Problem 80 - Physics for Scientists and Engineers with Modern Physics

Question

(a) Show that the mean life of a radioactive nuclide, defined as $$\tau=\frac{\int_{0}^{\infty} t N(t) d t}{\int_{0}^{\infty} N(t) d t}$$ is \(\tau=1 / \lambda .\) (b) What fraction of the original number of nuclei remains after one mean life?

Step-by-Step Solution

Answer

(a) The mean life \( \tau = \frac{1}{\lambda} \). (b) After one mean life, \( 36.79\% \) or \( e^{-1} \) of the original nuclei remain.

Step 1: Understanding Time-Dependent Nuclei Number

To find the mean life \( \tau \), we need to understand \( N(t) \). The number of nuclei remaining at time \( t \) can be expressed as \( N(t) = N_0 e^{-\lambda t} \), where \( \lambda \) is the decay constant and \( N_0 \) is the initial number of nuclei.

Step 2: Derive Mean Life Expression

The mean life \( \tau \) is defined by the formula:\[\tau = \frac{\int_{0}^{\infty} t N(t) dt}{\int_{0}^{\infty} N(t) dt}.\]We will substitute \( N(t) = N_0 e^{-\lambda t} \) into this expression.

Step 3: Compute the Numerator

The numerator is \( \int_{0}^{\infty} t N(t) dt = N_0 \int_{0}^{\infty} t e^{-\lambda t} dt \). Use integration by parts, where \( u = t \) and \( dv = e^{-\lambda t} dt \), yielding an integral \( \frac{N_0}{\lambda^2} \).

Step 4: Compute the Denominator

The denominator is \( \int_{0}^{\infty} N(t) dt = N_0 \int_{0}^{\infty} e^{-\lambda t} dt \). This integral simplifies to \( \frac{N_0}{\lambda} \).

Step 5: Simplify the Mean Life Expression

Substituting the results from Steps 3 and 4 back into the mean life expression:\[\tau = \frac{ \frac{N_0}{\lambda^2}}{ \frac{N_0}{\lambda} } = \frac{1}{\lambda}.\]

Step 6: Calculate Fraction Remaining After One Mean Life

After one mean life \( \tau = \frac{1}{\lambda} \), the fraction of origin likely remaining is obtained by substituting \( t = \tau \) in \( N(t) = N_0 e^{-\lambda t} \):\[N(\tau) = N_0 e^{-\lambda \frac{1}{\lambda}} = N_0 e^{-1}.\]

Step 7: Conclusion

Thus, after one mean life, the fraction of the original number remaining is \( e^{-1} \approx 0.3679 \).

Key Concepts

Exponential Decay

Exponential decay is a process that describes how a quantity decreases over time. In the context of radioactive substances, this decreases exponentially over time. Imagine

• an initial quantity, such as the number of atoms in a radioactive sample, decreasing at a rate proportional to its current value.
• This means that every fixed period of time, a consistent fraction of the substance remains.

A practical example is how a radioactive sample loses a certain percentage of the remaining atoms in each successive period.
This behavior is defined mathematically using the function:\[ N(t) = N_0 e^{-\lambda t} \]where:- \(N(t)\) is the number of atoms left at time \(t\),- \(N_0\) is the initial number of atoms, - \(\lambda\) is the decay constant,- \(e\) is the base of the natural logarithm (approximately 2.71828).
The exponential function \(e^{-\lambda t}\) allows us to easily predict how the amount will decrease over time by plugging in different values of \(t\).

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. It is a random and spontaneous event where nuclei transform into a more stable state. Key characteristics include:

• This decay leads to the formation of different elements or isotopes over time.
• Each radioactive substance has a unique decay constant (\(\lambda\)), determining how quickly it undergoes decay.

Understanding this process involves analyzing how individual atoms change. Each decay event is independent, meaning that previous decays do not affect future events. This randomness can be described statistically, rather than deterministically.
One of the mathematical models that describe radioactive decay is the exponential decay model we discussed earlier. This model helps in predicting how long a radioactive sample takes to reach a certain state, such as its half-life, which is the time required for half of the substance to decay. Radioactive decay is not only fundamental to nuclear physics but also crucial in fields such as archaeology, medicine, and energy.

Decay Constant

The decay constant, represented as \(\lambda\), is a crucial parameter in the study of radioactive decay. It is a probability per unit time that each nucleus will decay.
This constant provides insights into how `fast` or `slow` a specific nuclide will decay over time.Consider the following:

• A higher decay constant implies a faster rate of decay.
• Conversely, a lower decay constant indicates a slower decay process.

The decay constant allows for the calculation of important quantities such as the mean life \(\tau\) and half-life \(t_{1/2}\). The relationship between these is given by:\[ \tau = \frac{1}{\lambda} \]which logistically defines the average time a nucleus will remain before decaying. Another crucial aspect is the half-life, defined by:\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]These relationships underscore the importance of \(\lambda\) in predicting and understanding the behavior of radioactive materials over time. It serves as a gateway to not only measuring but also comprehending the broader impacts and applications of radioactive substances.