Question 302: Atoms & Nuclei

Question

A radioactive nucleus A with a half life T, decays into a nucleus B. At t = 0, there is no nucleus B. At
sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by :

Accepted Answer

Let initially there are total N
0
number of nuclei.
At time t $${{{N_B}} \over {{N_A}}}$$ = 0.3 (given)
$$ \Rightarrow $$ $${{N_B}}$$ = 0.3$${{N_A}}$$
N
0
= N
A
+ N
B
= N
A
+ 0.3N
A
$$ 	herefore $$ N
A
= $${{{N_0}} \over {1.3}}$$
As we know N
t
= N
0
e
Ã¢ÂÂ $$\lambda $$t
$$ \Rightarrow $$ $${{{N_0}} \over {1.3}}$$ = N
0
e
Ã¢ÂÂ $$\lambda $$t
$$ \Rightarrow $$ $${1 \over {1.3}}$$ = e
Ã¢ÂÂ $$\lambda $$t
$$ \Rightarrow $$ ln(1.3) = $$\lambda $$t
$$ \Rightarrow $$ t = $${{\ln \left( {1.3} ight)} \over \lambda }$$
If T is half-life, then $$\lambda $$ = $${{{{\ln} } 2} \over T}$$
$$ \Rightarrow $$ t = $${{\ln \left( {1.3} ight)} \over {{{\ln 2} \over T}}}$$ = $${{\ln \left( {1.3} ight)} \over {\ln 2}}T$$

Answer

$$t = {T \over {\log (1.3)}}$$

Answer

$$t = T\log (1.3)$$

Answer

$$t = {T \over 2}{{\log 2} \over {\log 1.3}}$$

Answer

$$t = T{{\log 1.3} \over {\log 2}}$$

Atoms & Nuclei

A radioactive nucleus A with a half life T, decays into a nucleus B. At t = 0, there is no nucleus B. At sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by :

Choose the correct answer: