Modern Physics for Scientists and Engineers
Modern Physics for Scientists and Engineers
4th Edition
ISBN: 9781133103721
Author: Stephen T. Thornton, Andrew Rex
Publisher: Cengage Learning
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Chapter 4, Problem 26P
To determine

The isotope shifts for each of the four visible Balmer series wavelengths for deuterium and tritium compared with hydrogen.

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Answer to Problem 26P

The isotope shifts for each of the four visible Balmer series wavelengths for deuterium and tritium compared with hydrogen is calculated below.

Explanation of Solution

The isotope shift of the spectral lines is defined as the shift in wavelength sue to different isotopic mass of given elements.

Value of Rydberg constant for atomic hydrogen is 0.99946R , for deuterium is  0.99973R for tritium is 0.99982R.

Here, R Rydberg constant for infinite nuclear mass and its value is 1.097373×107m1 .

Lower state of Balmer series is nl=2, The symbol α refers to nu=3, β refers to nu=4, γ refers to nu=5  and δ refers to nu=6.

Write the expression for isotope shifted wavelengths.

    1λ=R(1nl21nu2)                                                                           

Here, λ is the wavelength, nl is the number of lower energy state and nu is the number of upper energy state.

Wavelength of Hα line is calculated when nl=2 and nu=3.

    1λ=R(122132)=0.13889R        (I)

Wavelength of Hβ line is calculated when nl=2 and nu=4.

    1λ=R(122142)=0.1875R        (II)

Wavelength of Hγ line is calculated when nl=2 and nu=5.

    1λ=R(122152)=0.21R        (III)

Wavelength of Hδ line is calculated when nl=2 and nu=6.

    1λ=R(122162)=0.222R        (IV)

Conclusion:

Value of Rydberg constant for atomic hydrogen is 0.99946R , for deuterium is  0.99973R for tritium is 0.99982R.

 For λ(Hα,hydrogen):

Substitute 0.99946R for R and 1.097373×107m1   for R in equation (I).   

    1λ=0.13889(0.99946R)=0.13889(0.99946(1.097373×107m-1))=1523318.3m-1λ=656.47nm

Similarly value of λ(Hα,deuterium)=656.29nm and value of λ(Hα,tritium)=656.23nm

For λ(Hβ,hydrogen):

Substitute 0.99946R for R and 1.097373×107m1   for R in equation (II).   

    1λ=0.1875(0.99946R)=0.1875(0.99946(1.097373×107m-1))=2056463.3m-1λ=486.27nm

Similarly value of λ(Hβ,deuterium)=486.14nm and value of λ(Hβ,tritium)=486.10nm.

For λ(Hγ,hydrogen):

Substitute 0.99946R for R and 1.097373×107m1   for R in equation (II).   

    1λ=0.21(0.99946R)=0.21(0.99946(1.097373×107m-1))=2303238.88m-1λ=434.17nm

Similarly value of λ(Hγ,deuterium)=434.05nm and value of λ(Hγ,tritium)=434.02nm.

For λ(Hδ,hydrogen):

Substitute 0.99946R for R and 1.097373×107m1   for R in equation (II).   

    1λ=0.22(0.99946R)=0.22(0.99946(1.097373×107m-1))=2412916.9m-1λ=410.29nm

Similarly value of λ(Hδ,deuterium)=410.18nm and value of λ(Hδ,tritium)=410.15nm

Thus, the isotope shifts for each of the four visible Balmer series wavelengths for deuterium and tritium compared with hydrogen is calculated above.

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