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    4.4: Bohr's Theory of the Hydrogen Emission Spectrum

    4.4: Bohr's Theory of the Hydrogen Emission Spectrum

    Last updated Apr 27, 2016

    4.3: The Photoelectric Effect

    4.5: de Broglie's Postulate

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    To introduce the concept of absorption and emission line spectra and describe the Balmer equation to describe the visible lines of atomic hydrogen.

    Describe Rydberg's theory for the hydrogen spectra.

    Interpret the hydrogen spectrum in terms of the energy states of electrons.

    The first person to realize that white light was made up of the colors of the rainbow was Isaac Newton, who in 1666 passed sunlight through a narrow slit, then a prism, to project the colored spectrum on to a wall. This effect had been noticed previously, of course, not least in the sky, but previous attempts to explain it, by Descartes and others, had suggested that the white light became colored when it was refracted, the color depending on the angle of refraction. Newton clarified the situation by using a second prism to reconstitute the white light, making much more plausible the idea that the white light was composed of the separate colors. He then took a monochromatic component from the spectrum generated by one prism and passed it through a second prism, establishing that no further colors were generated. That is, light of a single color did not change color on refraction. He concluded that white light was made up of all the colors of the rainbow, and that on passing through a prism, these different colors were refracted through slightly different angles, thus separating them into the observed spectrum.

    Atomic Line Spectrum

    The spectrum of hydrogen, which turned out to be crucial in providing the first insight into atomic structure over half a century later, was first observed by Anders Angstrom in Uppsala, Sweden, in 1853. His communication was translated into English in 1855. Angstrom, the son of a country minister, was a reserved person, not interested in the social life that centered around the court. Consequently, it was many years before his achievements were recognized, at home or abroad (most of his results were published in Swedish).

    Most of what is known about atomic (and molecular) structure and mechanics has been deduced from spectroscopy. Figure 1.4.1 shows two different types of spectra. A continuous spectrum can be produced by an incandescent solid or gas at high pressure (blackbody radiation, for example, is a continuum). An emission spectrum can be produced by a gas at low pressure excited by heat or by collisions with electrons. An absorption spectrum results when light from a continuous source passes through a cooler gas, consisting of a series of dark lines characteristic of the composition of the gas.

    Figure 4.4.1 4.4.1

    : Continuous spectrum and two types of line spectra. from Wikipedia.

    In 1802, William Wollaston in England had discovered (perhaps by using a thinner slit or a better prism) that in fact the solar spectrum itself had tiny gaps - there were many thin dark lines in the rainbow of colors. These were investigated much more systematically by Joseph von Fraunhofer, beginning in 1814. He increased the dispersion by using more than one prism. He found an "almost countless number" of lines. He labeled the strongest dark lines A, B, C, D, etc. Frauenhofer between 1814 and 1823 discovered nearly 600 dark lines in the solar spectrum viewed at high resolution and designated the principal features with the letters A through K, and weaker lines with other letters (Table

    4.4.1 4.4.1

    ). Modern observations of sunlight can detect many thousands of lines. It is now understood that these lines are caused by absorption by the outer layers of the Sun.

    Table 4.4.1 4.4.1

    : Major Fraunhofer lines, and the elements they are associated with

    y O2 898.765 Z O2 822.696 A O2 759.370 B O2 686.719 C H 656.281 a O2 627.661 D1 Na 589.592 D2 Na 588.995 D3 or d He 587.5618

    Fraunhofer Absorption Lines

    The Fraunhofer lines are typical spectral absorption lines. These dark lines are produced whenever a cold gas is between a broad spectrum photon source and the detector. In this case, a decrease in the intensity of light in the frequency of the incident photon is seen as the photons are absorbed, then re-emitted in random directions, which are mostly in directions different from the original one. This results in an absorption line, since the narrow frequency band of light initially traveling toward the detector, has been turned into heat or re-emitted in other directions.

    Figure 4.4.2 4.4.2

    : Spectrum of blue sky. Dips are present at the Fraunhofer line wavelengths. from Wikipedia.

    By contrast, if the detector sees photons emitted directly from a glowing gas, then the detector often sees photons emitted in a narrow frequency range by quantum emission processes in atoms in the hot gas, resulting in an emission line. In the Sun, Fraunhofer lines are seen from gas in the outer regions of the Sun, which are too cold to directly produce emission lines of the elements they represent.

    Source : chem.libretexts.org

    Calculate the shortest and longest wavelengths of the class 12 physics CBSE

    Calculate the shortest and longest wavelengths of the Balmer series of hydrogen atoms Given R 1097 times 107m

    Calculate the shortest and longest wavelengths of the Balmer series of Hydrogen atoms. Given

    R=1.097× 10 7 /m. R=1.097×107/m. Verified 110k+ views 1 likes

    Hint: In this question use the direct formula for the Balmer series of hydrogen spectrum that is

    According to Balmer series of hydrogen

    ( H 2 ) (H2)

    spectrum the wavelength is given as,

    1 λ =R[ 1 n 2 1 − 1 n 2 2 ] 1λ=R[1n12−1n22]

    where R is molar gas constant,

    n 1 n1 and n 2 n2

    are the spectrum numbers where,

    n 1 n1 is always less than n 2 n2

    . Consider the fact that for shortest wavelength

    n 2 n2

    should be infinity and for longest wavelength values of

    n 1 n1 and n 2 n2

    are 1 and 2. This will help solve this problem.

    Formula used:

    1 λ =R[ 1 n 2 1 − 1 n 2 2 ] 1λ=R[1n12−1n22]

    ............................ (1)

    Where, λ λ

    = wavelength, R = molar gas constant

    =1.097× 10 7 /m =1.097×107/m and n 1 n1 and n 2 n2

    are the spectrum numbers where,

    n 1 n1 is always less than n 2 n2 . Where n 1 n1

    varies from 1 to infinity and

    n 2 n2

    varies from 2 to infinity.

    Complete step-by-step solution:For shorter wavelength: For shorter wavelengths the value of

    n 1 n1

    should be least and the value of

    n 2 n2

    should be maximum. The values of

    n 1 n1 and n 2 n2

    are 2 and infinity (

    ∞ ∞ ), as [ 1 ∞ =0] [1∞=0]

    Now substitute all the values in equation (1) we have,

    ⇒ 1 λ =1.097× 10 7 [ 1 2 2 − 1 ∞ 2 ]

    ⇒1λ=1.097×107[122−1∞2]

    Now simplify this we have,

    ⇒ 1 λ =1.097× 10 7 [ 1 4 −0]= 1.097× 10 7 4

    ⇒1λ=1.097×107[14−0]=1.097×1074

    ⇒λ= 4 1.097× 10 7 =3.646× 10 −7

    ⇒λ=41.097×107=3.646×10−7

    meter. ⇒λ=3646× 10 −10 ⇒λ=3646×10−10 meter.

    Now as we all know that

    1 A o =1× 10 −10 1Ao=1×10−10 m Therefore, λ=3646 A o λ=3646Ao ∴ ∴

    The shortest wavelength is

    3646 A o 3646Ao .

    For longest wavelength: For the longest wavelength the value of

    n 1 n1

    should be least and the value of

    n 2 n2

    also should be least.The values of

    n 1 n1 and n 2 n2

    are 2 and 3.Now substitute all the values in equation (1) we have,

    ⇒ 1 λ =1.097× 10 7 [ 1 2 2 − 1 3 2 ]

    ⇒1λ=1.097×107[122−132]

    Now simplify this we have,

    ⇒ 1 λ =1.097× 10 7 [ 1 4 − 1 9 ]=1.097× 10 7 ( 5 36 )

    ⇒1λ=1.097×107[14−19]=1.097×107(536)

    ⇒λ= 36 5 1.097× 10 7 =( 36 5 )0.91157× 10 −7 =6.563 × 10 −7

    ⇒λ=3651.097×107=(365)0.91157×10−7=6.563×10−7

    meter. ⇒λ=6563× 10 −10 ⇒λ=6563×10−10 meter.

    Now as we all know that

    1 A o =1× 10 −10 1Ao=1×10−10 m Therefore, λ=6563 A o λ=6563Ao ∴ ∴

    The longest wavelength is

    6563 A o 6563Ao .

    Note: Talking about a series of spectral emission lines for the hydrogen atom that can be achieved from the result of electron transmission from some higher level even down to the energy level of principal quantum 2 can be termed as a Balmer series.

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