Advertisements
Advertisements
प्रश्न
The earth revolves round the sun due to gravitational attraction. Suppose that the sun and the earth are point particles with their existing masses and that Bohr's quantization rule for angular momentum is valid in the case of gravitation. (a) Calculate the minimum radius the earth can have for its orbit. (b) What is the value of the principal quantum number n for the present radius? Mass of the earth = 6.0 × 10−24 kg. Mass of the sun = 2.0 × 1030 kg, earth-sun distance = 1.5 × 1011 m.
Advertisements
उत्तर १
Given:
Mass of the earth, me = 6.0 × 1024 kg
Mass of the sun, ms = 2.0 × 1030 kg
Distance between the earth and the sun, d = 1.5 × 1111 m
According to the Bohr's quantization rule,
Angular momentum, L =`(nh)/(2pi)`
⇒`mvr=(nh)/(2pi) ....(1)`
Here,
n = Quantum number
h = Planck's constant
m = Mass of electron
r = Radius of the circular orbit
v = Velocity of the electron
Squaring both the sides, we get
`m_e^2v^2r^2 = (n^2h^2)/(4pi^2)` ....(2)
Gravitational force of attraction between the earth and the sun acts as the centripetal force.
`F = (Gm_em_s)/r^2 = (m_ev^2)/r`
`rArr v^2 = (Gm_s)/r` ......(3)
Dividing (2) by (3), we get
`m_e^2 r = (n^2h2)/(4pi^2Gm_s)`
(a) For n = 1,
`r = sqrt((h^2)/(4pi^2Gm_sm_e^2))`
`r = sqrt((6.63xx10^-34)^2/(4xx(3.14)^2xx(6.67xx10^-11)xx(6xx10^24)^2xx(2xx10^30))`
`r = 2.29xx10^-138 m`
`r =2.3xx10^-138 m`
(b)
From (2), the value of the principal quantum number (n) is given by
`n^2 = (m_e^2xxrxx4xxpixxGxxm_s)/h^2`
`rArr n = sqrt(m_e^2xxrxx4xxpixxGxxm_s)/h^2`
`n=sqrt(((6xx10^24)^2xx(1.5xx10^11)xx4xx(3.14)^2xx(6.67xx10^-11)xx(2xx10^30))/(6.6xx10^-34))`
n = 2.5 ×1074
उत्तर २
Given:
Mass of the earth, me = 6.0 × 1024 kg
Mass of the sun, ms = 2.0 × 1030 kg
Distance between the earth and the sun, d = 1.5 × 1111 m
According to the Bohr's quantization rule,
Angular momentum, L =`(nh)/(2pi)`
⇒`mvr=(nh)/(2pi)`
....(1)
Here,
n = Quantum number
h = Planck's constant
m = Mass of electron
r = Radius of the circular orbit
v = Velocity of the electron
Squaring both the sides, we get
`m_e^2v^2r^2 = (n^2h^2)/(4pi^2)` ....(2)
Gravitational force of attraction between the earth and the sun acts as the centripetal force.
`F = (Gm_em_s)/r^2 = (m_ev^2)/r`
`rArr v^2 = (Gm_s)/r` ......(3)
Dividing (2) by (3), we get
`m_e^2 r = (n^2h2)/(4pi^2Gm_s)`
(a) For n = 1,
`r = sqrt((h^2)/(4pi^2Gm_s_e^2))`
`r = sqrt((6.63xx10^-34)^2/(4xx(3.14)^2xx(6.67xx10^-11)xx(6xx10^24)^2xx(2xx10^30))`
`r = 2.29xx10^-138 m`
`r =2.3xx10^-138 m`
(b)
`n^2 = (m_e^2xxrxx4xxpixxGxxm_s)/h^2`
`rArr n = sqrt(m_e^2xxrxx4xxpixxGxxm_s)/(6.6xx10^-34)^2`
n = 2.5 ×1074
APPEARS IN
संबंधित 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
Explain, giving reasons, which of the following sets of quantum numbers are not possible.
- n = 0, l = 0, ml = 0, ms = + ½
- n = 1, l = 0, ml = 0, ms = – ½
- n = 1, l = 1, ml = 0, ms = + ½
- n = 2, l = 1, ml = 0, ms = – ½
- n = 3, l = 3, ml = –3, ms = + ½
- n = 3, l = 1, ml = 0, ms = + ½
Using Bohr's postulates, derive the expression for the orbital period of the electron moving in the nth orbit of hydrogen atom ?
Which of the following parameters are the same for all hydrogen-like atoms and ions in their ground states?
When a photon stimulates the emission of another photon, the two photons have
(a) same energy
(b) same direction
(c) same phase
(d) same wavelength
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?
What is the energy in joules released when an electron moves from n = 2 to n = 1 level in a hydrogen atom?
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.
The wavelength of the first time line of Ballmer series is 6563 A°. The Rydberg constant for hydrogen is about:-
The energy of an electron in hth orbit of hydrogen atom is –13.6/n2ev energy required to excite the electron from the first orbit to the third orbit is
The simple Bohr model cannot be directly applied to calculate the energy levels of an atom with many electrons. This is because ______.
When an electron falls from a higher energy to a lower energy level, the difference in the energies appears in the form of electromagnetic radiation. Why cannot it be emitted as other forms of energy?
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")`.
An electron in H-atom makes a transition from n = 3 to n = 1. The recoil momentum of the H-atom will be ______.
The electron in a hydrogen atom first jumps from the third excited state to the second excited state and subsequently to the first excited state. The ratio of the respective wavelengths, λ1/λ2, of the photons emitted in this process is ______.
What is the energy of an electron in stationary state corresponding to n = 2?
According to Bohr's theory, the radius of the nth Bohr orbit of a hydrogen like atom of atomic number Z is proportional to ______.
The radius of the nth orbit in the Bohr model of hydrogen is proportional to ______.
The wavelength of the second line of the Balmer series in the hydrogen spectrum is 4861 Å. Calculate the wavelength of the first line of the same series.
The figure below is the Energy level diagram for the Hydrogen atom. Study the transitions shown and answer the following question:
- State the type of spectrum obtained.
- Name the series of spectrum obtained.
