Problem 29 - Calculus: Graphical, Numerical, Algebraic

Question

Mean Life of Radioactive Nuclei Physicists using the radioactive decay equation \(y=y_{0} e^{-k t}\) call the number 1\(/ k\) the mean life of a radioactive nucleus. The mean life of a radon- 222 nucleus is about 1\(/ 0.18 \approx 5.6\) days. The mean life of a carbon-1 4 nucleus is more than 8000 years. Show that 95\(\%\) of the radioactive nuclei originally present in any sample will disintegrate within three mean lifetimes, that is, by time \(t=3 / k .\) Thus, the mean life of a nucleus gives a quick way to estimate how long the radioactivity of a sample will last.

Step-by-Step Solution

Answer

After three mean lifetimes (i.e. when \(t=3 / k\)), approximately 95% of the radioactive nuclei originally present in any sample will have disintegrated.

Step 1: Understand the Decay Equation

The radioactive decay equation \(y=y_{0} e^{-kt}\) describes the amount of certain substance that remains after a period of time. Here, \(y_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time. The mean life of a substance is the average time it takes for a substance to decay, which is given as \(1/k\). We're asked to show that 95% of the starting amount of substance will have decayed after three mean lives, or \(t=3/k\).

Step 2: Substitute Given Values

We can substitute the given values into the decay equation. To find out what fraction remains after three mean lives, that is \(t=3/k\), we substitute \(t=3/k\) into the decay equation: \(y = y_0 e^{-k \cdot (3/k)}\).\nAfter simplifying, we obtain: \(y = y_0 e^{-3}\).

Step 3: Calculate the Remaining Amount

Now we find the remaining fraction of the substance, \(y/y_0\), which is equal to \(e^{-3}\). Using the constant value of euler's number \(e \approx 2.718\), the remaining amount is \(y/y_0 \approx 0.05\). Thus, only about 5% of the nuclei remain after three mean lives.

Step 4: Analyze the Result

Since only 5% of the nuclei remain, it means that 95% of the initial radioactivity has disintegrated within three mean lifetimes. So, we've proved our initial claim.

Key Concepts

Mean Life of Radioactive Nucleus

Understanding the mean life of a radioactive nucleus is crucial for grasping the nature of radioactive decay. The mean life, represented as \(1/k\), is the average time a single nucleus will take to decay. To put it simply, if you have a large number of nuclei, on average, they would all take about the mean life to disintegrate. However, this doesn't mean every nucleus will last exactly this long; due to the probabilistic nature of quantum mechanics, the actual lifespan of a single nucleus can vary widely.

Take radon-222, for instance. It has a mean life of approximately 5.6 days. This figure allows scientists, engineers, and safety officials to estimate how long a sample containing radon-222 is likely to remain hazardous. The calculation becomes particularly useful in fields such as medicine, where radioactive isotopes are used for treatments and diagnostics, and the timing of decay can influence treatment plans.

Exponential Decay

Exponential decay is a principle that applies to numerous processes in physics and other sciences, where the rate of change of a quantity decreases over time. In the context of radioactivity, it's reflected in the radioactive decay equation \( y = y_0 e^{-kt} \), which mathematically models how the quantity of radioactive material decreases over time.

This equation suggests that the decay process is continuous and never actually reaches zero, since the exponential function approach zero asymptotically – which means it gets close to zero but never quite touches it. For example, after one mean life \(1/k\), approximately 63.2% of the original sample will have decayed, leaving 36.8% remaining. After two mean lives, about 86.5% will have decayed, and so on. By understanding this exponential nature, students and professionals can better predict and handle radioactive materials.

Radioactive Nuclei Disintegration

Radioactive nuclei disintegration is the process by which unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. In the radioactive decay equation, \(y\) represents the amount of radioactive substance that remains, while \(y_0\) is the initial amount. Over time, the number of undecayed nuclei \(y\) decreases, demonstrating the disintegration process.

The exercise presented showed that after three mean lifetimes \(3/k\), 95% of the nuclei would have disintegrated. This is a practical rule of thumb for estimating how quickly a radioactive sample loses its radioactivity. The practical applications of understanding this process are vast, including managing nuclear waste, dating archaeological finds with techniques like carbon dating, and ensuring safety standards in industries that handle radioactive materials.

Calculus in Physics

Calculus is a branch of mathematics that deals with rates of change and accumulation of quantities and is an indispensable tool in physics, particularly when describing processes like radioactive decay. The radioactive decay equation illustrates how calculus can be used to model natural phenomena that change continuously over time.

In the case of our exercise, we use concepts from calculus to understand exponentially decaying processes like radioactive decay. Calculus allows us to find the remaining percentage of a substance after a certain time by integrating the decay rate over that time period. It also enables us to work with continuous growth and decay rates, which are commonly seen in not just physics but also in fields such as biology, economics, and engineering. For students grappling with these concepts, calculus provides the necessary framework to move beyond simple proportionality to more complex models of change in the natural world.