Why Can Hydrogen Atom Only Have Two Electrons to Be Stable

Chapter 6. Electronic Structure and Periodic Properties of Elements

half-dozen.ii The Bohr Model

Learning Objectives

Past the end of this section, y'all will be able to:

• Describe the Bohr model of the hydrogen atom
• Use the Rydberg equation to calculate energies of low-cal emitted or absorbed by hydrogen atoms

Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the movie of atoms consisting of tiny dense nuclei surrounded by lighter and fifty-fifty tinier electrons continually moving about the nucleus was well established. This picture was called the planetary model, since it pictured the atom as a miniature "solar organisation" with the electrons orbiting the nucleus like planets orbiting the sun. The simplest atom is hydrogen, consisting of a single proton equally the nucleus about which a single electron moves. The electrostatic strength alluring the electron to the proton depends but on the distance betwixt the two particles. The electrostatic forcefulness has the same form as the gravitational forcefulness between two mass particles except that the electrostatic force depends on the magnitudes of the charges on the particles (+one for the proton and −1 for the electron) instead of the magnitudes of the particle masses that govern the gravitational force. Since forces can be derived from potentials, it is convenient to piece of work with potentials instead, since they are forms of energy. The electrostatic potential is as well chosen the Coulomb potential. Because the electrostatic potential has the same class as the gravitational potential, according to classical mechanics, the equations of move should be like, with the electron moving around the nucleus in circular or elliptical orbits (hence the label "planetary" model of the atom). Potentials of the grade Five(r) that depend merely on the radial distance r are known every bit central potentials. Central potentials have spherical symmetry, and and so rather than specifying the position of the electron in the usual Cartesian coordinates (x, y, z), it is more user-friendly to use polar spherical coordinates centered at the nucleus, consisting of a linear coordinate r and 2 athwart coordinates, ordinarily specified past the Greek letters theta (θ) and phi (Φ). These coordinates are like to the ones used in GPS devices and virtually smart phones that track positions on our (virtually) spherical earth, with the ii angular coordinates specified by the latitude and longitude, and the linear coordinate specified by sea-level peak. Because of the spherical symmetry of central potentials, the energy and angular momentum of the classical hydrogen atom are constants, and the orbits are constrained to lie in a plane similar the planets orbiting the sunday. This classical mechanics description of the atom is incomplete, however, since an electron moving in an elliptical orbit would exist accelerating (by changing direction) and, according to classical electromagnetism, it should continuously emit electromagnetic radiation. This loss in orbital energy should result in the electron'south orbit getting continually smaller until it spirals into the nucleus, implying that atoms are inherently unstable.

In 1913, Niels Bohr attempted to resolve the diminutive paradox by ignoring classical electromagnetism's prediction that the orbiting electron in hydrogen would continuously emit light. Instead, he incorporated into the classical mechanics description of the cantlet Planck'due south ideas of quantization and Einstein's finding that lite consists of photons whose energy is proportional to their frequency. Bohr assumed that the electron orbiting the nucleus would not normally emit any radiation (the stationary state hypothesis), but information technology would emit or absorb a photon if information technology moved to a different orbit. The free energy absorbed or emitted would reverberate differences in the orbital energies co-ordinate to this equation:

[latex]|\Delta Eastward| = |E_\text{f} - E_\text{i}| = h\nu = \frac{hc}{\lambda}[/latex]

In this equation, h is Planck's constant and E_i and E_f are the initial and final orbital energies, respectively. The absolute value of the free energy difference is used, since frequencies and wavelengths are always positive. Instead of allowing for continuous values for the angular momentum, energy, and orbit radius, Bohr assumed that only discrete values for these could occur (actually, quantizing whatever one of these would imply that the other two are also quantized). Bohr's expression for the quantized energies is:

[latex]E_n = - \frac{chiliad}{n^two}, northward = i, 2, 3, \dots[/latex]

In this expression, chiliad is a constant comprising fundamental constants such as the electron mass and charge and Planck'due south constant. Inserting the expression for the orbit energies into the equation for ΔE gives

[latex]\Delta E = k(\frac{1}{due north^2_1} - \frac{i}{n^2_2}) = \frac{hc}{\lambda}[/latex]

or

[latex]\frac{1}{\lambda} = \frac{thousand}{hc} (\frac{one}{n^2_1} - \frac{i}{north^2_2})[/latex]

which is identical to the Rydberg equation for [latex]R_{\infty} = \frac{grand}{hc}[/latex]. When Bohr calculated his theoretical value for the Rydberg constant, [latex]R_{\infty}[/latex], and compared it with the experimentally accepted value, he got excellent agreement. Since the Rydberg constant was one of the most precisely measured constants at that time, this level of agreement was amazing and meant that Bohr's model was taken seriously, despite the many assumptions that Bohr needed to derive it.

The lowest few energy levels are shown in Effigy 1. One of the fundamental laws of physics is that affair is most stable with the everyman possible energy. Thus, the electron in a hydrogen atom commonly moves in the north = 1 orbit, the orbit in which it has the lowest energy. When the electron is in this everyman energy orbit, the cantlet is said to be in its ground electronic land (or merely ground state). If the atom receives energy from an exterior source, information technology is possible for the electron to move to an orbit with a higher n value and the atom is now in an excited electronic state (or but an excited country) with a college free energy. When an electron transitions from an excited state (higher energy orbit) to a less excited state, or ground state, the difference in energy is emitted as a photon. Similarly, if a photon is absorbed by an atom, the energy of the photon moves an electron from a lower energy orbit up to a more excited i. Nosotros can relate the free energy of electrons in atoms to what we learned previously about energy. The police force of conservation of energy says that nosotros can neither create nor destroy energy. Thus, if a certain amount of external energy is required to excite an electron from one energy level to some other, that same amount of energy will be liberated when the electron returns to its initial state (Figure 2). In effect, an atom can "store" energy by using it to promote an electron to a state with a higher free energy and release it when the electron returns to a lower state. The free energy can be released as i quantum of energy, as the electron returns to its ground state (say, from n = 5 to n = one), or information technology can exist released as two or more than smaller quanta every bit the electron falls to an intermediate state, then to the ground country (say, from n = v to northward = 4, emitting one breakthrough, then to n = 1, emitting a second quantum).

Since Bohr's model involved only a unmarried electron, it could as well be applied to the single electron ions He⁺, Liⁱⁱ⁺, Be³⁺, and so along, which differ from hydrogen only in their nuclear charges, and then one-electron atoms and ions are collectively referred to every bit hydrogen-similar atoms. The free energy expression for hydrogen-like atoms is a generalization of the hydrogen atom energy, in which Z is the nuclear charge (+one for hydrogen, +two for He, +3 for Li, and so on) and yard has a value of 2.179 × x^–18 J.

[latex]E_n = -\frac{kZ^two}{n^two}[/latex]

The sizes of the circular orbits for hydrogen-similar atoms are given in terms of their radii past the following expression, in which α0α0 is a constant called the Bohr radius, with a value of 5.292 × ten⁻¹¹ m:

[latex]r= \frac{due north^2}{Z}a_0[/latex]

The equation as well shows united states that as the electron's energy increases (as n increases), the electron is found at greater distances from the nucleus. This is unsaid by the inverse dependence on r in the Coulomb potential, since, as the electron moves abroad from the nucleus, the electrostatic allure between it and the nucleus decreases, and it is held less tightly in the cantlet. Notation that as n gets larger and the orbits get larger, their energies go closer to cipher, and and so the limits [latex]due north \longrightarrow \infty \;\; n \longrightarrow \infty[/latex], and [latex]r \longrightarrow \infty \;\; r \longrightarrow \infty[/latex] imply that E = 0 corresponds to the ionization limit where the electron is completely removed from the nucleus. Thus, for hydrogen in the footing land n = 1, the ionization energy would be:

[latex]\Delta E = E_{n \longrightarrow \infty} - E_1= 0 + thou = k[/latex]

With three extremely puzzling paradoxes now solved (blackbody radiations, the photoelectric effect, and the hydrogen cantlet), and all involving Planck'due south constant in a fundamental manner, information technology became articulate to most physicists at that time that the classical theories that worked so well in the macroscopic world were fundamentally flawed and could non exist extended down into the microscopic domain of atoms and molecules. Unfortunately, despite Bohr's remarkable accomplishment in deriving a theoretical expression for the Rydberg constant, he was unable to extend his theory to the adjacent simplest atom, He, which only has 2 electrons. Bohr'south model was severely flawed, since it was nonetheless based on the classical mechanics notion of precise orbits, a concept that was later found to be untenable in the microscopic domain, when a proper model of quantum mechanics was developed to supersede classical mechanics.

Effigy 1. Quantum numbers and free energy levels in a hydrogen atom. The more negative the calculated value, the lower the energy.

Instance 1

Calculating the Energy of an Electron in a Bohr Orbit
Early researchers were very excited when they were able to predict the energy of an electron at a particular distance from the nucleus in a hydrogen atom. If a spark promotes the electron in a hydrogen cantlet into an orbit with n = iii, what is the calculated energy, in joules, of the electron?

Solution
The energy of the electron is given by this equation:

[latex]East = \frac{-kZ^2}{due north^2}[/latex]

The atomic number, Z, of hydrogen is 1; k = 2.179 × 10^–18 J; and the electron is characterized by an n value of iii. Thus,

[latex]Due east = \frac{-(2.179 \times x^{-xviii} \;\text{J}) \times (1)^two}{(3)^2} = -2.421 \times 10^{-19} \;\text{J}[/latex]

Check Your Learning
The electron in Effigy 2 is promoted even further to an orbit with due north = six. What is its new energy?

Effigy 2. The horizontal lines show the relative free energy of orbits in the Bohr model of the hydrogen cantlet, and the vertical arrows depict the free energy of photons absorbed (left) or emitted (right) as electrons move between these orbits.

Case 2

Calculating the Free energy and Wavelength of Electron Transitions in a I–electron (Bohr) System
What is the energy (in joules) and the wavelength (in meters) of the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit with north = 4 to the orbit with n = half-dozen? In what office of the electromagnetic spectrum do we find this radiation?

Solution
In this case, the electron starts out with n = iv, so n ₁ = four. It comes to balance in the northward = 6 orbit, so n ₂ = 6. The divergence in energy betwixt the two states is given by this expression:

[latex]\begin{array}{r @{{}={}} l} \Delta E & E_1 - E_2 = 2.179 \times x^{-18} (\frac{one}{n^2_1} - \frac{ane}{n^2_2}) \\[1em] \Delta E & ii.179 \times 10^{-18} (\frac{ane}{4^ii} - \frac{1}{6^two}) \;\text{J} \\[1em] \Delta Due east & ii.179 \times 10^{-18} (\frac{one}{16} - \frac{1}{36}) \;\text{J} \\[1em] \Delta Due east & 7.566 \times 10^{-20} \;\text{J} \end{array}[/latex]

This energy difference is positive, indicating a photon enters the system (is absorbed) to excite the electron from the n = 4 orbit up to the n = 6 orbit. The wavelength of a photon with this energy is establish by the expression [latex]E = \frac{hc}{\lambda}[/latex]. Rearrangement gives:

[latex]\lambda = \frac{hc}{Eastward}[/latex]

[latex]\begin{assortment}{l} = (six.626 \times ten^{-34} \;\dominion[0.5ex]{0.6em}{0.1ex}\hspace{-0.6em}\text{J} \;\rule[0.5ex]{0.6em}{0.1ex}\hspace{-0.6em}\text{s}) \times \frac{ii.998 \times 10^viii \;\text{one thousand} \;\rule[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{s}^{-1}}{seven.566 \times ten^{-20} \;\rule[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{J}} \\[1em] = two.626 \times ten^{-6} \;\text{thousand} \terminate{array}[/latex]

From Figure two in Affiliate 6.ane Electromagnetic Free energy, we tin see that this wavelength is plant in the infrared portion of the electromagnetic spectrum.

Cheque Your Learning
What is the energy in joules and the wavelength in meters of the photon produced when an electron falls from the n = v to the n = iii level in a He⁺ ion (Z = two for He⁺)?

Answer:

6.198 × 10^–nineteen J; iii.205 × 10⁻⁷ m

Bohr's model of the hydrogen atom provides insight into the behavior of matter at the microscopic level, but it is does non account for electron–electron interactions in atoms with more than ane electron. Information technology does introduce several important features of all models used to describe the distribution of electrons in an atom. These features include the post-obit:

• The energies of electrons (energy levels) in an atom are quantized, described by quantum numbers: integer numbers having only specific allowed value and used to characterize the arrangement of electrons in an atom.
• An electron's energy increases with increasing distance from the nucleus.
• The discrete energies (lines) in the spectra of the elements result from quantized electronic energies.

Of these features, the well-nigh important is the postulate of quantized energy levels for an electron in an atom. As a issue, the model laid the foundation for the breakthrough mechanical model of the atom. Bohr won a Nobel Prize in Physics for his contributions to our agreement of the structure of atoms and how that is related to line spectra emissions.

Central Concepts and Summary

Bohr incorporated Planck's and Einstein's quantization ideas into a model of the hydrogen cantlet that resolved the paradox of atom stability and discrete spectra. The Bohr model of the hydrogen atom explains the connectedness between the quantization of photons and the quantized emission from atoms. Bohr described the hydrogen atom in terms of an electron moving in a circular orbit about a nucleus. He postulated that the electron was restricted to certain orbits characterized by discrete energies. Transitions between these allowed orbits consequence in the absorption or emission of photons. When an electron moves from a college-energy orbit to a more stable one, free energy is emitted in the class of a photon. To move an electron from a stable orbit to a more than excited 1, a photon of energy must be absorbed. Using the Bohr model, we can calculate the energy of an electron and the radius of its orbit in whatsoever one-electron system.

Central Equations

• [latex]E_n = -\frac{kZ^2}{n^2}, due north = 1, 2, 3, \dots[/latex]
• [latex]\Delta Eastward = kZ^2(\frac{1}{n^2_1} - \frac{i}{north^2_2})[/latex]
• [latex]r = \frac{north^two}{Z} \; a_0[/latex]

Chemistry Cease of Chapter Exercises

1. Why is the electron in a Bohr hydrogen cantlet bound less tightly when it has a quantum number of 3 than when it has a quantum number of one?
2. What does information technology mean to say that the energy of the electrons in an atom is quantized?
3. Using the Bohr model, determine the energy, in joules, necessary to ionize a ground-country hydrogen atom. Bear witness your calculations.
4. The electron volt (eV) is a convenient unit of energy for expressing atomic-scale energies. It is the amount of free energy that an electron gains when subjected to a potential of 1 volt; 1 eV = 1.602 × x^–19 J. Using the Bohr model, determine the energy, in electron volts, of the photon produced when an electron in a hydrogen cantlet moves from the orbit with due north = 5 to the orbit with n = 2. Show your calculations.
5. Using the Bohr model, determine the everyman possible free energy, in joules, for the electron in the Li^two+ ion.
6. Using the Bohr model, decide the lowest possible free energy for the electron in the He⁺ ion.
7. Using the Bohr model, make up one's mind the energy of an electron with n = 6 in a hydrogen atom.
8. Using the Bohr model, determine the energy of an electron with due north = 8 in a hydrogen atom.
9. How far from the nucleus in angstroms (i angstrom = i × 10^–10 m) is the electron in a hydrogen cantlet if it has an energy of –8.72 × 10^–20 J?
10. What is the radius, in angstroms, of the orbital of an electron with north = viii in a hydrogen atom?
11. Using the Bohr model, determine the energy in joules of the photon produced when an electron in a He⁺ ion moves from the orbit with n = 5 to the orbit with northward = 2.
12. Using the Bohr model, determine the energy in joules of the photon produced when an electron in a Li²⁺ ion moves from the orbit with n = two to the orbit with due north = 1.
13. Consider a large number of hydrogen atoms with electrons randomly distributed in the n = ane, 2, three, and iv orbits.

    (a) How many different wavelengths of low-cal are emitted by these atoms as the electrons fall into lower-free energy orbitals?

    (b) Calculate the lowest and highest energies of calorie-free produced by the transitions described in part (a).

    (c) Calculate the frequencies and wavelengths of the light produced by the transitions described in part (b).

14. How are the Bohr model and the Rutherford model of the cantlet similar? How are they unlike?
15. The spectra of hydrogen and of calcium are shown in Figure 12 in Chapter 6.i Electromagnetic Energy. What causes the lines in these spectra? Why are the colors of the lines different? Propose a reason for the observation that the spectrum of calcium is more than complicated than the spectrum of hydrogen.

Glossary

Bohr'south model of the hydrogen cantlet

structural model in which an electron moves around the nucleus merely in circular orbits, each with a specific allowed radius; the orbiting electron does non normally emit electromagnetic radiation, but does so when changing from i orbit to another.

excited country

state having an free energy greater than the ground-land energy

ground country

state in which the electrons in an cantlet, ion, or molecule have the everyman energy possible

breakthrough number

integer number having just specific allowed values and used to characterize the arrangement of electrons in an atom

Solutions

2. Quantized free energy means that the electrons tin possess only certain discrete energy values; values between those quantized values are not permitted.

4. [latex]\brainstorm{array}{r @{{}={}}fifty} Due east & E_2 - E_5 = two.179 \times 10^{-18} (\frac{1}{due north^2_2} - \frac{1}{n^2_5}) \;\text{J} \\[1em] & 2.179 \times 10^{-18} (\frac{1}{2^2} - \frac{1}{5^2}) = 4.576 \times ten^{-19} \;\text{J} \\[1em] & \frac{4.576 \times x^{-19} \;\dominion[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J}}{one.602 \times 10^{-19} \;\dominion[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J eV}^{-1}} = 2.856 \;\text{eV} \end{assortment}[/latex]

6. −8.716 × ten^−eighteen J

8. −three.405 × 10^−xx J

ten. 33.nine Å

12. ane.471 × x⁻¹⁷ J

fourteen. Both involve a relatively heavy nucleus with electrons moving around it, although strictly speaking, the Bohr model works only for one-electron atoms or ions. According to classical mechanics, the Rutherford model predicts a miniature "solar organisation" with electrons moving most the nucleus in circular or elliptical orbits that are bars to planes. If the requirements of classical electromagnetic theory that electrons in such orbits would emit electromagnetic radiation are ignored, such atoms would be stable, having abiding energy and angular momentum, simply would not emit any visible lite (contrary to ascertainment). If classical electromagnetic theory is applied, then the Rutherford atom would emit electromagnetic radiation of continually increasing frequency (contrary to the observed detached spectra), thereby losing energy until the cantlet collapsed in an absurdly short time (contrary to the observed long-term stability of atoms). The Bohr model retains the classical mechanics view of circular orbits bars to planes having abiding free energy and angular momentum, but restricts these to quantized values dependent on a single breakthrough number, n. The orbiting electron in Bohr's model is assumed non to emit whatever electromagnetic radiations while moving about the nucleus in its stationary orbits, but the atom can emit or absorb electromagnetic radiation when the electron changes from one orbit to another. Considering of the quantized orbits, such "quantum jumps" will produce discrete spectra, in agreement with observations.

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