You are watching: How did bohr explain the line spectrum of hydrogen?
The principle of the photon emerged from experimentation with thermal radiation, electromagnetic radiation emitted as the an outcome of a source’s temperature, which produces a constant spectrum that energies.The photoelectric effect detailed indisputable evidence for the presence of the photon and thus the particle-like habits of electromagnetic radiation. However, more direct proof was needed to verify the quantized nature of energy in every matter. In this section, we explain how monitoring of the communication of atoms v visible light noted this evidence.
Although objects in ~ high temperature emit a continuous spectrum that electromagnetic radiation, a various kind that spectrum is observed as soon as pure samples of individual elements are heated. Because that example, once a high-voltage electrical discharge is passed through a sample of hydrogen gas at low pressure, the result individual diverted hydrogen atoms brought about by the dissociation of H2 emit a red light. Uneven blackbody radiation, the color of the irradiate emitted by the hydrogen atoms does not depend greatly on the temperature the the gas in the tube. When the emitted light is passed v a prism, just a few narrow currently of particular wavelengths, referred to as a line spectrum, are observed rather than a constant range of wavelengths (Figure (PageIndex1)). The light emitted through hydrogen atoms is red because, the its four characteristic lines, the most intense heat in the spectrum is in the red part of the visible spectrum, at 656 nm. With sodium, however, us observe a yellow color due to the fact that the many intense present in that is spectrum are in the yellow part of the spectrum, at around 589 nm.
Such emission spectra were observed for plenty of other facets in the so late 19th century, which presented a major an obstacle because classic physics was unable to describe them. Component of the explanation is provided by Planck’s equation: the monitoring of just a few values that λ (or ( u )) in the heat spectrum supposed that only a few values that E to be possible. Hence the power levels the a hydrogen atom had actually to be quantized; in various other words, just states the had specific values of power were possible, or allowed. If a hydrogen atom could have any worth of energy, climate a continuous spectrum would have been observed, similar to blackbody radiation.
In 1885, a Swiss math teacher, Johann Balmer (1825–1898), proved that the frequencies of the lines it was observed in the visible region of the spectrum of hydrogen right a an easy equation that can be expressed as follows:
< u=constant; left ( dfrac12^2-dfrac1n^^2 ight ) label6.3.1>
where n = 3, 4, 5, 6. As a result, this lines are known as the Balmer series. The sweden physicist john Rydberg (1854–1919) consequently restated and expanded Balmer’s an outcome in the Rydberg equation:
< dfrac1lambda =Re; left ( dfrac1n^2_1-dfrac1n^2_2 ight ) label6.3.2>
where (n_1) and also (n_2) are positive integers, (n_2 > n_1), and ( Re ) the Rydberg constant, has a worth of 1.09737 × 107 m−1.
Johann Balmer (1825–1898)
A mathematics teacher in ~ a second school for girls in Switzerland, Balmer to be 60 year old when he composed the document on the spectral currently of hydrogen the made him famous.
Balmer published just one other document on the topic, which appeared when he to be 72 year old.
Like Balmer’s equation, Rydberg’s an easy equation explained the wavelengths of the visible lines in the emissions spectrum the hydrogen (with n1 = 2, n2 = 3, 4, 5,…). More important, Rydberg’s equation also predicted the wavelength of other collection of currently that would be it was observed in the emission spectrum of hydrogen: one in the ultraviolet (n1 = 1, n2 = 2, 3, 4,…) and one in the infrared (n1 = 3, n2 = 4, 5, 6). Unfortunately, scientists had not however developed any type of theoretical justification because that an equation that this form.
In 1913, a Danish physicist, Niels Bohr (1885–1962; Nobel compensation in Physics, 1922), propose a theoretical version for the hydrogen atom that defined its emissions spectrum. Bohr’s model compelled only one assumption: The electron moves about the cell nucleus in circular orbits that have the right to have only details allowed radii. Rutherford’s earlier model the the atom had likewise assumed the electrons relocated in circular orbits roughly the nucleus and that the atom was held together by the electrostatic attraction between the positively fee nucleus and also the negatively charged electron. Although us now understand that the assumption of one orbits to be incorrect, Bohr’s understanding was come propose that the electron could occupy only particular regions that space.
Using classic physics, Niels Bohr proved that the power of one electron in a specific orbit is provided by
< E_n=dfrac-Re hcn^2 label6.3.3>
where ( Re ) is the Rydberg constant, h is Planck’s constant, c is the rate of light, and also n is a optimistic integer corresponding to the number assigned come the orbit, v n = 1 matching to the orbit closest to the nucleus. In this version n = ∞ corresponds to the level whereby the power holding the electron and also the nucleus together is zero. In that level, the electron is unbound from the nucleus and also the atom has been separated into a negatively fee (the electron) and also a positively charged (the nucleus) ion. In this state the radius the the orbit is additionally infinite. The atom has been ionized.
In his last years, he dedicated himself to the tranquil application of atomic physics and also to addressing political problems arising from the advancement of atom weapons.
As n decreases, the energy holding the electron and also the nucleus with each other becomes increasingly negative, the radius of the orbit shrinks and an ext energy is necessary to ionize the atom. The orbit through n = 1 is the shortest lying and most tightly bound. The negative sign in Equation ( ef6.3.3) suggests that the electron-nucleus pair is an ext tightly bound (i.e. At a reduced potential energy) as soon as they are close to each various other than as soon as they are far apart. Because a hydrogen atom through its one electron in this orbit has actually the lowest possible energy, this is the ground state (the many stable setup of electrons for an aspect or a compound) for a hydrogen atom. Together n increases, the radius that the orbit increases; the electron is farther indigenous the proton, which results in a less stable plan with greater potential power (Figure (PageIndex2a)). A hydrogen atom through an electron in an orbit through n > 1 is thus in an excited state, characterized as any kind of arrangement of electrons that is higher in energy than the floor state. When an atom in one excited state undergoes a change to the ground state in a procedure called decay, that loses power by emitting a photon whose energy corresponds to the difference in energy in between the two claims (Figure (PageIndex1)).
Using atoms to Time
In modern applications, electron transitions are used in timekeeping that requirements to be exact. Telecommunications systems, such as cell phones, count on time signals that are precise to in ~ a millionth of a second per day, as space the gadgets that control the us power grid. Global positioning device (GPS) signals must be exact to within a billionth the a 2nd per day, i m sorry is identical to acquiring or losing no more than one second in 1,400,000 years. Quantifying time requires finding an event with one interval the repeats on a consistent basis.
To attain the accuracy forced for modern purposes, physicists have actually turned come the atom. The present standard provided to calibrate clocks is the cesium atom. Supercooled cesium atoms are placed in a vacuum chamber and bombarded through microwaves whose frequencies are closely controlled. Once the frequency is precisely right, the atom absorb enough energy to experience an electronic shift to a higher-energy state. Decay to a lower-energy state emits radiation. The microwave frequency is continually adjusted, serving together the clock’s pendulum.
In 1967, the 2nd was defined as the expression of 9,192,631,770 oscillations that the resonant frequency that a cesium atom, referred to as the cesium clock. Research study is right now under method to construct the following generation of atomic clocks the promise to be even an ext accurate. Such gadgets would permit scientists to screen vanishingly faint electromagnetic signals created by nerve pathways in the mind and geologists to measure variations in gravitational fields, which reason fluctuations in time, that would assist in the discovery of oil or minerals.
Example (PageIndex1): The Lyman Series
The so-called Lyman series of lines in the emissions spectrum that hydrogen corresponds to transitions from various excited claims to the n = 1 orbit. Calculate the wavelength of the lowest-energy heat in the Lyman collection to three far-reaching figures. In what an ar of the electromagnetic spectrum does the occur?
Given: lowest-energy orbit in the Lyman series
Asked for: wavelength of the lowest-energy Lyman line and corresponding an ar of the spectrum
Strategy:substitute the ideal values into Equation ef6.3.2(the Rydberg equation) and solve for (lambda). Use number 2.2.1 to locate the an ar of the electromagnetic spectrum corresponding to the calculate wavelength.
We have the right to use the Rydberg equation to calculation the wavelength:
< dfrac1lambda =-Re left ( dfrac1n_2^2 - dfrac1n_1^2 ight ) onumber >
A for the Lyman series, n1 = 1. The lowest-energy line is as result of a shift from the n = 2 to n = 1 orbit since they space the the next in energy.
< dfrac1lambda =-Re left ( dfrac1n_2^2 - dfrac1n_1^2 ight )=1.097 imes m^-1left ( dfrac11-dfrac14 ight )=8.228 imes 10^6; m^-1 onumber >
It turns out the spectroscopists (the people who study spectroscopy) use cm-1 quite than m-1 together a common unit. Wavelength is inversely proportional come energy but frequency is straight proportional as presented by Planck"s formula,(E=h u).
Spectroscopists often talk around energy and frequency together equivalent. The cm-1 unit is particularly convenient. The infrared variety is roughly 200 - 5,000 cm-1, the clearly shows from 11,000 to 25.000 cm-1 and also the UV in between 25,000 and 100,000 cm-1. The devices of cm-1 are called wavenumbers, although people often verbalize it as inverse centimeters. We can convert the prize in component A to cm-1.
< widetilde u =dfrac1lambda =8.228 imes 10^6cancelm^-1left (dfraccancelm100;cm ight )=82,280: cm^-1 onumber >
This emission line is referred to as Lyman alpha. That is the the strongest atomic emission heat from the sun and also drives the stclairdrake.netistry of the upper setting of all the planets, producing ions through stripping electron from atoms and also molecules. It is completely absorbed through oxygen in the top stratosphere, dissociating O2 molecule to O atoms which react with other O2 molecules to kind stratospheric ozone.
B This wavelength is in the ultraviolet region of the spectrum.
Exercise (PageIndex1): The Pfund Series
The Pfund collection of currently in the emissions spectrum the hydrogen coincides to transitions from greater excited states to the n = 5 orbit. Calculate the wavelength the the second heat in the Pfund collection to three significant figures. In which an ar of the spectrum does it lie?Answer
4.65 × 103 nm; infrared
Bohr’s model of the hydrogen atom gave specific explanation for its observed emission spectrum. The following are his vital contributions come our knowledge of atom structure:
Unfortunately, Bohr can not explain why the electron need to be limited to details orbits. Also, regardless of a an excellent deal the tinkering, such together assuming that orbits could be ellipses quite than circles, his model might not quantitatively explain the emission spectra of any kind of element various other than hydrogen (Figure (PageIndex5)). In fact, Bohr’s model worked only for varieties that consisted of just one electron: H, He+, Li2+, and so forth. Scientists required a basic change in their means of thinking about the electronic structure of atoms to advance beyond the Bohr model.
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The power States the the Hydrogen Atom
Thus much we have explicitly considered only the emissions of light by atoms in excited states, i m sorry produces an emission spectrum (a spectrum created by the emissions of irradiate by atoms in excited states). The converse, absorption of light by ground-state atom to create an excited state, can also occur, producing an absorb spectrum (a spectrum developed by the absorption of light by ground-state atoms).