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Atomic Spectra of GasesAll objects emit thermal radiation characterized by a continuous distribution of wavelengths. In sharp contrast to this continuousdistribution spectrum is the discrete line spectrum observed when a low-pressure

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Слайд 1Course of lectures «Contemporary Physics: Part2»
Lecture №10

Atomic Physics. Atomic Spectra

of Gases. Early Models of the Atom. Bohr’s Model of

the Hydrogen Atom. The Quantum Model of the Hydrogen Atom. The Wave Functions for Hydrogen.
Course of lectures «Contemporary Physics: Part2»Lecture №10Atomic Physics. Atomic Spectra of Gases. Early Models of the Atom.

Слайд 2Atomic Spectra of Gases
All objects emit thermal radiation characterized by

a continuous distribution of wavelengths. In sharp contrast to this

continuousdistribution spectrum is the discrete line spectrum observed when a low-pressure gas undergoes an electric discharge. Observation and analysis of these spectral lines is called emission spectroscopy.
Atomic Spectra of GasesAll objects emit thermal radiation characterized by a continuous distribution of wavelengths. In sharp

Слайд 3Atomic Spectra of Gases
Another form of spectroscopy very useful in

analyzing substances is absorption spectroscopy. An absorption spectrum is obtained

by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed. The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source.
Atomic Spectra of GasesAnother form of spectroscopy very useful in analyzing substances is absorption spectroscopy. An absorption

Слайд 4Atomic Spectra of Gases
In 1885, a Swiss schoolteacher, Johann Jacob

Balmer (1825–1898), found an empirical equation that correctly predicted the

wavelengths of four visible emission lines of hydrogen: Hα (red), Hβ (bluegreen), Hγ (blue-violet), and Hδ (violet). The four visible lines occur at the wavelengths 656.3 nm, 486.1 nm, 434.1 nm, and 410.2 nm. The complete set of lines is called the Balmer series.
Atomic Spectra of GasesIn 1885, a Swiss schoolteacher, Johann Jacob Balmer (1825–1898), found an empirical equation that

Слайд 5Atomic Spectra of Gases
Other lines in the spectrum of hydrogen

were found following Balmer’s discovery. These spectra are called the

Lyman, Paschen, and Brackett series after their discoverers. The wavelengths of the lines in these series can be calculated through the use of the following empirical equations:

No theoretical basis existed for these equations; they simply worked. The same constant RH appears in each equation, and all equations involve small integers.

Atomic Spectra of GasesOther lines in the spectrum of hydrogen were found following Balmer’s discovery. These spectra

Слайд 6Early Models of the Atom

Early Models of the Atom

Слайд 7Early Models of the Atom

Early Models of the Atom

Слайд 8Early Models of the Atom
Two basic difficulties exist with Rutherford’s

planetary model. As we previously an atom emits (and absorbs)

certain characteristic frequencies of electromagnetic radiation and no others, but the Rutherford model cannot explain this phenomenon. A second difficulty is that Rutherford’s electrons are undergoing a centripetal acceleration. According to Maxwell’s theory of electromagnetism, centripetally accelerated charges revolving with frequency f should radiate electromagnetic waves of frequency f. Unfortunately, this classical model leads to a prediction of self-destruction when applied to the atom. As the electron radiates, energy is carried away from the atom, the radius of the electron’s orbit steadily decreases, and its frequency of revolution increases. This process would lead to an ever-increasing frequency of emitted radiation and an ultimate collapse of the atom as the electron plunges into the nucleus .
Early Models of the AtomTwo basic difficulties exist with Rutherford’s planetary model. As we previously an atom

Слайд 9Bohr’s Model of the Hydrogen Atom
Bohr combined ideas from Planck’s

original quantum theory, Einstein’s concept of the photon, Rutherford’s planetary

model of the atom, and Newtonian mechanics to arrive at a semiclassical model based on some revolutionary ideas. The postulates of the Bohr theory as it applies to the hydrogen atom are as follows:

1. The electron moves in circular orbits around the proton under the influence of the electric force of attraction as shown in Figure.

Bohr’s Model of the Hydrogen AtomBohr combined ideas from Planck’s original quantum theory, Einstein’s concept of the

Слайд 10Bohr’s Model of the Hydrogen Atom
2. Only certain electron orbits

are stable. When in one of these stationary states, as

Bohr called them, the electron does not emit energy in the form of radiation, even though it is accelerating. Hence, the total energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion. Bohr’s model claims that the centripetally accelerated electron does not continuously emit radiation, losing energy and eventually spiraling into the nucleus, as predicted by classical physics in the form of Rutherford’s planetary model.
Bohr’s Model of the Hydrogen Atom2. Only certain electron orbits are stable. When in one of these

Слайд 11Bohr’s Model of the Hydrogen Atom
3. The atom emits radiation

when the electron makes a transition from a more energetic

initial stationary state to a lower-energy stationary state. This transition cannot be visualized or treated classically. In particular, the frequency f of the photon emitted in the transition is related to the change in the atom’s energy and is not equal to the frequency of the electron’s orbital motion. The frequency of the emitted radiation is found from the energy-conservation expression

where Ei is the energy of the initial state, Ef is the energy of the final state,

Bohr’s Model of the Hydrogen Atom3. The atom emits radiation when the electron makes a transition from

Слайд 12Bohr’s Model of the Hydrogen Atom
4. The size of an

allowed electron orbit is determined by a condition imposed on

the electron’s orbital angular momentum: the allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of

where me is the electron mass, v is the electron’s speed in its orbit, and r is the orbital radius.

Bohr’s Model of the Hydrogen Atom4. The size of an allowed electron orbit is determined by a

Слайд 13Bohr’s Model of the Hydrogen Atom
Using these four postulates, let’s

calculate the allowed energy levels and find quantitative values of

the emission wavelengths of the hydrogen atom.

the kinetic energy of the electron is

Bohr’s Model of the Hydrogen AtomUsing these four postulates, let’s calculate the allowed energy levels and find

Слайд 14Bohr’s Model of the Hydrogen Atom
The following expression for the

total energy of the atom:
The orbit with the smallest radius,

called the Bohr radius a0, corresponds to n =1 and has the value
Bohr’s Model of the Hydrogen AtomThe following expression for the total energy of the atom:The orbit with

Слайд 15Bohr’s Model of the Hydrogen Atom
The quantization of orbit radii

leads to energy quantization. Substituting
Inserting numerical values into this expression,

we find that
Bohr’s Model of the Hydrogen AtomThe quantization of orbit radii leads to energy quantization. SubstitutingInserting numerical values

Слайд 16We can calculate the frequency of the photon emitted when

the electron makes a transition from an outer orbit to

an inner orbit:

Bohr’s Model of the Hydrogen Atom

Because the quantity measured experimentally is wavelength, it is convenient to use c=fλ to express this Equation in terms of wavelength:

Remarkably, this expression, which is purely theoretical, is identical to the general form of the empirical relationships discovered by Balmer and Rydberg:

We can calculate the frequency of the photon emitted when the electron makes a transition from an

Слайд 17Bohr’s Model of the Hydrogen Atom
Soon after Bohr demonstrated that

these two quantities agree to within approximately 1%, this work

was recognized as the crowning achievement of his new quantum theory of the hydrogen atom. Furthermore, Bohr showed that all the spectral series for hydrogen have a natural interpretation in his theory. The different series correspond to transitions to different final states characterized by the quantum number nf.
Bohr’s Model of the Hydrogen AtomSoon after Bohr demonstrated that these two quantities agree to within approximately

Слайд 18Bohr’s Model of the Hydrogen Atom
Bohr showed that many mysterious

lines observed in the spectra of the Sun and several

other stars could not be due to hydrogen but were correctly predicted by his theory if attributed to singly ionized helium. In general, the number of protons in the nucleus of an atom is called the atomic number of the element and is given the symbol Z. To describe a single electron orbiting a fixed nucleus of charge +Ze, Bohr’s theory gives
Bohr’s Model of the Hydrogen AtomBohr showed that many mysterious lines observed in the spectra of the

Слайд 19Bohr’s Model of the Hydrogen Atom
Bohr’s Correspondence Principle
In our study

of relativity, we found that Newtonian mechanics is a special

case of relativistic mechanics and is usable only for speeds much less than c. Similarly,

quantum physics agrees with classical physics when the difference between quantized levels becomes vanishingly small.

This principle, first set forth by Bohr, is called the correspondence principle.

Bohr’s Model of the Hydrogen AtomBohr’s Correspondence PrincipleIn our study of relativity, we found that Newtonian mechanics

Слайд 20The Quantum Model of the Hydrogen Atom
The formal procedure for

solving the problem of the hydrogen atom is to substitute

the appropriate potential energy function into the Schrӧdinger equation, find solutions to the equation, and apply boundary conditions. The potential energy function for the hydrogen atom is that due to the electrical interaction between the electron and the proton:

The mathematics for the hydrogen atom is more complicated than that for the particle in a box because the atom is three-dimensional and U depends on the radial coordinate r. If the time-independent Schrцdinger equation is extended to three-dimensional rectangular coordinates, the result is

The Quantum Model of the Hydrogen AtomThe formal procedure for solving the problem of the hydrogen atom

Слайд 21The Quantum Model of the Hydrogen Atom
The first quantum number,

associated with the radial function R(r) of the full wave

function, is called the principal quantum number and is assigned the symbol n.

The orbital quantum number, symbolized l comes from the differential equation for f(u) and is associated with the orbital angular momentum of the electron. The orbital magnetic quantum number m, arises from the differential equation for g(f). Both l and m, are integers.

The Quantum Model of the Hydrogen AtomThe first quantum number, associated with the radial function R(r) of

Слайд 22The Quantum Model of the Hydrogen Atom
The application of boundary

conditions on the three parts of the full wave function

leads to important relationships among the three quantum numbers as well as certain restrictions on their values:
The Quantum Model of the Hydrogen AtomThe application of boundary conditions on the three parts of the

Слайд 23The Quantum Model of the Hydrogen Atom

The Quantum Model of the Hydrogen Atom

Слайд 24The Wave Functions for Hydrogen
Because the potential energy of the

hydrogen atom depends only on the radial distance r between

nucleus and electron, some of the allowed states for this atom can be represented by wave functions that depend only on r. For these states, f(u) and g(f) are constants. The simplest wave function for hydrogen is the one that describes the 1s state and is designated
The Wave Functions for HydrogenBecause the potential energy of the hydrogen atom depends only on the radial

Слайд 25The Wave Functions for Hydrogen
Therefore, the radial probability density function

is
the radial probability density function for the hydrogen atom in

its ground state:
The Wave Functions for HydrogenTherefore, the radial probability density function isthe radial probability density function for the

Слайд 26The Wave Functions for Hydrogen

The Wave Functions for Hydrogen

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