(D) The number of nuclei remaining at time $t$ is given by the radioactive decay law: $N(t) = N_0 e^{-\lambda t}$.For material $X_1$ with decay consta

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(D) The number of nuclei remaining at time $t$ is given by the radioactive decay law: $N(t) = N_0 e^{-\lambda t}$.For material $X_1$ with decay constant $10 \lambda$: $N_1 = N_0 e^{-10 \lambda t}$.For material $X_2$ with decay constant $\lambda$: $N_2 = N_0 e^{-\lambda t}$.Given that the ratio $N_1 / N_2 = 1 / e$ at time $t$:$\frac{N_1}{N_2} = \frac{N_0 e^{-10 \lambda t}}{N_0 e^{-\lambda t}} = e^{-10 \lambda t + \lambda t} = e^{-9 \lambda t}$.We are given $\frac{N_1}{N_2} = e^{-1}$.Equating the exponents: $-9 \lambda t = -1$.Therefore, $t = \frac{1}{9 \lambda}$.

Class 12 Physics Nuclei Law of Radioactivity by Rutherford and Soddy and Half Life and Mean Life

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Two radioactive materials $X_1$ and $X_2$ have decay constants $10 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $1 / e$ after a time:

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A
$1 / (10 \lambda)$
B
$1 / (11 \lambda)$
C
$11 / (10 \lambda)$
D
$1 / (9 \lambda)$

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Nuclei

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Law of Radioactivity by Rutherford and Soddy and Half Life and Mean Life

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$A$ radioactive sample is an $\alpha$-emitter with a half-life of $138.6$ days. $A$ student observes its activity to be $2000$ disintegrations per second. The number of radioactive nuclei for this given activity is:

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$A$ radioactive isotope $X$ has a half-life of $20$ years and decays into a stable element $Y$. In a given rock sample, the ratio of the amounts of $X$ and $Y$ is found to be $1:7$. The estimated age of the rock is ......... years.

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$A$ radioactive sample decays by $\beta$-emission. In the first $2 \ s$,$n$ $\beta$-particles are emitted,and in the next $2 \ s$,$0.25n$ $\beta$-particles are emitted. The half-life of the radioactive nuclei is ...... $s$.

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Two radioactive materials $X_1$ and $X_2$ have decay constants $10 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $1 / e$ after a time:

Similar Questions

The half-life of a radioactive element depends upon:

Two radioactive substances $A$ and $B$ have the same number of initial nuclei. If the half-lives of $A$ and $B$ are $1.5 \ days$ and $4.5 \ days$ respectively, then the ratio of the number of nuclei remaining in $A$ and $B$ after $9 \ days$ is

$A$ radioactive sample is an $\alpha$-emitter with a half-life of $138.6$ days. $A$ student observes its activity to be $2000$ disintegrations per second. The number of radioactive nuclei for this given activity is:

$A$ radioactive isotope $X$ has a half-life of $20$ years and decays into a stable element $Y$. In a given rock sample, the ratio of the amounts of $X$ and $Y$ is found to be $1:7$. The estimated age of the rock is ......... years.

$A$ radioactive sample decays by $\beta$-emission. In the first $2 \ s$,$n$ $\beta$-particles are emitted,and in the next $2 \ s$,$0.25n$ $\beta$-particles are emitted. The half-life of the radioactive nuclei is ...... $s$.

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