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

Define Mean Life

The mean life of a radioactive isotope is the average time that a single atom remains intact before decaying. Mathematically, it is given by the integral of the product of time \( t \) and the decay probability density function \( e^{-\lambda t} \) from 0 to infinity.

Set Up the Integral

The mean life \( T \) can be expressed as:\[T = \int_0^\infty t \cdot \lambda e^{-\lambda t} \, dt\]This integral represents the expected value of the time until decay.

Apply Integration by Parts

Use the integration by parts formula \( \int u \, dv = uv - \int v \, du \). Let \( u = t \) and \( dv = \lambda e^{-\lambda t} \, dt \). Then, \( du = dt \) and \( v = -e^{-\lambda t} \).

Compute the Integral

Substitute into the integration by parts formula:\[\int t \cdot \lambda e^{-\lambda t} \, dt = [-t e^{-\lambda t}]_0^\infty + \int e^{-\lambda t} \, dt_0^\infty\]Evaluate the boundary terms, \([-t e^{-\lambda t}]_0^\infty = 0 - 0\), and the remaining integral:\[\int e^{-\lambda t} \, dt = \frac{1}{\lambda}\]Thus, the entire expression equals \( 0 + \frac{1}{\lambda} \).

Conclusion

The mean life of the atoms is therefore calculated as:\[T = \frac{1}{\lambda}\]This shows that the mean life of the atoms of the radioactive isotope with decay constant \( \lambda \) is \( \lambda^{-1} \).

Key concepts

Decay Constant

The decay constant, represented by the Greek letter \( \lambda \), is a fundamental concept in radioactive decay. It defines the rate at which a radioactive isotope undergoes decay. Essentially, the decay constant is a probability per unit time that an atom will decay. In other words, it shows the fraction of atoms decaying in a given time period. The larger the decay constant, the faster the rate of decay. Since radioactive decay is an exponential process, the decay constant plays a crucial role in the relevant mathematical equations.

Understanding \( \lambda \):

• It is a measure of the stability of a nucleus; a small \( \lambda \) indicates a stable isotope, whereas a large \( \lambda \) indicates a less stable isotope.
• The decay constant is specific to each radioactive material.
• It forms the foundation for calculating the half-life and mean life of radioactive substances.

The relationship between the decay constant and the half-life of a radioactive substance can be expressed as \( t_{1/2} = \frac{\ln 2}{\lambda} \). This formula helps in understanding how long it will take for half of the initial quantity of the substance to decay.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It relies on the product rule for differentiation, and it's essential for solving problems where straightforward integration isn't feasible. The formula for integration by parts is given by \( \int u \ dv = uv - \int v \ du \). To apply this method, select one part of the integrand as \( u \) and another as \( dv \).

Applying the formula involves:

• Identifying \( u \) and \( dv \).
• Finding \( du \) by differentiating \( u \).
• Integrating \( dv \) to find \( v \).
• Substituting into the formula to solve the integral.

This approach allows us to transform products into more manageable integrals, an essential step in calculating mean life as shown in the solution. It reduces complex integrals into simpler components, making them easier to evaluate. Practicing with examples is beneficial for mastering this technique.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. A key feature is that it is a random process at the level of single atoms, but it leads to predictable and exponential decay at macroscopic levels. The fundamental formula representing radioactive decay is \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the number of undecayed nuclei at time \( t \), \( N_0 \) is the initial number of nuclei, and \( \lambda \) is the decay constant.

Key Characteristics:

• Exponential nature: The amount of substance decreases exponentially over time.
• Independence: It doesn’t depend on external factors like temperature or pressure.
• Half-Life: The time required for half of the radioactive atoms in a sample to decay.

The decay probability function \( e^{-\lambda t} \) is crucial, as it defines the likelihood that a given atom will remain undecayed up until time \( t \). This function is integral to determining the mean life of radioactive isotopes.

Expected Value Calculation

Expected value, in the context of radioactive decay, refers to the average expected time until an atom decays. It is calculated using the concept of probability and integrals. The mean life of a radioactive isotope is essentially the expected value of the time before decay. Mathematically, it is expressed through the integral \[ T = \int_0^\infty t \cdot \lambda e^{-\lambda t} \, dt \].

Steps to Calculate Expected Value:

• Set up the integral of the time \( t \) multiplied by the decay probability density \( \lambda e^{-\lambda t} \).
• Solve the integral using integration by parts to find a finite result.
• The result, \( \frac{1}{\lambda} \), represents the mean life of the isotope.

This calculation provides essential insights into how long, on average, a nucleus will survive before decaying. Mastering this concept is critical for anyone studying radioactive materials and their behavior over time.