Question

The Probability of survival of a radioactive nucleus for one mean life time is (A) \(1-(1 / \mathrm{e}) \mathrm{S}\) (B) \((1 / \mathrm{e})\) (C) \((2 / \mathrm{e})\) (D) \((3 / \mathrm{e})\)

Short answer

The probability of survival of a radioactive nucleus for one mean lifetime is given by the radioactive decay formula, P(t) = \(e^{-\lambda t}\). By substituting the mean lifetime τ for the time t and the decay constant λ as \(\frac{1}{\tau}\), we get P(τ) = \(e^{-1}\), which simplifies to P(τ) = \(\frac{1}{e}\). Therefore, the correct answer is (B) \(\frac{1}{e}\).

Step-by-step solution

Understand Radioactive Decay Probability Formula

The probability of survival of a radioactive nucleus for a given time 't' can be given by the radioactive decay formula: P(t) = \(e^{-\lambda t}\) where P(t) is the probability of survival, λ (lambda) is the decay constant, and t is the time. The decay constant λ is related to the mean lifetime (τ) as follows: λ = \(\frac{1}{\tau}\) To find the probability of survival for one mean life time, we will substitute t with τ (mean life time) in the radioactive decay probability formula.

Plug-in Mean Lifetime in the Probability Formula

Now, let's plug-in τ for the time t in the probability formula: P(τ) = \(e^{-\lambda \tau}\) Since λ = \(\frac{1}{\tau}\), we can replace λ in the formula: P(τ) = \(e^{-\frac{1}{\tau} \tau}\) As τ cancels out, we are left with: P(τ) = \(e^{-1}\)

Simplify the Expression

Now, we can simplify the expression: P(τ) = \(\frac{1}{e}\) The probability of survival of a radioactive nucleus for one mean lifetime is \(\frac{1}{e}\). Answer: (B) \(\frac{1}{e}\)