E of the electron in a hydrogen atom is the sum of its kinetic and

Md. Kamrul Hasan Reza

Department of Physics

Khulna University of Engineering & Technology Khulna-9203, Bangladesh

Tel.: +880-41-769468~75 Ext. 587(O), 588 (R)

e-mail: mkhreza@phy.kuet.ac.bd, mkhreza1@gmail.com Website : www.kuet.ac.bd/phy/reza/

Instagram: mkhreza1@ Md. Kamrul Hasan Reza

Twitter: mkhreza1@ Md. Kamrul Hasan Reza www.youtube.com/c/MdKamrulHasanReza

Welcome to my Class (Trial-II)

Physics Ph 1109

12:00 PM July 21, 2020

COVID-19 Precautions

Don’t be afraid

Be aware of the pandemic

Use appropriate outfits if you compelled to go out

Try to maintain proper diet

Do not forget to exercise (at least one hour) regularly

Try to follow the guidelines of WHO and Bangladesh Government

 Try to stay at home

Full name: Niels Henrik David Bohr

Born: October 7, 1885, Copenhagen, Denmark

Died: November 18, 1962, Carlsberg, Copenhagen, Denmark

Atom Model: 1913

Nobel Prize in Physics: 1922

for his services in the investigation of the structure of atoms and of the radiation emanating from them

Bohr Atom Model

Atomic Spectra

Fig. 1: An idealized spectrometer

Fig.2: Some of the principal lines in the emission spectra of hydrogen, helium, and mercury

Fig.3: The dark lines in the absorption spectrum of an element correspond to bright lines in its emission spectrum.

(1)

The quantity R, known as the Rydberg constant, has the value Rydberg constant R = 1.097 x 10⁷ m^-1 = 0.01097 nm^-1

Fig. 4: The Balmer series of hydrogen. The H_α line is red, the H_β line is blue, the H_γ and H_δ lines are violet, and the other lines are in the near ultraviolet.

(2)

In the ultraviolet the Lyman series contains the wavelengths specified by the formula

(3)

(4)

(5)

References:

Perspective of modern Physics- Arthur Beiser Concepts of modern Physics- Arthur Beiser and internet resources

In the infrared, three spectral series have been found whose lines have the wavelengths specified by the formulas

Fig. 5: The spectral series of hydrogen. The wavelengths in each series are related by simple formulas.

The Bohr Atom

Let us examining the wave behavior of an electron in orbit around a hydrogen nucleus The de Broglie wavelength of this electron

Fig.6: Force balance in the hydrogen atom

The condition for orbit stability F_c = F_e

The centripetal force, The electrostatic force,

…….(6)

…….(7)

By substituting 5.3 x 10^-11 m for the radius r of the electron

…….(8)

Now the electron wavelength

=33x10^-11m

This wavelength is exactly the same as the circumference of the electron orbit

2πr = 33x10^-11 m

Fig.7: The orbit of the electron in a hydrogen atom corresponds to a complete electron de Broglie wave joined on itself.

Circumference = 8 wavelengths

Circumference = 4 wavelengths Circumference = 4 wavelengths

Fig. 8: Some modes of vibration of a wire loop.

Fig.9: A fractional number of wavelengths cannot persist because destructive interference will occur.

An electron can circle a nucleus only if its orbit contains an integral number of de Broglie wavelengths.

Condition for orbit stability

n λ=2 π r_n n=1,2,3,…..

…….(9)

Using λ from eq. (8)

so the possible electron orbits are those whose radii are

The radius of the innermost orbit is customarily called the Bohr radius of the Hydrogen Atom

a₀= r₁= 5.292x 10^-11 m

The other radii are given in terms of a₀ by the formula r_n = n² a_o

…….(10)

…….(11)

A photon is emitted when an electron jumps from one energy level to a lower level

The total energy E of the electron in a hydrogen atom is the sum of its kinetic and potential energies

E = T + V

Energy Levels and Spectra

Kinetic energy, T=½mv² and potential energy,

The electron energy E_n is given in terms of the orbit radius r_n

…….(11)

…….(12)

Using v from eq. (7)

Substituting r_n from eq. (10), we see that

…….(13)

= 13.6 eV E₁ = -2.18x10^-18 jule = -13.6 eV

Fig. 10: Energy levels of the hydrogen atom.

Initial energy - final energy = photon energy E_i –E_f = hν …….(14)

Hence the energy difference between these states is

The frequency ν of the photon released in this transition is

…….(15)

…….(16)

…….(17)

In terms of photon wavelength λ

Since n_i is greater than n_f in each case, there be an excess of energy to be given off as a photon.

…….(17)

The calculated formulas for the first five series are

…….(18)

…….(19)

…….(20)

…….(21)

…….(22)

Our final step is to compare the value of the constant term in the above equations with that of the Rydberg constant in Eqs. (1) to (5). The value of the constant term is

= 1.097 x 10⁷ m^-1

Fig.11: Spectral lines originate in transitions between energy levels.

Bohr Correspondence Principle

The greater the quantum number, the closer quantum physics approaches classical physics

According to electromagnetic theory, an electron moving in a circular orbit radiates em waves whose frequencies are equal to its frequency of revolution and to harmonics (that is, integral multiples) of that frequency. In a hydrogen atom the electron’s speed is

Hence the frequency of revolution f of the electron is

The radius r_n of a stable orbit is given in terms of its quantum number n by Eq. (10)

and so the frequency of revolution is

…….(23)

a hydrogen atom dropping from the n_ith energy level to the n_fth energy levelemits a photon whose frequency is

Let us write n for the initial quantum number n_i and n – p (where p 1, 2, 3, . . .) for the final quantum number n_f.

With this substitution

When n_i and n_f are both very large, n is much greater than p, and

…….(24)

When p = 1, the frequency of the radiation is exactly the same as the frequency of rotation f of the orbital electron given in Eq. (23). Multiples of this frequency are radiated when p = 2, 3, 4, . . . .

Hence both quantum and classical pictures of the hydrogen atom make the same predictions in the limit of very large quantum numbers.

I Thank you