What Can Models Show Us Specifically About The Makeup Of Substances On The Atomic Level?

Affiliate 6. Electronic Structure and Periodic Backdrop of Elements

half-dozen.ii The Bohr Model

Learning Objectives

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

Describe 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 by lighter and even tinier electrons continually moving most the nucleus was well established. This picture was chosen the planetary model, since it pictured the atom every bit a miniature "solar organization" with the electrons orbiting the nucleus like planets orbiting the lord's day. The simplest atom is hydrogen, consisting of a unmarried proton as the nucleus about which a unmarried electron moves. The electrostatic strength attracting the electron to the proton depends only on the distance betwixt the ii particles. The electrostatic strength has the same course as the gravitational force between 2 mass particles except that the electrostatic force depends on the magnitudes of the charges on the particles (+one for the proton and −ane for the electron) instead of the magnitudes of the particle masses that govern the gravitational strength. Since forces tin be derived from potentials, it is convenient to work with potentials instead, since they are forms of energy. The electrostatic potential is as well called the Coulomb potential. Because the electrostatic potential has the same course every bit 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 class V(r) that depend merely on the radial distance r are known as primal potentials. Primal potentials take spherical symmetry, and so rather than specifying the position of the electron in the usual Cartesian coordinates (x, y, z), it is more convenient to use polar spherical coordinates centered at the nucleus, consisting of a linear coordinate r and two athwart coordinates, normally specified by the Greek letters theta (θ) and phi (Φ). These coordinates are similar to the ones used in GPS devices and well-nigh smart phones that track positions on our (most) spherical earth, with the two angular coordinates specified by the latitude and longitude, and the linear coordinate specified by sea-level elevation. Because of the spherical symmetry of central potentials, the energy and athwart momentum of the classical hydrogen atom are constants, and the orbits are constrained to lie in a airplane like the planets orbiting the sun. This classical mechanics description of the atom is incomplete, however, since an electron moving in an elliptical orbit would be accelerating (by changing direction) and, according to classical electromagnetism, it should continuously emit electromagnetic radiations. This loss in orbital energy should result in the electron's 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 calorie-free. Instead, he incorporated into the classical mechanics description of the cantlet Planck'south ideas of quantization and Einstein's 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 radiation (the stationary state hypothesis), but 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'south constant and E_i and E_f are the initial and last orbital energies, respectively. The absolute value of the free energy deviation is used, since frequencies and wavelengths are ever positive. Instead of allowing for continuous values for the athwart momentum, free 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'southward expression for the quantized energies is:

[latex]E_n = - \frac{k}{northward^2}, due north = i, 2, iii, \dots[/latex]

In this expression, k is a abiding 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 ΔEast gives

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

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

which is identical to the Rydberg equation for [latex]R_{\infty} = \frac{m}{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 constant was 1 of the most precisely measured constants at that time, this level of agreement was astonishing and meant that Bohr'due south model was taken seriously, despite the many assumptions that Bohr needed to derive information technology.

The lowest few free energy levels are shown in Figure 1. One of the fundamental laws of physics is that thing is most stable with the everyman possible energy. Thus, the electron in a hydrogen cantlet normally moves in the n = 1 orbit, the orbit in which it has the lowest free energy. When the electron is in this lowest free energy orbit, the cantlet is said to exist in its ground electronic state (or simply footing state). If the atom receives energy from an outside source, information technology is possible for the electron to move to an orbit with a college n value and the atom is now in an excited electronic country (or just an excited country) with a higher energy. When an electron transitions from an excited state (higher energy orbit) to a less excited land, or basis country, the divergence in energy is emitted as a photon. Similarly, if a photon is absorbed past an atom, the free energy of the photon moves an electron from a lower energy orbit upward to a more excited one. Nosotros can chronicle the energy of electrons in atoms to what we learned previously about energy. The law of conservation of 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 another, that aforementioned amount of free energy will be liberated when the electron returns to its initial land (Figure two). In effect, an atom tin can "shop" energy by using it to promote an electron to a country with a college energy and release it when the electron returns to a lower state. The energy tin exist released as 1 quantum of energy, as the electron returns to its ground country (say, from n = 5 to northward = i), or it can be released as two or more smaller quanta as the electron falls to an intermediate state, and so to the footing state (say, from n = 5 to n = four, emitting ane quantum, and so to n = 1, emitting a 2d breakthrough).

Since Bohr's model involved only a single electron, it could besides be applied to the single electron ions He⁺, Li²⁺, Be³⁺, and so along, which differ from hydrogen only in their nuclear charges, and and then one-electron atoms and ions are collectively referred to as hydrogen-like atoms. The energy expression for hydrogen-like atoms is a generalization of the hydrogen cantlet energy, in which Z is the nuclear accuse (+1 for hydrogen, +ii for He, +3 for Li, then on) and thou has a value of 2.179 × ten^–18 J.

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

The sizes of the circular orbits for hydrogen-like atoms are given in terms of their radii by the following expression, in which α0α0 is a constant chosen the Bohr radius, with a value of five.292 × ten⁻¹¹ yard:

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

The equation also shows us 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, every bit the electron moves away from the nucleus, the electrostatic attraction between it and the nucleus decreases, and it is held less tightly in the atom. Note that as n gets larger and the orbits get larger, their energies get closer to zero, and and then the limits [latex]northward \longrightarrow \infty \;\; n \longrightarrow \infty[/latex], and [latex]r \longrightarrow \infty \;\; r \longrightarrow \infty[/latex] imply that Eastward = 0 corresponds to the ionization limit where the electron is completely removed from the nucleus. Thus, for hydrogen in the ground state n = 1, the ionization free energy would be:

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

With three extremely puzzling paradoxes now solved (blackbody radiations, the photoelectric effect, and the hydrogen cantlet), and all involving Planck's constant in a fundamental fashion, information technology became clear to most physicists at that time that the classical theories that worked so well in the macroscopic world were fundamentally flawed and could not be extended downwards 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's model was severely flawed, since it was still based on the classical mechanics notion of precise orbits, a concept that was later constitute to be untenable in the microscopic domain, when a proper model of quantum mechanics was developed to supersede classical mechanics.

The figure includes a diagram representing the relative energy levels of the quantum numbers of the hydrogen atom. An upward pointing arrow at the left of the diagram is labeled, — **Figure 1.** Quantum numbers and free energy levels in a hydrogen atom. The more negative the calculated value, the lower the energy.

Case one

Calculating the Free 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 detail altitude from the nucleus in a hydrogen atom. If a spark promotes the electron in a hydrogen atom into an orbit with north = 3, what is the calculated 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 atomic number, Z, of hydrogen is 1; k = 2.179 × 10^–18 J; and the electron is characterized by an northward value of 3. Thus,

[latex]E = \frac{-(2.179 \times x^{-eighteen} \;\text{J}) \times (1)^2}{(3)^2} = -2.421 \times 10^{-xix} \;\text{J}[/latex]

Bank check Your Learning
The electron in Figure ii is promoted even further to an orbit with n = 6. What is its new free energy?

Case 2

Calculating the Free energy and Wavelength of Electron Transitions in a 1–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 motion of an electron from Bohr orbit with n = 4 to the orbit with n = 6? In what part of the electromagnetic spectrum practise we find this radiation?

Solution
In this case, the electron starts out with n = four, so due north _i = iv. It comes to remainder in the n = 6 orbit, then n ₂ = half-dozen. The difference in free energy between the two states is given past this expression:

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

This energy divergence is positive, indicating a photon enters the system (is captivated) to excite the electron from the northward = 4 orbit up to the n = half dozen orbit. The wavelength of a photon with this energy is found by the expression [latex]Due east = \frac{hc}{\lambda}[/latex]. Rearrangement gives:

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

[latex]\begin{assortment}{l} = (vi.626 \times x^{-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{2.998 \times 10^8 \;\text{m} \;\rule[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{due south}^{-1}}{vii.566 \times 10^{-twenty} \;\rule[0.25ex]{0.3em}{0.1ex}\hspace{-0.3em}\text{J}} \\[1em] = 2.626 \times x^{-half-dozen} \;\text{m} \end{array}[/latex]

From Effigy 2 in Chapter half-dozen.1 Electromagnetic Free energy, we can 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 north = 3 level in a He⁺ ion (Z = two for He⁺)?

Answer:

6.198 × ten^–19 J; 3.205 × 10⁻⁷ g

Bohr'south model of the hydrogen atom provides insight into the behavior of matter at the microscopic level, but it is does not business relationship for electron–electron interactions in atoms with more than i electron. It does introduce several of import features of all models used to describe the distribution of electrons in an atom. These features include the following:

The energies of electrons (free 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 free energy increases with increasing distance from the nucleus.
The discrete energies (lines) in the spectra of the elements issue from quantized electronic energies.

Of these features, the most 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 understanding of the structure of atoms and how that is related to line spectra emissions.

Key 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 detached spectra. The Bohr model of the hydrogen atom explains the connectedness betwixt the quantization of photons and the quantized emission from atoms. Bohr described the hydrogen cantlet in terms of an electron moving in a circular orbit about a nucleus. He postulated that the electron was restricted to certain orbits characterized past discrete energies. Transitions between these immune orbits result in the absorption or emission of photons. When an electron moves from a higher-energy orbit to a more stable one, energy is emitted in the form of a photon. To movement an electron from a stable orbit to a more excited one, a photon of free energy must exist absorbed. Using the Bohr model, nosotros can calculate the energy of an electron and the radius of its orbit in any one-electron system.

Key Equations

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

Chemical science End of Chapter Exercises

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 breakthrough number of one?
What does it mean to say that the energy of the electrons in an atom is quantized?
Using the Bohr model, decide the energy, in joules, necessary to ionize a ground-country hydrogen atom. Show your calculations.
The electron volt (eV) is a user-friendly 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 × 10^–xix 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 = ii. Show your calculations.
Using the Bohr model, determine the lowest possible energy, in joules, for the electron in the Li²⁺ ion.
Using the Bohr model, decide the everyman possible energy for the electron in the He⁺ ion.
Using the Bohr model, make up one's mind the energy of an electron with north = six in a hydrogen atom.
Using the Bohr model, decide the energy of an electron with north = 8 in a hydrogen atom.
How far from the nucleus in angstroms (ane angstrom = 1 × 10^–10 k) is the electron in a hydrogen atom if it has an energy of –8.72 × 10^–20 J?
What is the radius, in angstroms, of the orbital of an electron with n = eight in a hydrogen atom?
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 = 2.
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 due north = 2 to the orbit with north = 1.
Consider a large number of hydrogen atoms with electrons randomly distributed in the north = 1, 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 lite produced past the transitions described in part (a).

(c) Calculate the frequencies and wavelengths of the light produced by the transitions described in role (b).
How are the Bohr model and the Rutherford model of the atom like? How are they different?
The spectra of hydrogen and of calcium are shown in Figure 12 in Chapter half dozen.i Electromagnetic Energy. What causes the lines in these spectra? Why are the colors of the lines different? Suggest a reason for the ascertainment that the spectrum of calcium is more complicated than the spectrum of hydrogen.

Glossary

Bohr'southward model of the hydrogen atom: structural model in which an electron moves around the nucleus but in round orbits, each with a specific allowed radius; the orbiting electron does non commonly emit electromagnetic radiations, only does and so when changing from i orbit to another.

excited country: state having an free energy greater than the ground-state free energy

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

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

Solutions

2. Quantized free energy ways that the electrons can possess but certain discrete energy values; values between those quantized values are non permitted.

4. [latex]\begin{array}{r @{{}={}}l} E & E_2 - E_5 = ii.179 \times 10^{-18} (\frac{1}{n^2_2} - \frac{ane}{n^2_5}) \;\text{J} \\[1em] & 2.179 \times 10^{-18} (\frac{1}{2^two} - \frac{1}{five^2}) = 4.576 \times 10^{-19} \;\text{J} \\[1em] & \frac{iv.576 \times 10^{-xix} \;\rule[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J}}{i.602 \times 10^{-19} \;\rule[0.5ex]{0.4em}{0.1ex}\hspace{-0.4em}\text{J eV}^{-1}} = 2.856 \;\text{eV} \end{array}[/latex]

half dozen. −8.716 × 10⁻¹⁸ J

viii. −3.405 × 10⁻²⁰ J

ten. 33.9 Å

12. 1.471 × 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 nigh 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 radiations are ignored, such atoms would exist stable, having constant free energy and angular momentum, simply would non emit whatsoever visible light (opposite to observation). 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 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 abiding energy and athwart momentum, only restricts these to quantized values dependent on a single breakthrough number, n. The orbiting electron in Bohr's model is assumed not to emit any electromagnetic radiation while moving nearly 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" volition produce discrete spectra, in agreement with observations.