Advertisements
Advertisements
प्रश्न
State Bohr’s third postulate for hydrogen (H2) atom. Derive Bohr’s formula for the wave number. Obtain expressions for longest and shortest wavelength of spectral lines in ultraviolet region for hydrogen atom
Advertisements
उत्तर
Third postulate (Transition and frequency condition):-
As long as electron remains in one of the stationary orbits, it does not radiate energy. Whenever an electron jumps from higher stationary orbit to lower stationary orbit, it radiates energy equal to the difference in energies of the electron in the two orbits.
Bohr’s formula for spectral lines in hydrogen spectrum:-
i. Let, En = Energy of electron in nth higher orbit
Ep = Energy of electron in pth lower orbit
ii. According to Bohr’s third postulate,
En - Ep = hv
∴ v = (Ea - Ep)/h ...........(1)
iii. `"But "E_n=-(me^4)/(8epsilon_0^2"h"^2"n"^2)` ............(2)
`E_p=-(me^4)/(8epsilon_0^2"h"^2"p"^2` .............(3)
iv. From equations (1), (2) and (3),
`"v"=(((-me^4)/(8epsilon_0^2"h"^2"n"^2))-(-(me^4)/(8epsilon_0^2"h"^2"p"^2)))/h`
`therefore"v"=(me^4)/(8epsilon_0^2h^3)[-1/n^2+1/p^2]`
`thereforec/lambda=(me^4)/(8epsilon_0^2h^3)[1/p^2-1/n^2]` [∵ v = c/λ]
where, c = speed of electromagnetic radiation
`therefore1/lambda=(me^4)/(8epsilon_0^2h^3c)[1/p^2-1/n^2]`
v. `1/lambda=R[1/p^2-1/n^2]` ............(4)
`"where, "(me^4)/(8epsilon_0^2h^3c)=R="Rydberg’s constant"`
Equation (4) represents Bohr’s formula for hydrogen spectrum.
vi. 1/λ is called wave number (`bar"v"` ) of the line.
`thereforebar"v"=1/lambda=R(1/p^2-1/n^2)`
For longest wavelength in ultraviolet region (Lyman series),
p = 1; n = 2
`therefore1/lambda_(L_1)=R(1/1^2-1/2^2)=(3R)/4`
`thereforelambda_(L_1)=4/(3R)`
For shortest wavelength in ultraviolet region
P = 1; n = ∞
`therefore1/lambda_(L_2)=R(1/1^2-1/oo^2)=R`
`thereforelambda_(L_2)=1/R`
APPEARS IN
संबंधित प्रश्न
(i) State Bohr's quantization condition for defining stationary orbits. How does the de Broglie hypothesis explain the stationary orbits?
(ii) Find the relation between three wavelengths λ1, λ2 and λ3 from the energy-level diagram shown below.
Draw a neat, labelled energy level diagram for H atom showing the transitions. Explain the series of spectral lines for H atom, whose fixed inner orbit numbers are 3 and 4 respectively.
if `E_p` and `E_k` represent potential energy and kinetic energy respectively, of an orbital electron, then, according to B9hr's theory:
a)`E_k = -E_p"/"2`
b) `E_k = -E_p`
c) `E_k = -2E_p`
d) `E_k = 2E_p`
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 electron in hydrogen atom is initially in the third excited state. What is the maximum number of spectral lines which can be emitted when it finally moves to the ground state?
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.
The numerical value of ionization energy in eV equals the ionization potential in volts. Does the equality hold if these quantities are measured in some other units?
Suppose, the electron in a hydrogen atom makes transition from n = 3 to n = 2 in 10−8 s. The order of the torque acting on the electron in this period, using the relation between torque and angular momentum as discussed in the chapter on rotational mechanics is
Calculate the magnetic dipole moment corresponding to the motion of the electron in the ground state of a hydrogen atom.
Radiation coming from transition n = 2 to n = 1 of hydrogen atoms falls on helium ions in n = 1 and n = 2 states. What are the possible transitions of helium ions as they absorbs energy from the radiation?
The light emitted in the transition n = 3 to n = 2 in hydrogen is called Hα light. Find the maximum work function a metal can have so that Hα light can emit photoelectrons from it.
Obtain Bohr’s quantisation condition for angular momentum of electron orbiting in nth orbit in hydrogen atom on the basis of the wave picture of an electron using de Broglie hypothesis.
In which of the following systems will the wavelength corresponding to n = 2 to n = 1 be minimum?
According to Bohr's model of hydrogen atom, an electron can revolve round a proton indefinitely, if its path is ______.
If the radius of first electron orbit in hydrogen atom be r then the radius of the fourth orbit ill be ______.
The energy of an electron in an excited hydrogen atom is - 3.4 eV. Calculate the angular momentum of the electron according to Bohr's theory. (h = 6.626 × 10-34 Js)
When an electric discharge is passed through hydrogen gas, the hydrogen molecules dissociate to produce excited hydrogen atoms. These excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula
`bar(v) = 109677 1/n_1^2 - 1/n_f^2`
What points of Bohr’s model of an atom can be used to arrive at this formula? Based on these points derive the above formula giving description of each step and each term.
On the basis of Bohr's model, the approximate radius of Li++ ion in its ground state ifthe Bohr radius is a0 = 53 pm :
According to the Bohr theory of H-atom, the speed of the electron, its energy and the radius of its orbit varies with the principal quantum number n, respectively, as:
How will the energy of a hydrogen atom change if n increases from 1 to ∞?
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.
In Bohr's atomic model of hydrogen, let K. P and E are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level:
The energy of an electron in the first Bohr orbit of the H-atom is −13.6 eV. The energy value of an electron in the excited state of Li2+ is ______.
A 20% efficient bulb emits light of wavelength 4000 Å. If the power of the bulb is 1 W, the number of photons emitted per second is ______.
[Take, h = 6.6 × 10-34 J-s]
State three postulates of Bohr's theory of hydrogen atom.
Find the angular momentum of an electron revolving in the second orbit in Bohr's hydrogen atom.
How much is the angular momentum of an electron when it is orbiting in the second Bohr orbit of hydrogen atom?
Calculate the radius of the second orbit of He+.
