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.
Advertisements
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
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
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.
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
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?
Balmer series was observed and analysed before the other series. Can you suggest a reason for such an order?
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
Which of the following parameters are the same for all hydrogen-like atoms and ions in their ground states?
The Bohr radius is given by `a_0 = (∈_0h^2)/{pime^2}`. Verify that the RHS has dimensions of length.
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.
If l3 and l2 represent angular momenta of an orbiting electron in III and II Bohr orbits respectively, then l3: l2 is :
Use Bohr’s model of hydrogen atom to obtain the relationship between the angular momentum and the magnetic moment of the revolving electron.
What is the energy in joules released when an electron moves from n = 2 to n = 1 level in a hydrogen atom?
According to Bohr’s theory, the angular momentum of an electron in 5th orbit is ______.
Which of these statements correctly describe the atomic model according to classical electromagnetic theory?
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.
How will the energy of a hydrogen atom change if n increases from 1 to ∞?
On the basis of Bohr's theory, derive an expression for the radius of the nth orbit of an electron of hydrogen atom.
The de Broglie wavelength of an electron in the first Bohr’s orbit of hydrogen atom is equal to ______.
