Advertisements
Advertisements
प्रश्न
Consider a neutron and an electron bound to each other due to gravitational force. Assuming Bohr's quantization rule for angular momentum to be valid in this case, derive an expression for the energy of the neutron-electron system.
उत्तर
According to Bohr's quantization rule,
Angular momentum of electron, L = `(nh)/(2pi)`
`rArr mv_er = (nh)/(2pi)`
`rArr v_e = (nh)/(2pirm_e)`
...(1)
Here,
n = Quantum number
h = Planck's constant
m = Mass of the electron
r = Radius of the circular orbit
ve = Velocity of the electron
Let mn be the mass of neutron.
On equating the gravitational force between neutron and electron with the centripetal acceleration,
`(Gm_nm_e)/r^2 = m_ev^2`
`rArr (Gm)/r = v^2`
Squaring (1) and dividing it by (2), we have
`(m_e^2v^2r)/(v_2) = (n^2h^2r)/(4pi^2Gm_n)`
`rArr m_e^2r^2 = (n^2h^2r)/(4pi^2Gm_n)`
`rArr r = (n^2h^2)/(4piGm_nm_e^2)`
`v_e = (nh)/(2pirm_e`
`rArr v_e = (nh)/(2pim_e xx ((n^2h^2)/(4pi^2Gm_nm_e^2))`
`rArr v_e = (2piGm_nm_e)/(nh)`
Kinetic energy of the electrons, K = `1/2 m_ev_e^2`
=`1/2 m_e ((2piGm_nm_e)/(nh))^2`
`= (4pi^2G^2m_n^2m_e^3)/(2n^2h^2)`
Potential energy of the neutron,`p = (-Gm_nm_e)/r`
Substituting the value of r in the above expression,
`P = (-Gm_em_n4pi^2Gm_um_e^2 )/(n^2h^2)`
`P = (-4pi^2G^2m_n^2m_e^3)/(n^2h^2)`
Total energy = K + P =`-(2pi^2G^2m_n^2m_e^2)/(2n^2h^2)= -(pi^2G^2m_n^2m_e^2)/(n^2h^2)`
APPEARS IN
संबंधित प्रश्न
Explain, giving reasons, which of the following sets of quantum numbers are not possible.
(a) n = 0, l = 0, ml = 0, ms = + ½
(b) n = 1, l = 0, ml = 0, ms = – ½
(c) n = 1, l = 1, ml = 0, ms = + ½
(d) n = 2, l = 1, ml = 0, ms = – ½
(e) n = 3, l = 3, ml = –3, ms = + ½
(f) n = 3, l = 1, ml = 0, ms = + ½
Calculate the energy required for the process
\[\ce{He^+_{(g)} -> He^{2+}_{(g)} + e^-}\]
The ionization energy for the H atom in the ground state is 2.18 ×10–18 J atom–1
(a) Using the Bohr’s model calculate the speed of the electron in a hydrogen atom in the n = 1, 2, and 3 levels.
(b) Calculate the orbital period in each of these levels.
The radius of the innermost electron orbit of a hydrogen atom is 5.3 × 10−11 m. What are the radii of the n = 2 and n = 3 orbits?
State Bohr postulate of hydrogen atom that gives the relationship for the frequency of emitted photon in a transition.
Using Bohr's postulates, derive the expression for the total energy of the electron in the stationary states of the hydrogen atom ?
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.
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?
A neutron having kinetic energy 12.5 eV collides with a hydrogen atom at rest. Nelgect the difference in mass between the neutron and the hydrogen atom and assume that the neutron does not leave its line of motion. Find the possible kinetic energies of the neutron after the event.
A neutron moving with a speed υ strikes a hydrogen atom in ground state moving towards it with the same speed. Find the minimum speed of the neutron for which inelastic (completely or partially) collision may take place. The mass of neutron = mass of hydrogen = 1.67 × 10−27 kg.v
According to Bohr's theory, an electron can move only in those orbits for which its angular momentum is integral multiple of ____________.
The spectral line obtained when an electron jumps from n = 5 to n = 2 level in hydrogen atom belongs to the ____________ series.
A particle has a mass of 0.002 kg and uncertainty in its velocity is 9.2 × 10−6 m/s, then uncertainty in position is ≥ ____________.
(h = 6.6 × 10−34 J s)
Why was a change in the Bohr Model of atom required? Due to which important development (s), concept of movement of an electron in an orbit was replaced by, the concept of probability of finding electron in an orbital? What is the name given to the changed model of atom?
Derive an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n – 1). Also show that for large values of n, this frequency equals to classical frequency of revolution of an electron.
The wavelength of the first time line of Ballmer series is 6563 A°. The Rydberg constant for hydrogen is about:-
For the ground state, the electron in the H-atom has an angular momentum = h, according to the simple Bohr model. Angular momentum is a vector and hence there will be infinitely many orbits with the vector pointing in all possible directions. In actuality, this is not true ______.
The line at 434 nm in the Balmer series of the hydrogen spectrum corresponds to a transition of an electron from the nth to second Bohr orbit. The value of n is ______.
What is the velocity of an electron in the 3rd orbit of hydrogen atom if its velocity in the 1st orbit is v0?
