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7 Nuclear Physics – I
Fig. 7.18
7.86 The flow of heat in a material placed between the walls of a coaxial cylinder
is given by
dQ
dt
=
2π L
ln
r2
r1
(T 1 − T 2 )
( 1 )
Number of decays of radon atoms per second
dN /dt = 100 × 10
−3
× 3.7 × 10
10
= 3.7 × 10
9 disintegration/second
Energy deposited by α
s = 3.7 × 10
9
× 5.5 MeV/s
= 2.035 × 10
10 MeV/s = 3.256 × 10
−7 J = 0.779 × 10
−3 Cal/s
Using the values, k = 0.025 Cal cm
−2 s
−1 C
−1
, L = 5 cm, r 1 = 2 mm and
r 2 = 6 mm in (1), and solving for (T 1 − T 2 ) we find (T 1 − T 2 ) = 1.09
◦ C
7.87 In the series decay A→B→C, if λ A < λ B the transient equilibrium occurs
when
N B /N A = λ A /(λ B − λ A )
Here A = Uranium and B = Radium
λ A = 1/τ A = 0.693/4.5 × 10
9 year
−1
, λ B = 1/τ B = 0.693/1,620 year
−1
N (Rad)/N (U) ≈ 1,620/4.5 × 10
9
= 1/2.78 × 10
6
7.88 Use the law of radioactivity
N A = N A
0 exp(−λ A t)
( 1 )
N B = N B
0 exp(−λ B t)
( 2 )
Dividing (2) by (1)
N B /N A = exp(λ A − λ B )t
(Because N A
0
= N B
0 )
Take log e on both sides
t =
1
(λ A − λ B ) ln
N B
N A