Homework SourseIn the Bohr model of the hydrogen atom the energy levels are
In the Bohr model of the hydrogen atom, the energy levels are given by:
The orbital radii are given by:
The orbital speeds are given by:
a) What is the frequency of the radiation emitted by the atom when electron drops from the nth to the (n-1)th energy level?
b) What is the frequency with which an electron in the nth orbit circles the nucleus?
c) Which of th frequencies in (a) and (b) is a classical result, and which a quantum mechanical result?
d) Show that for n >> 1 (a) and (b) are the same.
Solution
a) En = - m e^4 /( 32 pi^2 e0^2 h^2 n^2)
E(n-1) = - m e^4 /( 32 pi^2 e0^2 h^2 (n-1)^2)
energy diff, deltaE = En - E(n-1) = - ((n-1)^2 - n^2 )(m e^4) / (32 pi^2 e0^2 h^2 n^2 (n-1)^2)
= (2n -1)(m e^4) / (32 pi^2 e0^2 h^2 n^2 (n-1)^2)
deltaE = h * f
(2n -1)(m e^4) / (32 pi^2 e0^2 h^2 n^2 (n-1)^2) = h * f
f = (2n -1)(m e^4) / (32 pi^2 e0^2 h^3 n^2 (n-1)^2)
b) f = v / 2pir
f = (e^2 / 4 pi e0 n h ) / 2pi(4 pi e0 h^2 n^2 / m e^2)
f = m e^4 / (32 pi^2 e0^2 n^3 h^3)
c) a - quantum mechanical
b - classical
d)
f = (2n -1)(m e^4) / (32 pi^2 e0^2 h^3 n^2 (n-1)^2)
f = n(2 - 1/n) (m e^4) / (32 pi^2 e0^2 h^3 n^4 (1 - 1/n)^2 )
f = (2 - 1/n) (m e^4) / (32 pi^2 e0^2 h^3 n^3 (1 - 1/n)^2 )
for n > > 1
1/n is aprroax zero.
f = m e^4 / (32 pi^2 e0^2 h^3 n^3 )
which is same as B.
