Activity of a radioactive sample is measured as N_0 counts per minute at t = 0 and N

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13.Nuclei

medium

The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5$ minutes. The time (in minutes) at which the activity reduces to half its value is

$log_e \;\frac{2}{5}$

$\frac{5}{{\log_e 2}}$

$5$$\log_{10}2$

$5$$\log_e2$

(AIPMT-2010)

Solution

According to activity law

$R=R_{0} e^{-\lambda t}$ ….. $(i)$

According to given problem,

$R_{0}=N_{0}$ counts per minute

$R=\frac{N_{0}}{e}$ counts per minute

$t=5\,minutes$

Substituting these values in equation $(i)$, we get

${\frac{N_{0}}{e}=N_{0} e^{-5 \lambda}}$

${e^{-1}=e^{-5 \lambda}}$

$5 \lambda=1$ or $\lambda=\frac{1}{5}$ per minute

At $t=T_{1/2},$ the activity $R$ reduces to $\frac{R_{0}}{2}$

where $T_{1 / 2}=$ half life of a radioactive sample From equation $(i)$, we get

$\frac{R_{0}}{2}=R_{0} e^{-\lambda T_{1 / 2}}$

$e^{\lambda T_{1 / 2}}=2$

Taking natural logarithms of both sides of above equation, we get

$\lambda T_{1 / 2}=\log _{e} 2$

or $T_{1 / 2}=\frac{\log _{e} 2}{\lambda}=\frac{\log _{e} 2}{\left(\frac{1}{5}\right)}=5 \log _{e} 2$ minutes

Standard 12

Physics

Following statements related to radioactivity are given below

$(A)$ Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.

$(B)$ The number of un-decayed nuclei in the radioactive sample decays exponentially with time.

$(C)$ Slope of the graph of $\log _{e}$ (no. of undecayed nuclei) $Vs$. time represents the reciprocal of mean life time $(\tau)$.

$(D)$ Product of decay constant ( $\lambda$ ) and half-life time $\left(T_{1 / 2}\right)$ is not constant.

Choose the most appropriate answer from the options given below

hard

(JEE MAIN-2022)

$\beta$- rays emitted by a radioactive material are

easy

(IIT-1983)

Match the nuclear processes given in column $I$ with the appropriate option$(s)$ in column $II$

column $I$ column $II$

$(A.)$Nuclear fusion $(P.)$ Absorption of thermal neutrons by ${ }_{92}^{213} U$

$(B.)$Fission in a nuclear reactor $(Q.)$ ${ }_{27}^{60} Co$ nucleus

$(C.)$ $\beta$-decay $(R.)$ Energy production in stars via hydrogen conversion to helium

$(D.)$ $\gamma$-ray emission $(S.)$ Heavy water

$(T.)$ Neutrino emission

normal

(IIT-2015)

The half life period of radioactive element ${x}$ is same as the mean life time of another radioactive element $y.$ Initially they have the same number of atoms. Then:

hard

(JEE MAIN-2021)

The half life period of a radioactive substance is 60 days. The time taken for $\frac{7}{8}$ th of its original mass to disintegrate will be $……days$