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प्रश्न
Using Bohr’s postulates, obtain the expression for the total energy of the electron in the stationary states of the hydrogen atom. Hence draw the energy level diagram showing how the line spectra corresponding to Balmer series occur due to transition between energy levels.
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उत्तर
According to Bohr’s postulates, in a hydrogen atom, a single alectron revolves around a nucleus of charge +e. For an electron moving with a uniform speed in a circular orbit os a given radius, the centripetal force is provided by Columb force of attraction between the electron and the nucleus. The gravitational attraction may be neglected as the mass of electron and proton is very small.
So,
`(mv^2)/r = (ke^2)/r^2`
or `mv^2 = (ke^2)/r .............. (1)`
where m = mass of electron
r = radius of electronic orbit
v = velocity of electron.
Again,
`mvr = (nh)/(2π)`
or `v = (nh)/(2πmr)`
From eq.(1), we get,
`m((nh)/(2πmr)^2) = (ke^2)/r`
`=> r = (n^2h^2)/(4π^2kme^2).....................(2)`
(i) Kinetic energy of electron,
`E_k = 1/2 mv^2 = (ke^2)/(2r)`
Using eq (2), we get
`Ek =ke^2/2 (4π^2kme^2)/(n^2h^2)`
=`(4π^2kme^2)/(n^2h^2)`
`(2π^2k^2me^4)/(n^2h^2)`
(ii) Potential energy
`E_p = -(k(e) xx (e))/r = - (ke^2) / r `
Using eq (2), we get
`E^p =-ke^2 xx (4π^2kme^2)/(n^2h^2)`
= `-(4π^2k^2me^4)/(n^2h^2)`
Hence, total energy of the electron in the nth orbit
`E =E_p+E_k =(4π^2k^2me^4)/(n^2h^2)+(2π^2k^2me^4)/(n^2h^2) =- (2π^2k^2me^4)/(n^2h^2) =- (13.6)/n^2 eV `
When the electron in a hydrogen atom jumps from higher energy level to the lower energy level, the difference of energies of the two energy levels is emitted as a radiation of particular wavelength. It is called a spectral line.
In H-atom, when an electron jumps from the orbit ni to orbit nf, the wavelength of the emitted radiation is given by,
`1/λ = R (1/n_f^2 -1/n_i^2)`
Where,
R → Rydberg’s constant = 1.09678 ×107 m−1
For Balmer series, nf = 2 and ni = 3, 4, 5, …
`1/λ = R (1/2^2 -1/n_i^2)`
Where, ni = 3, 4, 5, …
These spectral lines lie in the visible region.
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संबंधित प्रश्न
Using Bohr’s postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number ni) to the lower state, (nf).
When electron in hydrogen atom jumps from energy state ni = 4 to nf = 3, 2, 1, identify the spectral series to which the emission lines belong.
The difference in the frequencies of series limit of Lyman series and Balmer series is equal to the frequency of the first line of the Lyman series. Explain.
Which of the following is/are CORRECT according to Bohr's atomic theory?
(I) Energy is emitted when electron moves from a higher stationary state to a lower one.
(II) Orbits are arranged concentrically around the nucleus in an increasing order of energy.
(III) The energy of an electron in the orbit changes with time.
In form of Rydberg's constant R, the wave no of this first Ballmer line is
The Bohr model for the spectra of a H-atom ______.
- will not be applicable to hydrogen in the molecular from.
- will not be applicable as it is for a He-atom.
- is valid only at room temperature.
- predicts continuous as well as discrete spectral lines.
Taking the Bohr radius as a0 = 53 pm, the radius of Li++ ion in its ground state, on the basis of Bohr’s model, will be about ______.
The inverse square law in electrostatics is |F| = `e^2/((4πε_0).r^2)` for the force between an electron and a proton. The `(1/r)` dependence of |F| can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. If photons had a mass mp, force would be modified to |F| = `e^2/((4πε_0)r^2) [1/r^2 + λ/r]`, exp (– λr) where λ = mpc/h and h = `h/(2π)`. Estimate the change in the ground state energy of a H-atom if mp were 10-6 times the mass of an electron.
State Bohr's postulate to explain stable orbits in a hydrogen atom. Prove that the speed with which the electron revolves in nth orbit is proportional to `(1/"n")`.
The wavelength in Å of the photon that is emitted when an electron in Bohr orbit with n = 2 returns to orbit with n = 1 in H atom is ______ Å. The ionisation potential of the ground state of the H-atom is 2.17 × 10−11 erg.
What is the velocity of an electron in the 3rd orbit of hydrogen atom if its velocity in the 1st orbit is v0?
