Solved: PHYSICS Question for NEET

NEET PHYSICSMedium

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 $1/e$ will be:

$\frac{1}{11\lambda}$

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

$\frac{1}{6\lambda}$

$\frac{1}{5\lambda}$

Step-by-Step Solution

Identify the Decay Law: The number of undecayed nuclei $N$ at time $t$ is given by $N = N_0 e^{-\lambda t}$ .
Set up Equations for Both Elements:

Let the initial number of atoms for both be $N_0$ .
For $X_1$ : $N_1 = N_0 e^{-10\lambda t}$
For $X_2$ : $N_2 = N_0 e^{-\lambda t}$

Calculate the Ratio:

We are given that $\frac{N_1}{N_2} = \frac{1}{e} = e^{-1}$ .
Substitute the decay equations: $\frac{N_0 e^{-10\lambda t}}{N_0 e^{-\lambda t}} = e^{-1}$ .
Simplify the exponentials: $e^{(-10\lambda - (-\lambda))t} = e^{-9\lambda t}$ .

Solve for Time ( $t$ ):

Equate the exponents: $-9\lambda t = -1$ .
$t = \frac{1}{9\lambda}$ .

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