© examsnet.com
Question : 37
Total: 37
(a) Derive the law of radioactive decay N = N o e − λ t
(b) The half life of92 238 U α 4.5 × 10 9
(c) What fraction of the initial mass of a radioactive substance will decay in five half-life periods ?
OR
(a) State the postulates of Bohr's model of hydrogen atom and derive the expression for Bohr radius.
(b) Find the ratio of the longest and the shortest wavelengths amongst the spectral lines of Balmer series in the spectrum of hydrogen atom.
(b) The half life of
(c) What fraction of the initial mass of a radioactive substance will decay in five half-life periods ?
OR
(a) State the postulates of Bohr's model of hydrogen atom and derive the expression for Bohr radius.
(b) Find the ratio of the longest and the shortest wavelengths amongst the spectral lines of Balmer series in the spectrum of hydrogen atom.
Solution:
(a) Radioactive decay law : The rate of disintegration of a radioactive substance at an instant is directly proportional to the number of nuclei in the Radioactive substance at that time.
N = N o e − λ t
Let us consider a radioactive substance havingN o ( t = 0 )
After time( t )
Ifd N d t
−
α N
Or, −
= λ N
Here,λ
Equation (i) can also be written as
= − λ dt
Integrating
ln N = − λ t + K
Here,K
At t = 0 , N = N 0
∴ K = ln N 0
SubstitutingK
ln N = − λ t + ln N 0
ln
= − λ t
Or,
= e − λ t
∴ N = N 0 e − λ t
(b) Half-life= ( ) × ln 2
Or,4.5 × 10 9 = ( ) × ln 2
Or, Mean life= (
)
Or, Mean life= (
) = 6.5 × 10 9
(c) No. of half lives= 5
So,
= (
) 5 =
=
So, fraction of mass decayed= 1 −
=
OR
(a) Postulates of Bohr Model of Hydrogen atom:
Postulate - I: The electrons revolve in a circular orbit around the nucleus. The electrostatic force of attraction between the positively charged nucleus and negatively charged electrons provide necessary centripetal force for circular motion.
Postulate - II: The electrons can revolve only in certain selected orbits in which angular momentum of electrons is equal to the integral multiple
h
Postulate - III: When an electrons jumps from higher energy orbit to lower energy orbit, energy is radiated in the form of a quantum or photon of energyh v
Expression for Bohr radius :
Let us consider
m = mass of an electron
r = radius of the circular orbit in which the
electron is revolving
v = speed of electron
− e = charge of electron
From 1 st postulate
Centripetal force = Electrostatic force
=
∴ v 2 =
From2 nd
m v r =
Or,v =
Or,v 2 =
Comparing eqns (1) and (2),
=
Or,
=
∴ Bohr radius = r =
(b) Shortest wavelength in Balmer series:
= R (
−
)
∴ λ S =
Longest wavelength in Balmer series:
∴
= R (
−
)
∴ λ L =
So,
=
=
Let us consider a radioactive substance having
After time
If
Here,
Equation (i) can also be written as
Integrating
Here,
Substituting
Or,
(b) Half-life
Or,
Or, Mean life
Or, Mean life
(c) No. of half lives
So,
So, fraction of mass decayed
OR
(a) Postulates of Bohr Model of Hydrogen atom:
Postulate - I: The electrons revolve in a circular orbit around the nucleus. The electrostatic force of attraction between the positively charged nucleus and negatively charged electrons provide necessary centripetal force for circular motion.
Postulate - II: The electrons can revolve only in certain selected orbits in which angular momentum of electrons is equal to the integral multiple
Postulate - III: When an electrons jumps from higher energy orbit to lower energy orbit, energy is radiated in the form of a quantum or photon of energy
Expression for Bohr radius :
Let us consider
From
Or,
Or,
Comparing eqns (1) and (2),
(b) Shortest wavelength in Balmer series:
Longest wavelength in Balmer series:
So,
© examsnet.com
Go to Question: