Question

What is the half-life of radon-222 if a sample initially contains 150 mg and only 18.7 mg after 11.4 days?

Short answer

The half-life of radon-222 is approximately 5.03 days, calculated using the exponential decay formula and given values of initial and remaining quantities and time elapsed.

Step-by-step solution

Identify the given information and formula

The exponential decay formula is given by: \(N(t) = N_0 e^{-\lambda t}\), where: - \(N(t)\) is the remaining quantity after time t - \(N_0\) is the initial quantity - \(\lambda\) is the decay constant - \(t\) is the time In our case, we have the following information: - \(N(t) = 18.7\,mg\) - \(N_0 = 150\,mg\) - \(t = 11.4\,days\) Our goal is to find the half-life (\(t_{1/2}\)) of radon-222.

Solve for the decay constant \(\lambda\)

First, we need to solve for the decay constant (\(\lambda\)). Using the exponential decay formula, we can solve for \(\lambda\) by dividing both sides by \(N_0\) and taking the natural logarithm of the result: \(e^{-\lambda t} = \frac{N(t)}{N_0}\) \(-\lambda t = ln(\frac{N(t)}{N_0})\) \(\lambda = -\frac{ln(\frac{N(t)}{N_0})}{t}\) Now, plug in the given values: \(\lambda = -\frac{ln(\frac{18.7}{150})}{11.4}\) Calculate the decay constant: \(\lambda \approx 0.1378\,days^{-1}\)

Calculate the half-life

Now that we have the decay constant, we can use the following relationship between the half-life and the decay constant: \(t_{1/2} = \frac{ln(2)}{\lambda}\) Plug in the decay constant: \(t_{1/2} = \frac{ln(2)}{0.1378}\) Calculate the half-life: \(t_{1/2} \approx 5.03\,days\)

State the result

The half-life of radon-222 is approximately 5.03 days.

Key concepts

Exponential Decay

Understanding the nature of exponential decay is essential when studying processes like radioactive decay. At its core, exponential decay describes the process by which a quantity diminishes at a rate proportional to its current value. This means the larger the quantity, the faster it will decrease, but the decrease will slow down as the quantity gets smaller.

Mathematically, this rate of decline is represented by the formula: \(N(t) = N_0 e^{-\t(t)}\). In this formula, \(N(t)\) represents the amount of substance left after time \(t\), while \(N_0\) is the initial quantity. The base of the natural logarithm (e) is approximately 2.718, which is a constant that appears frequently in growth and decay problems. When working with exponential decay, one characteristic phenomenon to look out for is the half-life, which is the time it takes for half of the substance to decay. This principle is not only pivotal in nuclear physics but also in other fields such as finance, biology, and geology.

Decay Constant

The decay constant, denoted as \(\t\), is the cornerstone of understanding exponential decay. It signifies how quickly a substance decays and is essential in calculating the half-life of a radioactive element. A larger decay constant means a faster decay rate, and conversely, a smaller decay constant indicates a slower rate of decay.

To find the decay constant, one can rearrange the exponential decay formula to isolate \(\t\). By taking the natural logarithm of the ratio of the remaining quantity \(N(t)\) over the initial quantity \(N_0\) and dividing by time \(t\), we get \(\t = -\frac{ln(\frac{N(t)}{N_0})}{t}\). Having the decay constant allows us to understand and predict how a radioactive element or substance will behave over time.

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. This process follows the principles of exponential decay. Each radioactive isotope, like radon-222 in our exercise, has its specific half-life that can be used to determine the age of objects (in fields such as archaeology or geology) or in medical treatments by using isotopes to target and destroy diseased cells.

When a sample of a radioactive isotope decays, it transmutes into another element or a different isotope of the same element. This decay process continues until a stable isotope is formed. The rate of decay is not influenced by external factors, such as temperature or pressure, making it a predictable process, governed by the decay constant and exemplified by the exponential decay function.