoring, and biomedical laboratories.
Bohr's Solution for Hydrogen
Bohr's greatest achievement was showing that the observed hydrogen spectrum could be explained using a few simple physical ideas. Starting with Rutherford's nuclear model, Bohr proposed that electrons are not free to orbit the nucleus at arbitrary distances. Instead, they can occupy only certain allowed orbits, each with a specific energy.
This idea was revolutionary because it introduced the concept of quantization directly into atomic structure. Electrons can move between allowed energy levels by absorbing or emitting energy, but they cannot exist between those levels. As a result, atoms emit and absorb only specific amounts of energy, producing the discrete spectral lines observed experimentally.
Whenever an electron changes from one allowed energy level to another, the atom absorbs or emits a photon whose energy equals the difference between the two levels:
Here, [latex]\Delta E[/latex] is the change in the electron's energy, [latex]h[/latex] is Planck's constant, [latex]f[/latex] is the frequency of the emitted or absorbed photon, and [latex]E_i[/latex] and [latex]E_f[/latex] are the initial and final electron energies.
Energy is required to move an electron to a higher orbit, while energy is released when an electron falls to a lower orbit. Unlike planets orbiting the Sun, however, electrons cannot occupy just any orbit. Only specific, quantized orbits are permitted.
A convenient way to represent these allowed states is with an energy-level diagram. Instead of drawing electron orbits, the diagram displays the allowed energies as horizontal lines. The lowest level is called the ground state, while higher levels are called excited states. Vertical arrows represent electrons changing between these energy levels.
Energy-level diagrams are widely used not only for atoms but also for molecules, nuclei, semiconductors, and many other physical systems. Any successful theory must correctly predict the locations of these allowed energy levels.
Healthcare Connection: Fluorescence and Medical Imaging
Electron transitions between quantized energy levels are responsible for fluorescence and many forms of optical spectroscopy used in medicine. These principles are applied in fluorescence microscopy, flow cytometry, DNA sequencing, laser diagnostics, and numerous laboratory assays that detect disease by measuring the wavelengths of emitted light.
Bohr went one step further by proposing that the electron's angular momentum is also quantized. Instead of taking any value, it can occur only in discrete multiples of a fundamental constant:
In this equation, [latex]m_e[/latex] is the electron's mass, [latex]v[/latex] is its orbital speed, [latex]r_n[/latex] is the radius of the orbit, and [latex]n[/latex] is the principal quantum number. At the time, Bohr could not explain why angular momentum should be quantized, but this simple assumption successfully reproduced the observed hydrogen spectrum.
Using this idea together with Coulomb's law and the requirement for circular motion, Bohr derived the allowed sizes of electron orbits. The electrostatic attraction between the positively charged nucleus and the negatively charged electron provides the centripetal force required for circular motion.
The same derivation applies to any hydrogen-like atom—an atom or ion containing only one electron. Examples include neutral hydrogen ([latex]Z=1[/latex]), singly ionized helium ([latex]\mathrm{He}^+[/latex], [latex]Z=2[/latex]), and doubly ionized lithium ([latex]\mathrm{Li}^{2+}[/latex], [latex]Z=3[/latex]).
Setting the Coulomb force equal to the centripetal force gives
Combining this result with Bohr's angular momentum condition yields the radius of each allowed orbit:
The constant [latex]a_B[/latex] is called the Bohr radius. It is the radius of the ground-state orbit of hydrogen:
These equations accurately predict the size of hydrogen and show that electron orbit radii increase as the square of the principal quantum number. For example, the [latex]n=2[/latex] orbit has four times the radius of the ground-state orbit, while the [latex]n=3[/latex] orbit is nine times larger.
The electron's total energy is the sum of its kinetic energy and electric potential energy:
Using the expressions for kinetic energy, electric potential energy, and the previously derived orbit radius, Bohr obtained the allowed energy levels for any hydrogen-like atom:
Here, [latex]E_0[/latex] is the ground-state energy of hydrogen:
For hydrogen ([latex]Z=1[/latex]), the energy levels simplify to
The negative sign indicates that the electron is bound to the nucleus. Energy must be supplied to remove the electron completely from the atom. As [latex]n[/latex] approaches infinity, the electron becomes free and its total energy approaches zero.
Figure 107.7. Energy-level diagram for hydrogen. Transitions ending at [latex]n=1[/latex] form the Lyman series (ultraviolet), transitions ending at [latex]n=2[/latex] form the Balmer series (mostly visible), and transitions ending at [latex]n=3[/latex] form the Paschen series (infrared). Each spectral line corresponds to an electron transitioning between two allowed energy levels.The negative sign in the energy equation has an important physical meaning: it indicates that the electron is bound to the nucleus. Just as an object trapped at the bottom of a valley must gain energy to escape, an electron requires energy to leave the atom completely.
As the principal quantum number [latex]n[/latex] increases, the electron occupies larger orbits and its total energy approaches zero. When
the electron is no longer bound to the nucleus. Its electric potential energy becomes zero because it is effectively an infinite distance away, and its total energy is also zero.
For a hydrogen atom in its ground state, the electron has an energy of [latex]-13.6\ \text{eV}[/latex]. Therefore, an energy of
must be supplied to remove the electron completely from the atom. This process is called ionization, and the required energy is known as the ionization energy of hydrogen.
If the electron receives more than 13.6 eV, the excess energy becomes kinetic energy. For example, if a ground-state hydrogen atom absorbs 15.0 eV, the electron is ionized and leaves the atom with
of kinetic energy.
When an electron moves from a higher energy level to a lower one, the atom emits a photon whose energy equals the difference between the two energy levels:
Substituting Bohr's expression for the hydrogen energy levels,
gives the energy of the emitted photon:
Since the wavelength of light is related to its frequency by [latex]c=f\lambda[/latex], dividing both sides by [latex]hc[/latex] yields
The constant
is the Rydberg constant. Therefore, Bohr's model naturally reproduces the empirical equation that had been discovered experimentally years earlier:
This result was one of Bohr's greatest successes. What had previously been only an empirical relationship now had a physical explanation. The hydrogen spectrum consists of discrete wavelengths because electrons can occupy only discrete energy levels.
For an emission to occur, the electron must move from a higher energy level to a lower one, so
Transitions ending at different final energy levels produce the various spectral series. For example, all transitions ending at [latex]n_f=1[/latex] form the Lyman series, those ending at [latex]n_f=2[/latex] form the Balmer series, and those ending at [latex]n_f=3[/latex] form the Paschen series.
Triumphs and Limits of the Bohr Theory
Bohr's model represented a major breakthrough in atomic physics. For the first time, a theory successfully explained the observed spectrum of hydrogen while also predicting the correct size of the hydrogen atom. Many of its central ideas remain fundamental to modern physics.
Among Bohr's most important achievements were:
- Introducing the concept of quantized energy levels for electrons.
- Showing that atoms emit and absorb photons only during transitions between allowed energy states.
- Correctly predicting the hydrogen spectrum and the value of the Rydberg constant.
- Explaining why electrons do not continuously radiate energy and spiral into the nucleus, as classical physics predicted.
- Introducing the quantization of angular momentum, an idea that became an essential part of quantum mechanics.
Healthcare Connection: Why Quantized Energy Matters
The concept of quantized energy levels underlies many technologies used in healthcare. Lasers used in eye surgery, fluorescence imaging, pulse oximeters, DNA sequencing instruments, and many medical laboratory analyzers all rely on electrons making transitions between discrete energy levels and emitting or absorbing photons of specific wavelengths.
Despite its remarkable success, Bohr's theory has important limitations. It accurately describes only atoms and ions that contain a single electron, such as hydrogen, singly ionized helium ([latex]\mathrm{He}^+[/latex]), and other hydrogen-like ions. It cannot correctly predict the behavior of atoms with multiple electrons.
Bohr's model is also considered semiclassical. It combines the classical idea of electrons moving in circular orbits with the nonclassical assumption that only certain orbits are allowed. Modern quantum mechanics has shown that electrons do not travel along well-defined paths around the nucleus. Instead, they are described by probability distributions, often called atomic orbitals, which indicate where an electron is most likely to be found.
In addition, Bohr's model could not explain several experimental observations, including the fine splitting of spectral lines, the behavior of atoms in magnetic fields, or the spectra of more complex atoms.
Nevertheless, Bohr's work was a pivotal step in the development of quantum theory. Although later superseded by quantum mechanics, his model introduced the essential idea that energy in atoms is quantized and provided the first successful physical explanation for atomic spectra. Nearly every modern theory of atomic structure builds upon the concepts that Bohr introduced.
Interactive Exploration: Models of the Hydrogen Atom
Our understanding of the atom did not emerge all at once. Instead, it developed through a series of increasingly accurate scientific models, each proposed to explain new experimental observations. In this interactive simulation, you will compare several historical models of the hydrogen atom and investigate how each predicts the interaction between atoms and light.
As you explore, observe how different models explain—or fail to explain—the emission and absorption of light. Notice how experimental evidence eventually led scientists from classical descriptions of the atom to the modern quantum mechanical model.
Guided Exploration
Use the simulation to investigate how different atomic models explain the behavior of hydrogen. As you work through the activity, consider the following questions:
- Compare the different atomic models. Which models successfully reproduce the observed hydrogen spectrum, and which do not?
- Shine light on the hydrogen atom. Under what conditions does the atom absorb or emit a photon?
- Observe the colors of the emitted light. Why are only certain wavelengths produced instead of a continuous rainbow of colors?
- Compare the Bohr model with the quantum mechanical model. What important ideas do they share, and what are the major differences in how they describe the electron?
- How can experimental spectra be used to test whether an atomic model is accurate?
- Based on your observations, explain why the quantum mechanical model ultimately replaced earlier descriptions of the atom.
Reflection: After completing the simulation, compare your observations with the concepts presented in this chapter. Notice that only models with quantized energy levels correctly reproduce the observed hydrogen spectrum. This quantization explains why atoms absorb and emit light only at specific wavelengths and provided some of the strongest experimental evidence for the development of quantum mechanics.
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Section Summary
- Bohr combined Rutherford's nuclear model with the idea that electron energies are quantized, allowing him to develop the first successful theory of the hydrogen atom.
- The wavelengths of light emitted by hydrogen are described by the Rydberg equation:
where [latex]\lambda[/latex] is the wavelength of the emitted photon and the Rydberg constant is
- The integers [latex]n_i[/latex] and [latex]n_f[/latex] identify the electron's initial and final energy levels. For an emitted photon, the electron must move from a higher level to a lower one, so
- Electrons absorb or emit photons only when transitioning between allowed energy levels. The energy of the photon equals the difference between the electron's initial and final energies:
- An energy-level diagram provides a convenient way to visualize the allowed energy states of an atom and the transitions responsible for its spectral lines.
- Bohr proposed that the electron's angular momentum is quantized according to
- For any hydrogen-like atom containing a single electron, the allowed orbital radii are
where [latex]Z[/latex] is the atomic number and the Bohr radius is
- The allowed energy levels for hydrogen-like atoms are
where
For hydrogen ([latex]Z=1[/latex]), this becomes
- Bohr's model successfully explained the hydrogen spectrum, predicted the correct size of the hydrogen atom, and introduced the concept of quantized energy levels. Although modern quantum mechanics provides a more complete description of atoms, Bohr's theory remains an important milestone in the development of atomic physics.
Conceptual Questions
- 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.
- Explain how Bohr's rule for the quantization of electron orbital angular momentum differs from the actual rule.
- What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?
Problems & Exercises
- By calculating its wavelength, show that the first line in the Lyman series is UV radiation.
- Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.
- Look up the values of the quantities in [latex]{a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}[/latex], and verify that the Bohr radius [latex]{a}_{\text{B}}[/latex] is [latex]0.529\times10^{-10}\ \text{m}[/latex].
- Verify that the ground-state energy [latex]{E}_{0}[/latex] is 13.6 eV by using [latex]{E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}[/latex].
- If a hydrogen atom has its electron in the [latex]n=4[/latex] state, how much energy in eV is needed to ionize it?
- A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is [latex]n[/latex] for a hydrogen atom if 0.850 eV of energy can ionize it?
- Find the radius of a hydrogen atom in the [latex]n=2[/latex] state according to Bohr's theory.
- Show that [latex]\left(13.6\ \text{eV}\right)/hc=1.097\times10^7\ \text{m}^{-1}=R[/latex] (Rydberg's constant), as discussed in the text.
- What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?
- 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.
- Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.
- (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? (c) How many are in the UV?
- A wavelength of [latex]4.653\ \mu\text{m}[/latex] is observed in a hydrogen spectrum for a transition that ends in the [latex]n_f=5[/latex] level. What was [latex]n_i[/latex] for the initial level of the electron?
- A singly ionized helium ion has only one electron and is denoted [latex]\mathrm{He}^{+}[/latex]. What is the ion's radius in the ground state compared to the Bohr radius of the hydrogen atom?
- A beryllium ion with a single electron (denoted [latex]\mathrm{Be}^{3+}[/latex]) is in an excited state with radius the same as that of the ground state of hydrogen.
- (a) What is [latex]n[/latex] for the [latex]\mathrm{Be}^{3+}[/latex] ion? (b) How much energy in eV is needed to ionize the ion from this excited state?
- Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is [latex]\mathrm{C}^{+5}[/latex], a carbon atom with only a single electron. (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? (c) What type of EM radiation is this?
- Verify Equations [latex]{r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}[/latex] and [latex]{a}_{B}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=0.529\times10^{-10}\ \text{m}[/latex] using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.
- The wavelengths of the four Balmer-series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength values and those predicted by [latex]\frac{1}{\lambda}=R\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)[/latex]? It is remarkable how accurately this simple empirical equation reproduced the experimental data.
Glossary
- hydrogen spectrum wavelengths
- The wavelengths of light emitted or absorbed by hydrogen atoms. They are described by the Rydberg equation, which relates each wavelength to transitions between quantized electron energy levels.
- Rydberg constant
- A fundamental physical constant used to calculate the wavelengths of spectral lines in hydrogen and hydrogen-like atoms, with the value [latex]1.097\times10^7\ \text{m}^{-1}[/latex].
- double-slit interference
- An interference phenomenon produced when waves or particles passing through two closely spaced openings combine to form alternating bright and dark regions due to constructive and destructive interference.
- energy-level diagram
- A graphical representation of the allowed energy states of a system, showing the energies of electrons and the transitions responsible for emission and absorption spectra.
- Bohr radius
- The radius of the lowest-energy (ground-state) electron orbit in a hydrogen atom, equal to [latex]0.529\times10^{-10}\ \text{m}[/latex].
- hydrogen-like atom
- An atom or ion containing only a single electron, such as hydrogen, [latex]\mathrm{He}^{+}[/latex], or [latex]\mathrm{Li}^{2+}[/latex].
- energies of hydrogen-like atoms
- The quantized electron energies predicted by Bohr's model, given by [latex]E_n=-\frac{Z^2}{n^2}E_0[/latex], where [latex]E_0=13.6\ \text{eV}[/latex] and [latex]Z[/latex] is the atomic number.
The wavelengths of light emitted or absorbed by hydrogen atoms. They are described by the Rydberg equation, which relates each wavelength to transitions between quantized electron energy levels.
A fundamental physical constant used to calculate the wavelengths of spectral lines in hydrogen and hydrogen-like atoms, with the value [latex]1.097\times10^7\ \text{m}^{-1}[/latex].
An interference phenomenon produced when waves or particles passing through two closely spaced openings combine to form alternating bright and dark regions due to constructive and destructive interference.
A graphical representation of the allowed energy states of a system, showing the energies of electrons and the transitions responsible for emission and absorption spectra.
The radius of the lowest-energy (ground-state) electron orbit in a hydrogen atom, equal to [latex]0.529\times10^{-10}\ \text{m}[/latex].
An atom or ion containing only a single electron, such as hydrogen, [latex]\mathrm{He}^{+}[/latex], or [latex]\mathrm{Li}^{2+}[/latex].
The quantized electron energies predicted by Bohr's model, given by [latex]E_n=-\frac{Z^2}{n^2}E_0[/latex], where [latex]E_0=13.6\ \text{eV}[/latex] and [latex]Z[/latex] is the atomic number.
