Using the Bohr model, we can calculate the energy of an electron and the radius of its orbit in any one-electron system. Quantized energy means that the electrons can possess only certain discrete energy values; values between those quantized values are not permitted.
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. 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 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 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, but restricts these to quantized values dependent on a single quantum number, n. Skip to content Chapter 6. Electronic Structure and Periodic Properties of Elements. 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 light emitted or absorbed by hydrogen atoms.
Example 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. Answer: 6. Chemistry End of Chapter Exercises Why is the electron in a Bohr hydrogen atom bound less tightly when it has a quantum number of 3 than when it has a quantum number of 1?
What does it mean to say that the energy of the electrons in an atom is quantized? Using the Bohr model, determine the energy, in joules, necessary to ionize a ground-state hydrogen atom.
Show your calculations. The electron volt eV is a convenient unit of energy for expressing atomic-scale energies.
Check Your Understanding What are the limits of the Lyman series? Can you see these spectral lines? The key to unlocking the mystery of atomic spectra is in understanding atomic structure. Scientists have long known that matter is made of atoms. According to nineteenth-century science, atoms are the smallest indivisible quantities of matter. This scientific belief was shattered by a series of groundbreaking experiments that proved the existence of subatomic particles, such as electrons, protons, and neutrons.
The electron was discovered and identified as the smallest quantity of electric charge by J. Around , E. In , Rutherford and Thomas Royds used spectroscopy methods to show that positively charged particles of -radiation called -particles are in fact doubly ionized atoms of helium. In the Rutherford gold foil experiment also known as the Geiger—Marsden experiment , -particles were incident on a thin gold foil and were scattered by gold atoms inside the foil see Types of Collisions.
The outgoing particles were detected by a scintillation screen surrounding the gold target for a detailed description of the experimental setup, see Linear Momentum and Collisions. When a scattered particle struck the screen, a tiny flash of light scintillation was observed at that location. By counting the scintillations seen at various angles with respect to the direction of the incident beam, the scientists could determine what fraction of the incident particles were scattered and what fraction were not deflected at all.
If the plum pudding model were correct, there would be no back-scattered -particles. However, the results of the Rutherford experiment showed that, although a sizable fraction of -particles emerged from the foil not scattered at all as though the foil were not in their way, a significant fraction of -particles were back-scattered toward the source. This kind of result was possible only when most of the mass and the entire positive charge of the gold atom were concentrated in a tiny space inside the atom.
In , Rutherford proposed a nuclear model of the atom. The atom also contained negative electrons that were located within the atom but relatively far away from the nucleus. Ten years later, Rutherford coined the name proton for the nucleus of hydrogen and the name neutron for a hypothetical electrically neutral particle that would mediate the binding of positive protons in the nucleus the neutron was discovered in by James Chadwick.
Rutherford is credited with the discovery of the atomic nucleus; however, the Rutherford model of atomic structure does not explain the Rydberg formula for the hydrogen emission lines. In the same way as Earth revolves around the sun, the negative electron in the hydrogen atom can revolve around the positive nucleus. However, an accelerating charge radiates its energy.
Classically, if the electron moved around the nucleus in a planetary fashion, it would be undergoing centripetal acceleration, and thus would be radiating energy that would cause it to spiral down into the nucleus.
Such a planetary hydrogen atom would not be stable, which is contrary to what we know about ordinary hydrogen atoms that do not disintegrate. Moreover, the classical motion of the electron is not able to explain the discrete emission spectrum of hydrogen.
These three postulates of the early quantum theory of the hydrogen atom allow us to derive not only the Rydberg formula, but also the value of the Rydberg constant and other important properties of the hydrogen atom such as its energy levels, its ionization energy, and the sizes of electron orbits.
The hydrogen atom, as an isolated system, must obey the laws of conservation of energy and momentum in the way we know from classical physics. Having this theoretical framework in mind, we are ready to proceed with our analysis. As a charged particle, the electron experiences an electrostatic pull toward the positively charged nucleus in the center of its circular orbit.
This electrostatic pull is the centripetal force that causes the electron to move in a circle around the nucleus. Therefore, the magnitude of centripetal force is identified with the magnitude of the electrostatic force:. Here, denotes the value of the elementary charge. The negative electron and positive proton have the same value of charge, When Figure is combined with the first quantization condition given by Figure , we can solve for the speed, and for the radius,.
We see from Figure that the size of the orbit grows as the square of n. This means that the second orbit is four times as large as the first orbit, and the third orbit is nine times as large as the first orbit, and so on. The radius of the first Bohr orbit is called the Bohr radius of hydrogen , denoted as Its value is obtained by setting in Figure :.
We can substitute in Figure to express the radius of the n th orbit in terms of. This result means that the electron orbits in hydrogen atom are quantized because the orbital radius takes on only specific values of given by Figure , and no other values are allowed. The total energy of an electron in the n th orbit is the sum of its kinetic energy and its electrostatic potential energy Utilizing Figure , we find that.
Recall that the electrostatic potential energy of interaction between two charges and that are separated by a distance is Here, is the charge of the nucleus in the hydrogen atom the charge of the proton , is the charge of the electron and is the radius of the n th orbit. Now we use Figure to find the potential energy of the electron:. The total energy of the electron is the sum of Figure and Figure :. Note that the energy depends only on the index n because the remaining symbols in Figure are physical constants.
The value of the constant factor in Figure is. Now we can see that the electron energies in the hydrogen atom are quantized because they can have only discrete values of given by Figure , and no other energy values are allowed. This set of allowed electron energies is called the energy spectrum of hydrogen Figure. We identify the energy of the electron inside the hydrogen atom with the energy of the hydrogen atom.
Note that the smallest value of energy is obtained for so the hydrogen atom cannot have energy smaller than that. This smallest value of the electron energy in the hydrogen atom is called the ground state energy of the hydrogen atom and its value is.
The lowest value of n is 1; this gives a smallest possible orbital radius of 0. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions.
However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels. Bohr did not believe in the existence of photons. These jumps reproduce the frequency of the k- th harmonic of orbit n. For sufficiently large values of n so-called Rydberg states , the two orbits involved in the emission process have nearly the same rotation frequency so that the classical orbital frequency is not ambiguous.
But for small n or large k , the radiation frequency has no unambiguous classical interpretation. Use the link below to answer the following questions:. Skip to main content. Electrons in Atoms. Search for:. Figure 1. Summary The Bohr model postulates that electrons orbit the nucleus at fixed energy levels.
Orbits further from the nucleus exist at higher energy levels. When electrons return to a lower energy level, they emit energy in the form of light. What was the Bohr model based on? What did Bohr believe about the orbits?
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