Question 288: Atoms & Nuclei

Question

At a given instant, say t = 0, two radioactive substance A and B have  equal activities. the ratio $${{{R_B}} \over {{R_A}}}$$ of their activities after time t itself decays with time t as e$$-$$3t. If the half-life of A is ln2, the half-life of B is :

Accepted Answer

We know,
Activity (R) = R
0
e
$$-$$$$\lambda $$t
Given that,
at  t = 0
R
A
= R
B
$$ \Rightarrow $$  R
0A
e
$$-$$$$\lambda $$
A
x0
=  R
0
B
e
$$-$$ $$\lambda $$
B
x0
$$ \Rightarrow $$  R
0A
= R
0B
Given that at time t,
$${{{R_B}} \over {{R_A}}} = {e^{^{ - 3t}}}$$
$$ \Rightarrow $$   $${{{R_{0B}}\,{e^{ - {\lambda _B}t}}} \over {{R_{0A}}\,{e^{ - {\lambda _A}t}}}}$$ = e
$$-$$3t
$$ \Rightarrow $$  $${e^{\left( {{\lambda _A} - {\lambda _B}} ight)t}} = {e^{ - 3t}}$$
$$ \Rightarrow $$  ($$\lambda $$
A
$$-$$ $$\lambda $$
B
)t = $$-$$3t
$$ \Rightarrow $$ $$\lambda $$
A
$$-$$ $$\lambda $$
B
= $$-$$ 3 . . . . .   (1)
We know half life, $${t_{{1 \over 2}}} = {{{{\ln }^2}} \over \lambda }$$
$$ 	herefore $$  Half life of A is, $${{t_{{1 \over 2}}} = {{{{\ln }^2}} \over {{\lambda _A}}}}$$
$$ 	herefore $$  $$\lambda $$
A
= $${{\ln 2} \over {{t_{{1 \over 2}}}}}$$
= $${{\ln 2} \over {\ln 2}}$$
= 1
From equation (1)     we get,
$$\lambda $$
A
$$-$$ $$\lambda $$
B
= $$-$$ 3
$$ \Rightarrow $$  1 $$-$$ $$\lambda $$
B
= $$-$$ 3
$$ \Rightarrow $$  $$\lambda $$
B
= 4
$$ 	herefore $$  Half life of B is $$\left( {{t_{{1 \over 2}}}} ight) = {{\ln 2} \over {{\lambda _B}}} = {{\ln 2} \over 4}$$

Answer

4ln2

Answer

$${{\ln 2} \over 2}$$

Answer

$${{\ln 2} \over 4}$$

Answer

2ln2

Atoms & Nuclei

At a given instant, say t = 0, two radioactive substance A and B have equal activities. the ratio ${{{R_B}} \over {{R_A}}} of their activities after time t itself decays with time t as e-$3t. If the half-life of A is ln2, the half-life of B is :

Choose the correct answer: