(D) Given: $(T_{1/2})_A = (\tau)_B$, where $\tau$ is the mean life.We know that $T_{1/2} = \frac{0.693}{\lambda}$ and $\tau = \frac{1}{\lambda}$.There

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$B$ will decay at a faster rate than $A.$

(D) Given: $(T_{1/2})_A = (\tau)_B$, where $\tau$ is the mean life.We know that $T_{1/2} = \frac{0.693}{\lambda}$ and $\tau = \frac{1}{\lambda}$.Therefore, $\frac{0.693}{\lambda_A} = \frac{1}{\lambda_B}$.This implies $\lambda_A = 0.693 \lambda_B$, which means $\lambda_A The rate of decay is given by $R = \lambda N$.Initially, the number of atoms $N$ is the same for both elements.Since $\lambda_B > \lambda_A$, the decay rate $R_B = \lambda_B N$ will be greater than $R_A = \lambda_A N$.Thus, $B$ will decay at a faster rate than $A$.

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

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The half-life of a radioactive element $A$ is the same as the mean-life of another radioactive element $B.$ Initially, both substances have the same number of atoms, then

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A
$A$ and $B$ decay at the same rate always.
B
$A$ and $B$ decay at the same rate initially.
C
$A$ will decay at a faster rate than $B.$
D
$B$ will decay at a faster rate than $A.$

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

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Two radioactive isotopes $P$ and $Q$ have half-lives $10 \text{ minutes}$ and $15 \text{ minutes}$ respectively. Freshly prepared samples of each isotope initially contain the same number of atoms. After $30 \text{ minutes}$,the ratio $\frac{\text{number of atoms of } P}{\text{number of atoms of } Q}$ will be

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The half-life of a radioactive sample,where the initial activity of the material was $8 \text{ counts}$ and after $3 \text{ hours}$ it becomes $1 \text{ count}$,is ............... $hours$.

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$A$ radioactive element emits $\alpha$ and $\beta$ particles. Its mean lives for $\alpha$ and $\beta$ decay are $1620$ years and $405$ years,respectively. After how many years will the activity be reduced to $1/4$ of its initial value?

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The decay constants of two radioactive elements are $15x$ and $3x$ respectively. Initially,they have the same number of nuclei. After a time of $\frac{1}{6x}$,the ratio of the number of their nuclei will be ........

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The activity of a radioactive sample is measured as $N_0$ counts per minute at time $t=0$,and $\frac{N_0}{e}$ counts per minute at time $t=3$ minutes. The activity reduces to half its value in time (in minutes) is:

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The half-life of a radioactive element $A$ is the same as the mean-life of another radioactive element $B.$ Initially, both substances have the same number of atoms, then

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The half-life of a radioactive sample,where the initial activity of the material was $8 \text{ counts}$ and after $3 \text{ hours}$ it becomes $1 \text{ count}$,is ............... $hours$.

$A$ radioactive element emits $\alpha$ and $\beta$ particles. Its mean lives for $\alpha$ and $\beta$ decay are $1620$ years and $405$ years,respectively. After how many years will the activity be reduced to $1/4$ of its initial value?

The decay constants of two radioactive elements are $15x$ and $3x$ respectively. Initially,they have the same number of nuclei. After a time of $\frac{1}{6x}$,the ratio of the number of their nuclei will be ........

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