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) \(\frac{1}{e}\)

Key concepts

Probability of Survival

In the realm of radioactive decay, one important aspect to understand is the probability of survival of a radioactive nucleus. This concept refers to the likelihood that a particular nucleus remains "undecayed" over a given period.
Picturing the survival probability helps us comprehend how quickly a substance decays and how long it may be expected to remain in a "stable" state.

• This probability decreases exponentially over time, signifying that the chances of a nucleus not decaying reduce steadily.
• The formula representing this is encapsulated in the radioactive decay law: \[P(t) = e^{-\lambda t} \]
• In this equation, \( P(t) \) is the probability of survival at time \( t \), \( \lambda \) is the decay constant, and \( t \) represents time.

Understanding these elements helps predict the behavior of radioactive substances more accurately.

Mean Lifetime

The concept of mean lifetime is crucial as it provides a good measure of how long, on average, atoms of a radioactive substance will remain undecayed.
This statistical measure represents the average time a nucleus exists before decaying. It's a more direct representation of decay activity over time than half-life because it incorporates exponential decline.

• Mean lifetime is closely related to the decay constant. The relationship between the two is given by the formula: \[ \tau = \frac{1}{\lambda} \]
• Here, \( \tau \) is the mean lifetime, and \( \lambda \) is the decay constant.

By knowing the mean lifetime, scientists can estimate the time behavior of a large collection of nuclei, offering insights into radioactive decay over extended periods.

Decay Constant

The decay constant, denoted by \( \lambda \), is a fundamental characteristic in radioactive decay equations. It's essentially a parameter that quantifies the rate at which a particular radioactive substance decays.
Understanding the decay constant is key to making predictions about how quickly or slowly a radioactive sample will lose its radioactivity.

• Mathematically, the decay constant is defined as the inverse of the mean lifetime: \[ \lambda = \frac{1}{\tau} \]
• It incorporates the probability of decay per unit time, reflecting that if \( \lambda \) is large, the substance decays quickly, and if it's small, decay occurs more slowly.

This constant is vital in computing various decay metrics, including activity and mean lifetime, providing comprehensive insights into nuclear processes.

Exponential Decay Formula

At the heart of understanding radioactive decay is the exponential decay formula, which effectively describes the process by which unstable isotopes lose energy and transform into different isotopes.
This formula is pivotal in both theoretical and practical applications in nuclear physics, affecting fields such as nuclear medicine and radiocarbon dating.

• The formula is given by: \[ N(t) = N_0 e^{-\lambda t} \]
• In this equation, \( N(t) \) represents the number of undecayed nuclei at time \( t \), \( N_0 \) is the initial quantity of nuclei, and \( \lambda \) is the decay constant.
• The exponential nature indicates that as time passes, fewer and fewer nuclei remain undecayed, leading to an ever-decreasing rate of decay.

By understanding and applying the exponential decay formula, one can predict how a radioactive sample will behave over time, aiding in diverse scientific and industrial applications.