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NEET PHYSICSMedium

A sample of radioactive element has a mass of 10 gm at an instant t=0t = 0. The approximate mass of this element in the sample after two mean lives is:

A

3.70 gm

B

6.30 gm

C

1.35 gm

D

2.50 gm

Step-by-Step Solution

  1. Formula: The mass MM of a radioactive substance remaining after time tt follows the exponential decay law: M=M0e−λtM = M_0 e^{-\lambda t}, where M0M_0 is the initial mass and λ\lambda is the decay constant .
  2. Mean Life: The mean life (τ\tau) is the reciprocal of the decay constant: τ=1/λ\tau = 1/\lambda .
  3. Substitution: We are given t=2τ=2/λt = 2\tau = 2/\lambda. Substituting this into the mass equation: M=10e−λ(2/λ)=10e−2M = 10 e^{-\lambda (2/\lambda)} = 10 e^{-2}
  4. Calculation: Using the value of Euler's number e≈2.718e \approx 2.718: M=10(2.718)2≈107.389≈1.353 gmM = \frac{10}{(2.718)^2} \approx \frac{10}{7.389} \approx 1.353 \text{ gm}
  5. Conclusion: The approximate mass remaining is 1.35 gm.
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