At any instant,two elements X_1 and X_2 have | vedclass

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

Question

(D) Let $N_0$ be the initial number of radioactive atoms for both elements at $t = 0$.According to the law of radioactive decay,the number of atoms re

Vedclass · Accepted Answer

$\frac{1}{9\lambda}$

Vedclass · Answer

(D) Let $N_0$ be the initial number of radioactive atoms for both elements at $t = 0$.According to the law of radioactive decay,the number of atoms remaining at time $t$ is given by $N(t) = N_0 e^{-\lambda t}$.For element $X_1$ with decay constant $\lambda_1 = 10\lambda$,the number of atoms is $N_1 = N_0 e^{-10\lambda t}$.For element $X_2$ with decay constant $\lambda_2 = \lambda$,the number of atoms is $N_2 = N_0 e^{-\lambda t}$.We are given that the ratio $\frac{N_1}{N_2} = \frac{1}{e} = e^{-1}$.Substituting the expressions for $N_1$ and $N_2$:$\frac{N_0 e^{-10\lambda t}}{N_0 e^{-\lambda t}} = e^{-1}$$e^{-10\lambda t + \lambda t} = e^{-1}$$e^{-9\lambda t} = e^{-1}$Comparing the exponents:$-9\lambda t = -1$$t = \frac{1}{9\lambda}$.

At any instant,two elements $X_1$ and $X_2$ have the same number of radioactive atoms. If the decay constants of $X_1$ and $X_2$ are $10\lambda$ and $\lambda$ respectively,then the time when the ratio of their atoms becomes $\frac{1}{e}$ will be:

Similar Questions

The half-life of a radioactive substance is $40 \, years$. How long will it take to reduce to one-fourth of its original amount, and what is the value of the decay constant?

$A$ radioactive source has a half-life of $6 \,h$. $A$ freshly prepared sample of the same exhibits radioactivity $32$ times the permissible safe value. The minimum time after which it would be possible to work safely with the source is (in $\,h$)

The nuclide $^{133}I$ is radioactive,with a half-life of $8.04 \, days$. At noon on $January \, 1$,the activity of a certain sample is $600 \, Bq$. The activity at noon on $January \, 24$ will be

$A$ radioactive sample decays at a rate of $5000$ disintegrations/minute at a certain instant. After $5$ minutes, the rate becomes $1250$ disintegrations/minute. The decay constant is ....... (in $\ln 2$)

In a mean life of a radioactive sample,

Vedclass Test Series

Exam Paper Generator

Online Exam Module