(B) Let $N_0$ be the initial number of nuclei for both materials $A$ and $B$.The number of nuclei remaining at time $t$ is given by $N(t) = N_0 e^{-\l

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(B) Let $N_0$ be the initial number of nuclei for both materials $A$ and $B$.The number of nuclei remaining at time $t$ is given by $N(t) = N_0 e^{-\lambda t}$.For material $A$: $N_A(t) = N_0 e^{-(7 \lambda) t}$.For material $B$: $N_B(t) = N_0 e^{-\lambda t}$.We are given that the ratio $\frac{N_B(t)}{N_A(t)} = e$.Substituting the expressions: $\frac{N_0 e^{-\lambda t}}{N_0 e^{-7 \lambda t}} = e$.Simplifying the ratio: $e^{-\lambda t + 7 \lambda t} = e^1$.$e^{6 \lambda t} = e^1$.Equating the exponents: $6 \lambda t = 1$.Therefore,$t = \frac{1}{6 \lambda}$.

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

Medium

Two radioactive materials $A$ and $B$ having decay constants $7 \lambda$ and $\lambda$ respectively,initially have the same number of nuclei. The time taken for the ratio of the number of nuclei of material $B$ to that of $A$ to be $e$ is:

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A
$\frac{1}{\lambda}$
B
$\frac{1}{6 \lambda}$
C
$\frac{1}{7 \lambda}$
D
$\frac{1}{8 \lambda}$

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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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The graph represents the decay of a newly prepared sample of radioactive nuclide $X$ to a stable nuclide $Y$. The half-life of $X$ is $\tau$. The growth curve for $Y$ intersects the decay curve for $X$ after time $T$. What is the time $T$?

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Carbon-$14$ decays with a half-life of about $5,800$ years. In a sample of bone,the ratio of Carbon-$14$ to Carbon-$12$ is found to be $\frac{1}{4}$ of what it is in free air. This bone may belong to a period about $x$ centuries ago,where $x$ is nearest to:

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After five half-lives,what will be the fraction of the initial substance remaining?

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At $t = 0$,the counting rate from a radioactive source is $1600 \text{ counts/s}$,and at $t = 8 \text{ s}$,it is $100 \text{ counts/s}$. The counting rate at $t = 6 \text{ s}$ will be:

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$A$ radioactive substance is being produced at a constant rate of $10 \text{ nuclei/s}$. The decay constant of the substance is $0.5 \text{ s}^{-1}$. After what time will the number of radioactive nuclei become $10$? Initially,there are no nuclei present. Assume the decay law holds for the sample.

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Two radioactive materials $A$ and $B$ having decay constants $7 \lambda$ and $\lambda$ respectively,initially have the same number of nuclei. The time taken for the ratio of the number of nuclei of material $B$ to that of $A$ to be $e$ is:

Similar Questions

The graph represents the decay of a newly prepared sample of radioactive nuclide $X$ to a stable nuclide $Y$. The half-life of $X$ is $\tau$. The growth curve for $Y$ intersects the decay curve for $X$ after time $T$. What is the time $T$?

Carbon-$14$ decays with a half-life of about $5,800$ years. In a sample of bone,the ratio of Carbon-$14$ to Carbon-$12$ is found to be $\frac{1}{4}$ of what it is in free air. This bone may belong to a period about $x$ centuries ago,where $x$ is nearest to:

After five half-lives,what will be the fraction of the initial substance remaining?

At $t = 0$,the counting rate from a radioactive source is $1600 \text{ counts/s}$,and at $t = 8 \text{ s}$,it is $100 \text{ counts/s}$. The counting rate at $t = 6 \text{ s}$ will be:

$A$ radioactive substance is being produced at a constant rate of $10 \text{ nuclei/s}$. The decay constant of the substance is $0.5 \text{ s}^{-1}$. After what time will the number of radioactive nuclei become $10$? Initially,there are no nuclei present. Assume the decay law holds for the sample.

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