GATE-2016
53. The correct relationship between the residual amount P of the
parent radionuclide and amount D of the daughter product in a
radioactive decay is
A)
D= P ( eλt-1)
B)
D = P ( 1-eλt)
C)
D = P / (eλt-1)
D)
D= P / (1-eλt)
Solution:
If
at any time we have ‘P’ parent nuclei the number that decay in
the follwoing interval ‘dt’ is P( λ*dt). ( λ- decay
rate)
The
change ‘dp’ in the number of ‘P’ parent nuclei in a time
interval ‘dt’ due to spontaneous decay is
dP
= (-λ * P * dt) -------------------(1)
(dP/
dt) = -λ*P -----------(2)
which
has a solution
P=
(P0
*
e-λt)
--------------(3)
D
(daughter nuclies) is the difference between ‘P’ and ‘P0’
P0
-parent nuclides at initial stage
D=
P0
-
P --------------(4)
substitute
eq(3) value in eq(4)
D=
P0
-(P0
*
e-λt)
D=
P0
( 1- e-λt)
--------------(5)
The
original amout ‘P0’
of
the parent nuclide is not known, a rock
sample contains a residual
amount ‘P’ of the parent nuclide an
amount ‘D’ of the
daughter product eliminate
‘P0’.
from
eq (3)
P=
P0
e-λt
P0
= Peλt
-------------------(6)
substitute
eq(6) in eq(4)
D=
P0-
P
D=
Peλt
– P
D=
P ( eλt
– 1)
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