Physics
Bohr Model of the Atom
The Bohr Model of the Atom, proposed by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by orbiting electrons in specific energy levels. This model successfully explained the spectral lines of hydrogen and laid the foundation for understanding atomic structure and quantum mechanics. It was a significant advancement in the field of atomic physics.
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11 Key excerpts on "Bohr Model of the Atom"
eBook - PDF- Mark Fox(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
2 Hydrogen The quantum theory of hydrogen is the starting point for the whole subject of atomic physics. Bohr’s derivation of the quantized energies was one of the tri-umphs of early quantum theory, and makes a useful introduction to the notion of quantized energies and angular momenta. We, therefore, give a brief review of the Bohr model before moving to the main subject of the chapter, namely: the solution of the Schr¨ odinger equation for the electron-nucleus system. 2.1 The Bohr Model of Hydrogen The Bohr model is part of the “old” quantum theory of the atom (i.e., pre-quantum mechanics). It includes the quantization of energy and angular momentum, but uses classical mechanics to describe the motion of the electron. With the advent of quantum mechanics, we realize that this is an inconsistent approach, and therefore should not be pushed too far. Nevertheless, the Bohr model does give the correct quantized energy levels of hydrogen, and also gives a useful parameter (the Bohr radius) for quantifying the size of atoms. Hence, it remains a useful starting point to understand the basic structure of atoms. It is well known from classical physics that planetary orbits are characterized by their energy and angular momentum. We shall see that these are also key quantities in the quantum theory of the hydrogen atom. In 1911, Rutherford discovered the nucleus, which led to the idea of atoms consisting of electrons in classical orbits where the central forces are provided by the Coulomb attraction to the positive nucleus, as shown in Figure 2.1 . The problem with this idea is that the electron in the orbit is constantly accelerating. Accelerating charges emit radiation called bremsstrahlung , and so the electrons should be radiating all the time, losing energy. This would cause the electron to spiral into the 20 2.1 The Bohr Model of Hydrogen 21 -e + Ze v F r Figure 2.1 The Bohr Model of the Atom considers the electrons to be in orbit around the nucleus.
eBook - PDFChemistry
The Molecular Nature of Matter
- Neil D. Jespersen, Alison Hyslop(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
b. The Bohr Model of Hydrogen The first theoretical model of the hydrogen atom that successfully accounted for the Rydberg equation was proposed in 1913 by Niels Bohr (1885–1962), a Danish physicist. In his model, Bohr likened the electron moving around the nucleus to a planet circling the sun. He sug- gested that the electron moves around the nucleus along fixed paths, or orbits. This model broke with the classical laws of physics by placing restrictions on the sizes of the orbits and the NOTE Niels Bohr won the 1922 Nobel Prize in physics for his work on his model of the hydrogen atom. 7.3 The Bohr Theory 323 7.3 The Bohr Theory 323 energies that electrons could have in given orbits. This ultimately led Bohr to propose an equa- tion that described the energy of the electron in the atom. The equation includes a number of physical constants such as the mass of the electron, its charge, and Planck’s constant. It also contains an integer, n, that Bohr called a quantum number. Each of the orbits is identified by its value of n. When all the constants are combined, Bohr’s equation becomes E = − b _ n 2 (7.3) where E is the energy of the electron and b is the combined constant (its value is 2.18 × 10 –18 J). The allowed values of n are whole numbers that range from 1 to ∞. From this equation, the energy of the electron in any particular orbit could be calculated. Because of the negative sign in Equation 7.3, the lowest (most negative) energy value occurs when n = 1, which corresponds to the first Bohr orbit. The energy of the electron is negative because the electron is attracted to the nucleus, and energy is released due to the attraction. The lowest energy state of an atom that is, the most negative, is the most stable one and is called the ground state. For hydrogen, the ground state occurs when its electron has n = 1. According to Bohr’s theory, this orbit brings the electron closest to the nucleus.
eBook - PDF- George C. King(Author)
- 2023(Publication Date)
- Wiley(Publisher)
There is no justification for the postulates of fixed stable orbits, the absence of energy loss Atoms, the constituents of matter 15 of the circulating electrons, or for the quantization of angular momentum except for the fact that the model accurately agrees with experimental data obtained for the hydrogen spectrum. Furthermore, the Bohr model does not explain the spectra of atoms that are more complicated than hydrogen; even the spectrum of helium that has just two electrons. A correct description of the atomic structure had to wait for the arrival of quantum mechanics. But despite its shortcomings, the Bohr model gives a useful pictorial representation of the atom that is easy to visualise and, moreover, the mathematics involved are easy to understand. And indeed, the Bohr model is often useful as a first step in explaining a variety of phenomena in matter. Worked example In one type of extrinsic semiconductor, a tiny amount of phosphorous is added to a crystal of silicon. The phosphorous atoms occupy sites in the crystal lattice normally occupied by silicon atoms. Silicon has four electrons that are involved in bonding with other silicon atoms. But phosphorous has five available elec- trons, only four of which can bond to the silicon atoms. The remaining electron is then only weakly bound to the phosphorous atom and can easily be removed from it. We can view this situation as a single electron bound to a positively and singly charged phosphorous atom, analogous to the hydrogen atom. Use the Bohr model to determine the radius of the electron’s orbit in a phosphorous atom and the amount of energy required to remove it from the atom. Note that when an electron moves in a crystal, it has an effective mass m e due to the periodic nature of the electrical potential it experiences in the crystal lattice. The relative permittivity ε r of silicon is 11.6 and the effective mass m e is 0.26m, where m is the mass of an electron.
eBook - PDF- Kenneth S. Krane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
Such an atom would therefore violate the uncertainty relationship 202 Chapter 6 The Rutherford-Bohr Model of the Atom ΔzΔp z ≥ − h. In fact, as we discuss in Chapter 7, quantum mechanics intro- duces a degree of “fuzziness” to the behavior of electrons in atoms that is not consistent with any orbit in a single plane. In spite of its successes, the Bohr model is at best an incomplete model. It is useful only for atoms that contain one electron (hydrogen, singly ionized helium, doubly ionized lithium, and so forth), but not for atoms with two or more electrons, because we have considered only the force on the electron due to the nucleus, and not the force of each electron on the others. Furthermore, if we look very carefully at the emission spectrum, we find that many lines are in fact not single lines, but very closely spaced combinations of two or more lines; the Bohr model is unable to account for these doublets of spectral lines. The model is also limited in its usefulness as a basis from which to calculate other properties of the atom; although we can accurately calculate the energies of the spectral lines, we cannot calculate their intensities. For example, how often will an electron in the n = 3 state jump directly to the n = 1 state, emitting the corresponding photon, and how often will it jump first to the n = 2 state and then to the n = 1 state, emitting two photons? A complete theory should provide a way to calculate this property. We do not wish, however, to discard the model completely. The Bohr model provides a useful starting point in our study of atoms, and Bohr introduced several ideas (stationary states, quantization of angular momentum, corre- spondence principle) that carry over into the correct quantum-mechanical cal- culation. There are many atomic properties, especially those associated with magnetism, that can be simply modeled on the basis of Bohr orbits.
eBook - PDFElectrons, Neutrons and Protons in Engineering
A Study of Engineering Materials and Processes Whose Characteristics May Be Explained by Considering the Behavior of Small Particles When Grouped Into Systems Such as Nuclei, Atoms, Gases, and Crystals
- J. R. Eaton(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
The development of this concept of the nature of the atom proved to be of great importance in the advancement of scientific knowledge. Many of the salient features of the Bohr-Rutherford atom may be illustrated by a mathematical analysis of a simplified one-electron system. This simple model is very valuable as a means of introducing certain concepts of quantum theory and will be discussed in detail. 5.2. ONE-ELECTRON ATOMS The one-electron Bohr-Rutherford atom formed by a nucleus and a single electron is very readily analyzed by classical methods yielding information which reflects, at least in a qualitative way, much of the performance of multi-electron atoms which must be handled by advanced mathematical methods. For STRUCTURE OF THE ATOM 57 the present analysis, the one-electron atom is further simplified by the reasonable assumption that since almost all of the mass is concentrated in the nucleus, the nucleus can be assumed fixed while the electron executes its orbital motion. Further, it is assumed that the electron moves in a circular path around the nucleus. Two forces must be considered in the analysis: (a) the electric field force between the positively charged nucleus and the negative electron, which obeys Coulomb's law, and (b) the centripetal force which is necessary to hold a moving object in a circular path. Obviously, these two forces are the same. It is of interest to note that while the analysis of this one-electron system is re-latively simple, a similar analysis on a two-electron system has never been car-ried to satisfactory completion.
- Robert K Logan(Author)
- 2010(Publication Date)
- World Scientific(Publisher)
In classical electromagnetic theory, on the other hand, the frequency of Bohr’s Atom 179 the periodic motion and the frequency of the subsequent electromagnetic radiation are identical. Finally, in classical mechanics, an electron must orbit the nucleus in an infinite number of paths, differing by only an infinitesimal amount of energy. In Bohr’s scheme, however, the number of orbits is severely limited by restricting the allowed orbits to those for which the angular momentum is equal to an integer times Planck’s constant, h, divided by 2 π . If we represent the angular momentum by L, then L = l h/2 π where l is an integer. The angular momentum of the electron is equal approximately to the product of its momentum times the radius of its orbit. This definition is exact if the orbit is a perfect circle. By placing this restriction on the angular momentum, Bohr was able to obtain Balmer’s formulae for the radiated frequencies of the hydrogen atom. Bohr was also able to calculate Rydberg’s constant, R y and showed that it is simply related to the mass of the electron, m e , the charge of the electron, e and Planck’s constant, h, by the formula R y = 2 π 2 m e e 4 /h 3 . This result, in which one of the fundamental constants of nature was related to the others, was a great success and insured the acceptance of Bohr’s model. This model not only explained Balmer’s formula for the hydrogen atom but it also explained Ritz’s combination principle. Let us label the quantum states or energy levels of the atom by E 1 , E 2 , ... , E n where E 1 is the energy of the ground state, E 2 is the energy of the first excited state, ... , and E n is the energy of the (n-1) th excited state. (See Fig. 19.1). Here, we refer to the higher energy orbits of the electrons of the atom as excited states. These electrons have absorbed energy, but do not retain the additional energy very long.
eBook - PDF- Morris Hein, Susan Arena, Cary Willard(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
He also visualized the electrons as revolving in orbits around the nucleus, like planets rotating around the sun, as shown in Figure 10.4. Bohr’s first paper in this field dealt with the hydrogen atom, which he described as a single electron revolving in an orbit about a relatively heavy nucleus. He applied the concept of energy quanta, proposed in 1900 by the German physicist Max Planck (1858–1947), to the observed line spectrum of hydrogen. Planck stated that energy is never emitted in a continu- ous stream but only in small, discrete packets called quanta (Latin, quantus, “how much”). From this, Bohr theorized that electrons have several possible energies corresponding to sev- eral possible orbits at different distances from the nucleus. Therefore, an electron has to be in one specific energy level; it cannot exist between energy levels. In other words, the energy of the electron is said to be quantized. Bohr also stated that when a hydrogen atom absorbed one or more quanta of energy, its electron would “jump” to a higher energy level. Bohr was able to account for spectral lines of hydrogen this way. A number of energy lev- els are available, the lowest of which is called the ground state. When electrons are located in these lowest energy orbitals, the atom is said to be in the ground state. Any other electron arrangement would be called an excited state of the atom. When an electron falls from a high energy level to a lower one (say, from the fourth to the second), a quantum of energy is emitted as light at a specific frequency, or wavelength (Figure 10.5). This light corresponds to one of the lines visible in the hydrogen spectrum (Figure 10.3). Several lines are visible in this spectrum, each one corresponding to a specific electron energy-level shift within the hydrogen atom. 410 434 486 656 (in nanometers) FIGURE 10.3 Line spectrum of hydrogen.
eBook - PDFIntroductory Chemistry
An Active Learning Approach
- Mark Cracolice, Edward Peters, Mark Cracolice(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Figure 11.18 is designed to help you think about how electrons behave in atoms. The Bohr model is an analogy to the solar system. Electrons orbit the nucleus like planets orbit the sun. This model is incorrect. The quantum model, on the right of Figure 11.18, is the mental model you should form. Each blue dot represents the position of a single electron at one point in time. Are you wondering why the dots appear as a scattering with areas that are dense and areas that are diffuse? Imagine that you can take a photograph of the single electron in a hydrogen atom. You center your camera so that the nucleus is at the convergence of the x, y, and z axes in the figure. When you take a picture, you make an overhead transparency of the single blue dot. Now you overlap a few hundred such transparencies. This will result in the quantum model illustration you see. The dashed line that is labeled as the r 90 distance is the radius that encloses 90% of the dots. In other words, 90% of the time, you will find the electron within this radius. Now look at the s orbital in Figure 11.19. It represents the same thing as the quantum model illustration in Figure 11.18. The radius of the sphere in Figure 11.19 corresponds to the r 90 radius of the circle in Figure 11.18. It illustrates in three dimensions the limit of a region in space, inside of which there is a 90% probability of finding an electron at any given instant. The p and d orbitals of Figure 11.19 also represent the regions in which the p and d electrons may be found 90% of the time. Your mental models of the s and p electron orbitals, plus your model of the behavior of an electron in an atom, combine to make the quantum mechanical model you should hold in your mind at this point in your chemistry studies. Thinking About Your Thinking Active Example 11.4 The Quantum Mechanical Model of the Atom Identify the true statements and explain, in general terms, why they are true.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
In his theory, Bohr made a number of assumptions and combined the new quantum ideas of Planck and Einstein with the traditional description of a particle in uniform circular motion. Adopting Planck’s idea of quantized energy levels (see Section 29.2), Bohr hypothe- sized that in a hydrogen atom there can be only certain values of the total energy (electron kinetic energy plus potential energy). These allowed energy levels correspond to different orbits for the electron as it moves around the nucleus, the larger orbits being associated with larger total energies. Animated Figure 30.5 illustrates two of the orbits. In addi- tion, Bohr assumed that an electron in one of these orbits does not radiate electromagnetic waves. For this reason, the orbits are called stationary orbits or stationary states. Bohr recognized that radiationless orbits violated the laws of physics, as they were then known. But the assumption of such orbits was necessary, because the traditional laws indicated that an electron radiates electromagnetic waves as it accelerates around a circular path, and the loss of the energy carried by the waves would lead to the collapse of the orbit. To incorporate Einstein’s photon concept (see Section 29.3), Bohr theorized that a photon is emitted only when the electron changes orbits from a larger one with a higher energy to a smaller one with a lower energy, as Animated Figure 30.5 indicates. How do electrons get into the higher-energy orbits in the first place? They get there by picking up energy when atoms collide, which happens more often when a gas is heated, or by acquiring energy when a high voltage is applied to a gas. When an electron in an initial orbit with a larger energy E i changes to a final orbit with a smaller energy E f , the emitted photon has an energy of E i − E f , consistent with the law of conservation of energy. But according to Einstein, the energy of a photon is hf, where f is its frequency and h is Planck’s constant.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The Bohr model does not correctly represent this aspect of reality at the atomic level. Check Your Understanding (The answers are given at the end of the book.) 4. In the Bohr model for the hydrogen atom, the closer the electron is to the nucleus, the smaller is the total energy of the electron. Is this also true in the quantum mechanical picture of the hydrogen atom? 5. In the quantum mechanical picture of the hydrogen atom, the orbital angular momentum of the electron may be zero in any of the possible energy states. For which energy state must the orbital angular momentum be zero? 6. Consider two different hydrogen atoms. The electron in each atom is in a different excited state, so that each electron has a different total energy. Is it possible for the electrons to have the same orbital angular momentum L, according to (a) the Bohr model and (b) quantum mechanics? 7. The magnitude of the orbital angular momentum of the electron in a hydrogen atom is observed to increase. According to (a) the Bohr model and (b) quantum mechanics, does this necessarily mean that the total energy of the electron also increases? 30.6 | The Pauli Exclusion Principle and the Periodic Table of the Elements Except for hydrogen, all electrically neutral atoms contain more than one electron, with the number given by the atomic number Z of the element. In addition to being attracted by the nucleus, the electrons repel each other. This repulsion contributes to the total energy of a multiple-electron atom. As a result, the one-electron energy expression for hydrogen, E n 5 2(13.6 eV) Z 2 /n 2 , does not apply to other neutral atoms. However, the simplest approach for dealing with a multiple-electron atom still uses the four quantum numbers n, ,, m , , and m s . n = 2 = 1 m = 0 (a) (b) n = 2 = 0 m = 0 Figure 30.13 The electron probability clouds for the hydrogen atom when (a) n 5 2, , 5 0, m , 5 0 and (b) n 5 2, , 5 1, m , 5 0.
eBook - PDFChemistry
Structure and Dynamics
- James N. Spencer, George M. Bodner, Lyman H. Rickard(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
To measure the position of an object on the macroscopic scale, the object must be illuminated with light so that we can see it. The photons of illuminating light are scattered by the object and strike our eye, allowing us to measure the position of the object. However, in the atomic world the energy of a photon of light can cause electrons to change energy states. This means that the electron is no longer in its original energy state, which means that either its velocity or its position is no longer the same. Therefore, if we precise know the electron’s posi- tion, we cannot know its velocity, and vice versa. The German physicist Werner Heisenberg formulated this idea in an uncer- tainty principle, which states that the better the position of an electron is known, the less well its velocity is known. For this reason we often describe an electron as a cloud of electron density within the atom instead of describing it as a sin- gle particle. Schrödinger’s model describes the regions in space, or orbitals, where elec- trons are most likely to be found. Instead of trying to tell us where the electron is at any time, this model gives the probability that an electron can be found in a given region of space. The model no longer tells us where the electron is; it only tells us where it might be. The Schrödinger model uses three coordinates, or three quantum numbers, to describe the orbitals in which electrons can be found. These coordinates are known as the principal (n), angular (l), and magnetic (m l ) quantum numbers. The quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom. The principal quantum number (n) describes the size of the orbital––orbitals for which n = 2 are larger than those for which n = 1, for example. As we have 3.14 ALLOWED COMBINATIONS OF QUANTUM NUMBERS 95
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