What Can You Assume Has Happened if an Electron Moves to a Higher Energy Level?

Chapter half-dozen. Electronic Structure and Periodic Properties of Elements

vi.2 The Bohr Model

Learning Objectives

By the end of this section, you will be able to:

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

Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded past lighter and even 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 system" with the electrons orbiting the nucleus like planets orbiting the lord's day. The simplest atom is hydrogen, consisting of a single proton as the nucleus about which a single electron moves. The electrostatic forcefulness attracting the electron to the proton depends only on the distance between the two particles. The electrostatic force 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 (+1 for the proton and −i for the electron) instead of the magnitudes of the particle masses that govern the gravitational force. Since forces tin can exist derived from potentials, it is user-friendly to work with potentials instead, since they are forms of energy. The electrostatic potential is also chosen the Coulomb potential. Because the electrostatic potential has the aforementioned form as the gravitational potential, co-ordinate to classical mechanics, the equations of motion should be similar, with the electron moving around the nucleus in circular or elliptical orbits (hence the label "planetary" model of the atom). Potentials of the course V(r) that depend just on the radial distance r are known as cardinal potentials. Fundamental potentials have spherical symmetry, and then rather than specifying the position of the electron in the usual Cartesian coordinates (x, y, z), it is more user-friendly to utilize polar spherical coordinates centered at the nucleus, consisting of a linear coordinate r and two athwart coordinates, unremarkably specified past the Greek letters theta (θ) and phi (Φ). These coordinates are similar to the ones used in GPS devices and most smart phones that track positions on our (nearly) spherical earth, with the two angular coordinates specified past the latitude and longitude, and the linear coordinate specified past bounding main-level summit. Considering of the spherical symmetry of central potentials, the energy and angular momentum of the classical hydrogen cantlet are constants, and the orbits are constrained to lie in a plane similar the planets orbiting the sun. This classical mechanics description of the cantlet is incomplete, however, since an electron moving in an elliptical orbit would exist accelerating (by irresolute direction) and, co-ordinate to classical electromagnetism, it should continuously emit electromagnetic radiations. This loss in orbital energy should effect in the electron'due 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 atomic 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 atom Planck's ideas of quantization and Einstein'due south finding that light consists of photons whose energy is proportional to their frequency. Bohr assumed that the electron orbiting the nucleus would non normally emit any radiation (the stationary state hypothesis), just information technology would emit or absorb a photon if information technology moved to a different orbit. The energy absorbed or emitted would reflect differences in the orbital energies according to this equation:

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

In this equation, h is Planck's abiding 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 e'er positive. Instead of assuasive for continuous values for the angular momentum, energy, and orbit radius, Bohr causeless that only detached values for these could occur (actually, quantizing any one of these would imply that the other two are also quantized). Bohr's expression for the quantized energies is:

[latex]E_n = - \frac{yard}{north^2}, northward = i, 2, 3, \dots[/latex]

In this expression, one thousand is a constant comprising fundamental constants such as the electron mass and accuse and Planck'south constant. Inserting the expression for the orbit energies into the equation for ΔEastward gives

[latex]\Delta E = 1000(\frac{1}{n^2_1} - \frac{1}{n^2_2}) = \frac{hc}{\lambda}[/latex]

or

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

which is identical to the Rydberg equation for [latex]R_{\infty} = \frac{k}{hc}[/latex]. When Bohr calculated his theoretical value for the Rydberg constant, [latex]R_{\infty}[/latex], and compared information technology with the experimentally accepted value, he got excellent agreement. Since the Rydberg abiding was 1 of the most precisely measured constants at that time, this level of agreement was astonishing 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 Figure ane. One of the fundamental laws of physics is that matter is most stable with the lowest possible energy. Thus, the electron in a hydrogen atom ordinarily moves in the north = one orbit, the orbit in which information technology has the lowest energy. When the electron is in this lowest free energy orbit, the cantlet is said to exist in its ground electronic land (or simply ground state). If the cantlet receives energy from an exterior source, it 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 just an excited state) with a college energy. When an electron transitions from an excited state (higher energy orbit) to a less excited state, or footing state, the difference in energy is emitted as a photon. Similarly, if a photon is captivated by an cantlet, the energy of the photon moves an electron from a lower energy orbit up to a more excited ane. Nosotros can chronicle the energy of electrons in atoms to what we learned previously about energy. The law of conservation of free energy says that we can neither create nor destroy energy. Thus, if a certain amount of external energy is required to excite an electron from i energy level to some other, that same amount of free energy will exist liberated when the electron returns to its initial state (Figure two). In upshot, an atom can "shop" energy by using it to promote an electron to a state with a higher energy and release it when the electron returns to a lower country. The energy tin exist released as one quantum of free energy, every bit the electron returns to its footing land (say, from due north = 5 to n = 1), or it can be released every bit two or more smaller quanta equally the electron falls to an intermediate land, and then to the ground country (say, from north = v to n = four, emitting one quantum, then to due north = 1, emitting a 2nd quantum).

Since Bohr's model involved only a single electron, it could also be applied to the single electron ions He⁺, Li^two+, Be³⁺, and so forth, which differ from hydrogen only in their nuclear charges, and so i-electron atoms and ions are collectively referred to as 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 (+1 for hydrogen, +2 for He, +3 for Li, and then on) and k has a value of 2.179 × 10^–18 J.

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

The sizes of the circular orbits for hydrogen-like atoms are given in terms of their radii by the post-obit expression, in which α0α0 is a abiding chosen the Bohr radius, with a value of 5.292 × 10⁻¹¹ m:

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

The equation also shows us that as the electron's energy increases (as northward increases), the electron is institute at greater distances from the nucleus. This is implied past the inverse dependence on r in the Coulomb potential, since, as the electron moves abroad from the nucleus, the electrostatic attraction betwixt it and the nucleus decreases, and information technology is held less tightly in the atom. Note that as n gets larger and the orbits go larger, their energies go closer to zero, then the limits [latex]n \longrightarrow \infty \;\; due north \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 ground state n = i, the ionization energy would exist:

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

With three extremely puzzling paradoxes at present solved (blackbody radiation, the photoelectric issue, and the hydrogen atom), and all involving Planck'south constant in a central manner, it became clear to well-nigh physicists at that fourth dimension 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 achievement in deriving a theoretical expression for the Rydberg constant, he was unable to extend his theory to the next simplest cantlet, He, which only has two electrons. Bohr'due south model was severely flawed, since it was even so 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 adult to supersede classical mechanics.

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

Case 1

Calculating the Free energy of an Electron in a Bohr Orbit
Early on researchers were very excited when they were able to predict the energy of an electron at a particular altitude from the nucleus in a hydrogen atom. If a spark promotes the electron in a hydrogen atom into an orbit with due north = 3, what is the calculated free energy, in joules, of the electron?

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

[latex]E = \frac{-kZ^2}{n^2}[/latex]

The diminutive number, Z, of hydrogen is 1; k = ii.179 × 10^–18 J; and the electron is characterized past an n value of iii. Thus,

[latex]E = \frac{-(ii.179 \times ten^{-eighteen} \;\text{J}) \times (1)^two}{(3)^2} = -two.421 \times 10^{-19} \;\text{J}[/latex]

Check Your Learning
The electron in Effigy two is promoted fifty-fifty farther to an orbit with n = 6. What is its new energy?

Figure two. The horizontal lines bear witness the relative energy of orbits in the Bohr model of the hydrogen atom, and the vertical arrows depict the energy of photons absorbed (left) or emitted (right) as electrons move between these orbits.

Case 2

Calculating the Energy and Wavelength of Electron Transitions in a Ane–electron (Bohr) Arrangement
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 n = iv to the orbit with north = 6? In what part of the electromagnetic spectrum do nosotros find this radiation?

Solution
In this case, the electron starts out with n = 4, so northward _one = iv. It comes to balance in the n = 6 orbit, and then n _two = 6. The difference in energy between the two states is given by this expression:

[latex]\brainstorm{array}{r @{{}={}} fifty} \Delta E & E_1 - E_2 = 2.179 \times 10^{-18} (\frac{1}{n^2_1} - \frac{i}{n^2_2}) \\[1em] \Delta Due east & 2.179 \times 10^{-18} (\frac{1}{4^2} - \frac{one}{6^two}) \;\text{J} \\[1em] \Delta East & 2.179 \times ten^{-18} (\frac{i}{16} - \frac{ane}{36}) \;\text{J} \\[1em] \Delta E & vii.566 \times 10^{-20} \;\text{J} \terminate{array}[/latex]

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

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

[latex]\begin{assortment}{l} = (half dozen.626 \times 10^{-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{two.998 \times ten^eight \;\text{one thousand} \;\rule[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{south}^{-i}}{7.566 \times x^{-20} \;\rule[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{J}} \\[1em] = 2.626 \times 10^{-half dozen} \;\text{thousand} \end{array}[/latex]

From Figure 2 in Chapter 6.1 Electromagnetic Energy, we can see that this wavelength is plant in the infrared portion of the electromagnetic spectrum.

Cheque Your Learning
What is the free energy in joules and the wavelength in meters of the photon produced when an electron falls from the n = 5 to the north = 3 level in a He⁺ ion (Z = two for He⁺)?

Answer:

6.198 × 10^–19 J; 3.205 × ten⁻⁷ m

Bohr's model of the hydrogen cantlet provides insight into the behavior of thing at the microscopic level, just it is does non account for electron–electron interactions in atoms with more than than ane electron. It does introduce several important features of all models used to describe the distribution of electrons in an atom. These features include the following:

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

Of these features, the nearly important is the postulate of quantized energy levels for an electron in an atom. Equally a consequence, the model laid the foundation for the breakthrough mechanical model of the cantlet. 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'south quantization ideas into a model of the hydrogen cantlet that resolved the paradox of cantlet stability and discrete spectra. The Bohr model of the hydrogen cantlet explains the connection 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 well-nigh a nucleus. He postulated that the electron was restricted to sure orbits characterized by discrete energies. Transitions between these immune orbits result in the absorption or emission of photons. When an electron moves from a college-energy orbit to a more stable ane, energy is emitted in the grade of a photon. To move an electron from a stable orbit to a more excited one, a photon of free energy must be absorbed. Using the Bohr model, we can calculate the free energy of an electron and the radius of its orbit in whatsoever 1-electron system.

Primal Equations

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

Chemistry End of Chapter Exercises

1. Why is the electron in a Bohr hydrogen cantlet spring less tightly when it has a quantum number of three than when it has a quantum number of 1?
2. What does it mean to say that the free energy of the electrons in an atom is quantized?
3. Using the Bohr model, determine the energy, in joules, necessary to ionize a basis-state hydrogen atom. Testify your calculations.
4. The electron volt (eV) is a convenient unit of energy for expressing atomic-scale energies. It is the corporeality of energy that an electron gains when subjected to a potential of 1 volt; ane eV = 1.602 × ten^–19 J. Using the Bohr model, determine the energy, in electron volts, of the photon produced when an electron in a hydrogen atom moves from the orbit with n = five to the orbit with n = two. Prove your calculations.
5. Using the Bohr model, determine the lowest possible energy, in joules, for the electron in the Liⁱⁱ⁺ ion.
6. Using the Bohr model, decide the lowest possible energy for the electron in the He⁺ ion.
7. Using the Bohr model, determine the energy of an electron with n = six in a hydrogen atom.
8. Using the Bohr model, decide the energy of an electron with n = 8 in a hydrogen atom.
9. How far from the nucleus in angstroms (1 angstrom = ane × x^–10 k) is the electron in a hydrogen atom if it has an free energy of –8.72 × 10^–20 J?
10. What is the radius, in angstroms, of the orbital of an electron with n = 8 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 n = two.
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 northward = 2 to the orbit with n = ane.
13. Consider a large number of hydrogen atoms with electrons randomly distributed in the n = i, 2, iii, and 4 orbits.

    (a) How many different wavelengths of light are emitted by these atoms equally the electrons autumn into lower-energy orbitals?

    (b) Calculate the everyman and highest energies of lite produced past the transitions described in function (a).

    (c) Summate the frequencies and wavelengths of the low-cal produced past the transitions described in office (b).

14. How are the Bohr model and the Rutherford model of the cantlet similar? How are they different?
15. The spectra of hydrogen and of calcium are shown in Figure 12 in Chapter 6.1 Electromagnetic Energy. What causes the lines in these spectra? Why are the colors of the lines dissimilar? Suggest a reason for the observation that the spectrum of calcium is more complicated than the spectrum of hydrogen.

Glossary

Bohr's model of the hydrogen atom

structural model in which an electron moves around the nucleus only in round orbits, each with a specific allowed radius; the orbiting electron does not normally emit electromagnetic radiations, simply does so when changing from i orbit to another.

excited country

state having an energy greater than the ground-state energy

ground country

state in which the electrons in an atom, ion, or molecule have the lowest energy possible

quantum number

integer number having only specific allowed values and used to characterize the organisation of electrons in an cantlet

Solutions

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

4. [latex]\begin{array}{r @{{}={}}fifty} E & E_2 - E_5 = 2.179 \times 10^{-18} (\frac{1}{n^2_2} - \frac{1}{n^2_5}) \;\text{J} \\[1em] & 2.179 \times 10^{-18} (\frac{1}{2^2} - \frac{1}{5^2}) = four.576 \times ten^{-19} \;\text{J} \\[1em] & \frac{4.576 \times 10^{-19} \;\rule[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J}}{ane.602 \times 10^{-19} \;\dominion[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J eV}^{-1}} = 2.856 \;\text{eV} \end{array}[/latex]

6. −8.716 × x⁻¹⁸ J

8. −3.405 × 10⁻²⁰ J

x. 33.9 Å

12. ane.471 × 10⁻¹⁷ J

14. Both involve a relatively heavy nucleus with electrons moving around information technology, 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 system" with electrons moving about the nucleus in circular or elliptical orbits that are confined 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 constant energy and angular momentum, but would not emit any visible light (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 discrete spectra), thereby losing energy until the cantlet complanate 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 confined to planes having constant energy and angular momentum, only restricts these to quantized values dependent on a single quantum number, north. The orbiting electron in Bohr's model is assumed not to emit whatever electromagnetic radiation while moving well-nigh the nucleus in its stationary orbits, simply the cantlet can emit or absorb electromagnetic radiation when the electron changes from one orbit to another. Because of the quantized orbits, such "quantum jumps" volition produce discrete spectra, in agreement with observations.

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Source: https://opentextbc.ca/chemistry/chapter/6-2-the-bohr-model/

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