If T is the half-life of a radioactive substa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The activity $A$ of a radioactive substance at any time $t$ is given by $A = A_0 e^{-\lambda t}$,where $\lambda$ is the decay constant.The decay c

Vedclass · Accepted Answer

$T^{-2}$

Vedclass · Answer

(D) The activity $A$ of a radioactive substance at any time $t$ is given by $A = A_0 e^{-\lambda t}$,where $\lambda$ is the decay constant.The decay constant $\lambda$ is related to the half-life $T$ by the relation $\lambda = \frac{\ln 2}{T}$.The rate of change of activity is given by $\frac{dA}{dt} = \frac{d}{dt}(A_0 e^{-\lambda t}) = -\lambda A_0 e^{-\lambda t} = -\lambda A$.The magnitude of the rate of change of activity is $|\frac{dA}{dt}| = \lambda A$.Since $\lambda = \frac{\ln 2}{T}$,we have $|\frac{dA}{dt}| = \frac{\ln 2}{T} A$.Thus,the rate of change of activity is proportional to $T^{-1}$. However,looking at the instantaneous rate of change of activity $\frac{dA}{dt} = -\lambda^2 N_0 e^{-\lambda t}$,where $N = N_0 e^{-\lambda t}$.Since $A = \lambda N$,then $\frac{dA}{dt} = -\lambda A = -\lambda (\lambda N) = -\lambda^2 N$.Substituting $\lambda = \frac{\ln 2}{T}$,we get $\frac{dA}{dt} \propto \lambda^2 \propto (\frac{1}{T})^2 = T^{-2}$.Therefore,the rate of change of activity is proportional to $T^{-2}$.

If $T$ is the half-life of a radioactive substance,then its instantaneous rate of change of activity is proportional to

Similar Questions

The decay constant of a radioisotope is $\lambda$. If its activity at times $t_1$ and $t_2$ are $A_1$ and $A_2$ respectively,then the number of nuclei that have decayed during the time interval $(t_1 - t_2)$ is:

At time $t = 0$,a radioactive element has a mass of $10 \, gm$. What mass in $gm$ will remain after two mean lifetimes?

$A$ and $B$ are two radioactive elements. Their half-lives are $1 \, year$ and $2 \, years$ respectively. Initially,$10 \, g$ of $A$ and $1 \, g$ of $B$ are taken. After how many years will their remaining quantities be equal?

In a mean life of a radioactive sample,

Vedclass Test Series

Exam Paper Generator

Online Exam Module