what wavelength photon would be required to ionize li2+

Chapter 6. Electronic Construction and Periodic Properties of Elements

6.2 The Bohr Model

Learning Objectives

By the terminate of this department, you will be able to:

• Draw the Bohr model of the hydrogen atom
• Use the Rydberg equation to calculate energies of lite emitted or absorbed by hydrogen atoms

Following the piece of 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 near the nucleus was well established. This motion-picture show was called the planetary model, since it pictured the cantlet 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 as the nucleus nearly which a unmarried electron moves. The electrostatic force attracting the electron to the proton depends only on the distance betwixt the two particles. The electrostatic force has the same form as the gravitational force between 2 mass particles except that the electrostatic force depends on the magnitudes of the charges on the particles (+1 for the proton and −one for the electron) instead of the magnitudes of the particle masses that govern the gravitational forcefulness. Since forces tin can be derived from potentials, it is convenient to work with potentials instead, since they are forms of energy. The electrostatic potential is also called the Coulomb potential. Because the electrostatic potential has the same form as the gravitational potential, co-ordinate to classical mechanics, the equations of motion should be similar, with the electron moving effectually the nucleus in round or elliptical orbits (hence the label "planetary" model of the atom). Potentials of the grade V(r) that depend only on the radial distance r are known as key potentials. Central potentials have spherical symmetry, and and so rather than specifying the position of the electron in the usual Cartesian coordinates (10, y, z), it is more convenient to use polar spherical coordinates centered at the nucleus, consisting of a linear coordinate r and two angular coordinates, ordinarily specified by 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 (almost) spherical earth, with the two angular coordinates specified past the latitude and longitude, and the linear coordinate specified by ocean-level elevation. Because 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 aeroplane like the planets orbiting the dominicus. This classical mechanics description of the atom is incomplete, yet, since an electron moving in an elliptical orbit would be accelerating (by changing management) and, according to classical electromagnetism, it should continuously emit electromagnetic radiation. This loss in orbital energy should consequence in the electron's orbit getting continually smaller until information technology spirals into the nucleus, implying that atoms are inherently unstable.

In 1913, Niels Bohr attempted to resolve the diminutive paradox past ignoring classical electromagnetism'due south 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'southward finding that light consists of photons whose energy is proportional to their frequency. Bohr assumed that the electron orbiting the nucleus would not normally emit whatsoever radiations (the stationary land hypothesis), but it would emit or absorb a photon if it moved to a unlike orbit. The free 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 constant and Eastward_i and East_f are the initial and final orbital energies, respectively. The accented value of the energy difference is used, since frequencies and wavelengths are always positive. Instead of assuasive for continuous values for the angular momentum, free energy, and orbit radius, Bohr assumed that merely discrete values for these could occur (really, quantizing any one of these would imply that the other two are as well quantized). Bohr's expression for the quantized energies is:

[latex]E_n = - \frac{one thousand}{n^2}, n = i, 2, 3, \dots[/latex]

In this expression, g is a constant comprising fundamental constants such equally the electron mass and accuse and Planck's abiding. Inserting the expression for the orbit energies into the equation for ΔE gives

[latex]\Delta East = chiliad(\frac{i}{northward^2_1} - \frac{one}{n^2_2}) = \frac{hc}{\lambda}[/latex]

or

[latex]\frac{1}{\lambda} = \frac{k}{hc} (\frac{i}{n^2_1} - \frac{1}{due north^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 it with the experimentally accepted value, he got splendid agreement. Since the Rydberg constant was one of the most precisely measured constants at that fourth dimension, this level of agreement was astonishing and meant that Bohr'southward model was taken seriously, despite the many assumptions that Bohr needed to derive it.

The lowest few energy levels are shown in Figure i. One of the fundamental laws of physics is that matter is well-nigh stable with the lowest possible energy. Thus, the electron in a hydrogen atom unremarkably moves in the n = 1 orbit, the orbit in which it has the lowest energy. When the electron is in this everyman energy orbit, the atom is said to be in its ground electronic state (or simply ground state). If the atom receives energy from an outside 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 simply an excited state) with a higher energy. When an electron transitions from an excited state (higher energy orbit) to a less excited state, or ground land, the difference in energy is emitted equally a photon. Similarly, if a photon is absorbed by an cantlet, the energy of the photon moves an electron from a lower energy orbit up to a more excited ane. We can relate the free energy of electrons in atoms to what nosotros learned previously most energy. The law of conservation of energy says that we can neither create nor destroy energy. Thus, if a sure corporeality of external energy is required to excite an electron from ane energy level to another, that same amount of free energy will exist liberated when the electron returns to its initial state (Figure 2). In result, an atom can "store" free energy by using information technology to promote an electron to a state with a higher energy and release it when the electron returns to a lower state. The energy can be released as one breakthrough of energy, equally the electron returns to its ground country (say, from n = 5 to due north = 1), or it can be released as ii or more smaller quanta as the electron falls to an intermediate state, then to the ground land (say, from n = 5 to n = iv, emitting one quantum, and so to n = ane, emitting a second quantum).

Since Bohr's model involved simply a single electron, it could also be practical to the single electron ions He⁺, Li²⁺, Be³⁺, and so forth, which differ from hydrogen but in their nuclear charges, and so i-electron atoms and ions are collectively referred to as hydrogen-like atoms. The free energy expression for hydrogen-like atoms is a generalization of the hydrogen atom free energy, in which Z is the nuclear accuse (+1 for hydrogen, +2 for He, +three for Li, and so on) and yard has a value of ii.179 × 10^–eighteen J.

[latex]E_n = -\frac{kZ^2}{n^two}[/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 constant called the Bohr radius, with a value of 5.292 × 10⁻¹¹ thou:

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

The equation besides shows the states that every bit the electron's energy increases (equally n increases), the electron is found at greater distances from the nucleus. This is implied by 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 it is held less tightly in the atom. Note that equally n gets larger and the orbits get larger, their energies get closer to zero, and so the limits [latex]n \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 ground state north = 1, the ionization free energy would be:

[latex]\Delta E = E_{north \longrightarrow \infty} - E_1= 0 + 1000 = k[/latex]

With 3 extremely puzzling paradoxes now solved (blackbody radiation, the photoelectric outcome, and the hydrogen atom), and all involving Planck's constant in a key fashion, it became clear to most physicists at that time that the classical theories that worked and then well in the macroscopic earth were fundamentally flawed and could not 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 atom, He, which only has two electrons. Bohr'south model was severely flawed, since it was still 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.

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

Example ane

Computing 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 item distance from the nucleus in a hydrogen atom. If a spark promotes the electron in a hydrogen cantlet into an orbit with due north = 3, what is the calculated energy, in joules, of the electron?

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

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

The atomic number, Z, of hydrogen is ane; k = ii.179 × 10^–18 J; and the electron is characterized by an due north value of iii. Thus,

[latex]E = \frac{-(two.179 \times 10^{-18} \;\text{J}) \times (1)^2}{(3)^2} = -2.421 \times 10^{-19} \;\text{J}[/latex]

Bank check Your Learning
The electron in Effigy two is promoted even further to an orbit with northward = vi. What is its new energy?

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

Example ii

Calculating the Energy and Wavelength of Electron Transitions in a I–electron (Bohr) System
What is the free energy (in joules) and the wavelength (in meters) of the line in the spectrum of hydrogen that represents the motility of an electron from Bohr orbit with due north = four to the orbit with north = 6? In what function of the electromagnetic spectrum practice we notice this radiation?

Solution
In this case, the electron starts out with n = four, and then n ₁ = four. It comes to rest in the due north = vi orbit, so north ₂ = 6. The divergence in energy between the two states is given by this expression:

[latex]\begin{array}{r @{{}={}} fifty} \Delta E & E_1 - E_2 = 2.179 \times 10^{-18} (\frac{1}{northward^2_1} - \frac{i}{n^2_2}) \\[1em] \Delta East & 2.179 \times 10^{-18} (\frac{one}{4^2} - \frac{i}{6^ii}) \;\text{J} \\[1em] \Delta E & ii.179 \times 10^{-18} (\frac{ane}{16} - \frac{ane}{36}) \;\text{J} \\[1em] \Delta Due east & vii.566 \times 10^{-twenty} \;\text{J} \finish{array}[/latex]

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

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

[latex]\begin{array}{fifty} = (half dozen.626 \times 10^{-34} \;\rule[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{2.998 \times 10^eight \;\text{m} \;\dominion[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{s}^{-ane}}{7.566 \times 10^{-20} \;\dominion[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{J}} \\[1em] = 2.626 \times 10^{-6} \;\text{m} \end{array}[/latex]

From Figure 2 in Chapter 6.ane Electromagnetic Energy, nosotros tin come across that this wavelength is found in the infrared portion of the electromagnetic spectrum.

Check Your Learning
What is the energy in joules and the wavelength in meters of the photon produced when an electron falls from the n = 5 to the northward = 3 level in a He⁺ ion (Z = 2 for He⁺)?

Answer:

vi.198 × ten^–xix J; three.205 × ten⁻⁷ one thousand

Bohr'south model of the hydrogen atom provides insight into the behavior of matter at the microscopic level, simply information technology is does not account for electron–electron interactions in atoms with more than 1 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 atom are quantized, described past breakthrough numbers: integer numbers having but specific immune 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 outcome from quantized electronic energies.

Of these features, the most of import is the postulate of quantized energy levels for an electron in an atom. As a consequence, 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 understanding of the structure of atoms and how that is related to line spectra emissions.

Cardinal Concepts and Summary

Bohr incorporated Planck's and Einstein's quantization ideas into a model of the hydrogen atom that resolved the paradox of cantlet stability and discrete spectra. The Bohr model of the hydrogen atom 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 about a nucleus. He postulated that the electron was restricted to certain orbits characterized by detached energies. Transitions betwixt these allowed orbits result in the absorption or emission of photons. When an electron moves from a higher-energy orbit to a more stable 1, free energy is emitted in the class of a photon. To move an electron from a stable orbit to a more excited one, a photon of free energy must be captivated. Using the Bohr model, we can summate the energy of an electron and the radius of its orbit in any one-electron system.

Key Equations

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

Chemistry Finish of Chapter Exercises

1. Why is the electron in a Bohr hydrogen cantlet bound less tightly when it has a breakthrough number of 3 than when it has a quantum number of 1?
2. What does it hateful to say that the free energy of the electrons in an atom is quantized?
3. Using the Bohr model, determine the free energy, in joules, necessary to ionize a footing-state hydrogen atom. Show your calculations.
4. The electron volt (eV) is a convenient unit of energy for expressing atomic-scale energies. It is the amount of energy that an electron gains when subjected to a potential of 1 volt; one eV = i.602 × ten^–19 J. Using the Bohr model, make up one's mind the energy, in electron volts, of the photon produced when an electron in a hydrogen atom moves from the orbit with n = 5 to the orbit with due north = 2. Evidence your calculations.
5. Using the Bohr model, determine the lowest possible energy, in joules, for the electron in the Li^two+ ion.
6. Using the Bohr model, determine the lowest possible free energy for the electron in the He⁺ ion.
7. Using the Bohr model, determine the free energy of an electron with n = 6 in a hydrogen cantlet.
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 (one angstrom = 1 × 10^–x m) is the electron in a hydrogen cantlet if it has an energy of –eight.72 × 10^–twenty J?
10. What is the radius, in angstroms, of the orbital of an electron with n = viii in a hydrogen atom?
11. Using the Bohr model, decide the energy in joules of the photon produced when an electron in a He⁺ ion moves from the orbit with n = v to the orbit with n = two.
12. Using the Bohr model, make up one's mind the energy in joules of the photon produced when an electron in a Li²⁺ ion moves from the orbit with n = 2 to the orbit with north = 1.
13. Consider a big number of hydrogen atoms with electrons randomly distributed in the n = i, ii, three, and four orbits.

    (a) How many dissimilar wavelengths of light are emitted past these atoms as the electrons fall into lower-energy orbitals?

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

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

14. How are the Bohr model and the Rutherford model of the atom like? How are they different?
15. The spectra of hydrogen and of calcium are shown in Figure 12 in Chapter 6.one Electromagnetic Energy. What causes the lines in these spectra? Why are the colors of the lines different? Advise 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 atom

structural model in which an electron moves around the nucleus but in circular orbits, each with a specific allowed radius; the orbiting electron does not normally emit electromagnetic radiation, but does and so when changing from ane orbit to another.

excited state

state having an energy greater than the ground-country energy

ground state

state in which the electrons in an cantlet, ion, or molecule take the lowest energy possible

quantum number

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

Solutions

2. Quantized energy ways that the electrons can possess simply sure discrete energy values; values betwixt those quantized values are non permitted.

4. [latex]\brainstorm{array}{r @{{}={}}l} E & E_2 - E_5 = ii.179 \times 10^{-18} (\frac{ane}{n^2_2} - \frac{1}{n^2_5}) \;\text{J} \\[1em] & 2.179 \times 10^{-18} (\frac{1}{2^two} - \frac{one}{5^two}) = 4.576 \times 10^{-nineteen} \;\text{J} \\[1em] & \frac{4.576 \times 10^{-19} \;\rule[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J}}{1.602 \times ten^{-xix} \;\rule[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J eV}^{-1}} = 2.856 \;\text{eV} \terminate{array}[/latex]

six. −eight.716 × 10⁻¹⁸ J

8. −3.405 × x^−xx J

10. 33.9 Å

12. 1.471 × x⁻¹⁷ J

14. Both involve a relatively heavy nucleus with electrons moving around it, although strictly speaking, the Bohr model works only for ane-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 bars to planes. If the requirements of classical electromagnetic theory that electrons in such orbits would emit electromagnetic radiations are ignored, such atoms would exist stable, having constant energy and angular momentum, but would not emit any visible light (contrary to observation). If classical electromagnetic theory is practical, and so the Rutherford atom would emit electromagnetic radiation of continually increasing frequency (reverse to the observed discrete spectra), thereby losing free energy until the atom collapsed in an absurdly brusk 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 free energy and angular momentum, simply restricts these to quantized values dependent on a single quantum number, n. The orbiting electron in Bohr's model is assumed not to emit whatsoever electromagnetic radiation 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. Because of the quantized orbits, such "quantum jumps" will produce discrete spectra, in agreement with observations.

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