Question

Show that the mean life of the atoms of a radioactive isotope with decay constant \(\lambda\) is \(\lambda^{-1}\).

Short answer

The mean life of the atoms is \( \lambda^{-1} \).

Step-by-step solution

Understanding Radioactive Decay

Radioactive decay follows an exponential model described by the function \( N(t) = N_0 e^{- au} \), where \( N_0 \) is the initial number of atoms, and \( \tau = \lambda t \) is the decay constant multiplied by time.

Define Mean Lifetime

The mean lifetime of atoms can be defined as the expected value of time until a decay event occurs. In mathematical terms, it is \( \langle t \rangle = \int_0^\infty t P(t) \, dt \), where \( P(t) \) is the probability density function of decay.

Probability Density Function

The probability density function of decay is given by \( P(t) = \lambda e^{- au} = \lambda e^{-\lambda t} \). This is because the probability of decay per unit time is proportional to the number of atoms remaining at time \( t \).

Set Up the Integral for Mean Lifetime

We integrate \( t P(t) \) over time to find the mean lifetime: \( \langle t \rangle = \int_0^\infty t (\lambda e^{-\lambda t}) \, dt \).

Evaluate the Integral

Evaluate \( \langle t \rangle = \lambda \int_0^\infty t e^{-\lambda t} \, dt \). Use integration by parts, letting \( u = t \) and \( dv = e^{-\lambda t} \, dt \), which gives \( du = dt \) and \( v = -\frac{1}{\lambda}e^{-\lambda t} \), leading to \( \int t e^{-\lambda t} \, dt = \left[-\frac{t}{\lambda} e^{-\lambda t}\right]_0^\infty + \frac{1}{\lambda} \int e^{-\lambda t} \, dt \).

Simplify the Integral

Simplifying, we find \( = 0 + \frac{1}{\lambda^2} \right . \left [e^{-\lambda t} \right ]_0^\infty = \frac{1}{\lambda^2} (0 - (-1)) = \frac{1}{\lambda^2} \).

Result of the Integration

Finally yield \( \langle t \rangle = \lambda \times \frac{1}{\lambda^2} = \frac{1}{\lambda} \). This shows that the mean lifetime of the atoms is \( \lambda^{-1} \).

Key concepts

Decay Constant

The decay constant, denoted by \( \lambda \), is a fundamental parameter in the study of radioactive decay. It represents the probability per unit time that a given atom will decay. In practical terms, this constant is crucial because it dictates how fast a radioactive sample will lose its atoms due to decay.
- A higher \( \lambda \) means more rapid decay, as more atoms are likely to decay in a short period.
- Conversely, a smaller \( \lambda \) indicates a longer-lasting material, with fewer atoms decaying over the same time frame.
This constant is central to the exponential decay model, expressed as \( N(t) = N_0 e^{-\lambda t} \). This equation captures how the initial quantity of atoms, \( N_0 \), decreases exponentially over time due to the decay constant.
Understanding \( \lambda \) helps scientists predict the behavior of radioactive materials, which has vast applications in fields like nuclear medicine, archaeology, and energy production.

Mean Lifetime

In the context of radioactive decay, the mean lifetime of an isotope is the average time it takes for the atoms to decay. It is mathematically expressed as the expected value of the time until a decay event occurs. This mean lifetime can be calculated using the integral \( \langle t \rangle = \int_0^\infty t P(t) \, dt \), where \( P(t) \) is the probability density function of decay.
- For radioactive isotopes with a decay constant \( \lambda \), the mean lifetime is given by \( \langle t \rangle = \frac{1}{\lambda} \).
This relationship indicates that the mean lifetime is inversely proportional to the decay constant. If an isotope has a larger decay constant, it will have a shorter mean lifetime, signifying that the atoms decay rapidly. Conversely, a smaller decay constant suggests a longer mean lifetime, with atoms decaying more slowly. Understanding this concept is key in applications that require precise knowledge of how long a radioactive substance will remain active.

Probability Density Function

The probability density function (PDF) in the context of radioactive decay provides a mathematical description of the likelihood that an atom will decay at a specific time \( t \). The function is given by \( P(t) = \lambda e^{-\lambda t} \), where \( \lambda \) is the decay constant. This PDF plays a crucial role in determining the statistical behavior of decaying particles over time.
- \( P(t) \) embodies the property that the probability of decay per unit time is dependent on the number of atoms remaining at any time \( t \).
- The exponential nature of the function reflects how the decay likelihood diminishes over time, as fewer atoms remain undecayed.
This function helps in calculating important metrics such as the mean lifetime and half-life of radioactive materials. For instance, integrating \( tP(t) \) from 0 to infinity provides the mean lifetime, supporting the conclusion that \( \langle t \rangle = \frac{1}{\lambda} \). Understanding the PDF of decay fosters insight into the probabilistic nature of radioactive decay processes.