Question

The radon isotope \({ }^{222} \mathrm{Rn}\), which has a half-life of 3.825 days, is used for medical purposes such as radiotherapy. How long does it take until \({ }^{222} \mathrm{Rn}\) decays to \(10.00 \%\) of its initial quantity?

Short answer

Answer: It takes approximately 16.195 days for the radon isotope ${ }^{222} \mathrm{Rn}$ to decay to 10.00% of its initial quantity.

Step-by-step solution

Understand the radioactive decay formula

The radioactive decay formula is: \(N(t) = N_0 e^{-\lambda t}\) Where: - \(N(t)\) is the remaining quantity of radioactive material at time \(t\) - \(N_0\) is the initial quantity of radioactive material - \(\lambda\) is the decay constant - \(t\) is the time elapsed

Calculate the decay constant with the given half-life

The decay constant can be calculated using the half-life formula: \(\lambda = \frac{\ln 2}{T_{1/2}}\) Where: - \(T_{1/2}\) is the half-life of the radioactive material In this case, \(T_{1/2} = 3.825\) days. Plugging this value into the formula, we get: \(\lambda = \frac{\ln 2}{3.825} \approx 0.1812 \, \mathrm{day}^{-1}\)

Set the equation N(t) = 10% of N0 and solve for t

We are given that the remaining quantity should be 10% of the initial quantity which translates to the equation: \(N(t) = 0.1 N_0\) To solve for \(t\), we will use both equations and substitute like this: \(0.1 N_0 = N_0 e^{-\lambda t}\) The initial quantity \(N_0\) will cancel out, and we are left with: \(0.1 = e^{-0.1812 t}\)

Solve for t using logarithms

To solve for \(t\), take the natural logarithm of both sides: \(\ln 0.1 = -0.1812 t\) Now, isolate \(t\) by dividing both sides by \(-0.1812\): \(t = \frac{\ln 0.1 }{-0.1812} \approx 16.195 \, \mathrm{days}\)

Final answer

So, it takes approximately \(16.195\) days for the radon isotope \({ }^{222} \mathrm{Rn}\) to decay to 10.00% of its initial quantity.

Key concepts

Radon Isotope

Radon isotopes are forms of the radon element that vary in neutron number. The isotope \({ }^{222} \mathrm{Rn}\) is the most stable and prominent in discussions of radioactive decay. Radon is naturally occurring and is produced from the decay of uranium. It is a noble gas, which means it is colorless and odorless, making it hard to detect without specialized equipment. Due to its radioactive properties, \(^{222} \mathrm{Rn}\) is used in various medical applications, such as cancer treatment. In these applications, radon's capacity to emit alpha particles makes it effective in targeting cancer cells while minimizing damage to surrounding tissues. Understanding how radon isotopes decay is crucial for effectively using and managing this potent yet dangerous gas.

Half-Life

The concept of half-life is key to understanding radioactive decay. In simple terms, the half-life of a substance is the time required for half of the substance to decay. For \(^{222} \mathrm{Rn}\), this period is 3.825 days. This means that after 3.825 days, only half of the initial radon quantity remains.
Here are some important points about half-life:

• The half-life remains constant regardless of the initial amount of material.
• It is not affected by external conditions such as temperature or pressure.
• Knowing the half-life allows us to calculate the decay constant, which is vital for calculating how long it takes to reach a certain level of decay.

Half-life is a robust measure used frequently in radiometric dating and assessing the longevity of radioactive materials.

Decay Constant

The decay constant, represented by \(\lambda\), is an essential factor in the radioactive decay formula. It indicates the probability per unit time that an atom will decay. Calculating the decay constant helps in predicting how quickly a substance will decay over time. To find \(\lambda\), you use the formula \(\lambda = \frac{\ln 2}{T_{1/2}}\), where \(T_{1/2}\) is the half-life.
In the case of \(^{222} \mathrm{Rn}\), with a known half-life of 3.825 days:

• Plug in the half-life into the decay constant formula: \(\lambda = \frac{\ln 2}{3.825}\).
• The result is \(\lambda \approx 0.1812 \, \mathrm{day}^{-1}\), indicating how quickly the radon isotope is expected to decay daily.

A higher decay constant means a faster decay rate. For practical purposes, the decay constant is invaluable when calculating the percentage of a substance left after a certain time.

Logarithms in Decay Calculation

Logarithms play a key role in solving decay-related equations. They help us move from the exponential form, such as \(N(t) = N_0 e^{-\lambda t}\), to a format that allows for solving unknown variables like time. In solving for the time it takes \(^{222} \mathrm{Rn}\) to decay to 10% of its initial value, we use simple logarithmic operations. The procedure involves taking the natural logarithm (ln) of both sides of the equation: \

• After substituting 10% of initial quantity, the equation becomes \(0.1 = e^{-0.1812 t}\).
• Applying the natural logarithm gives \(\ln(0.1) = -0.1812 t\).
• Solving for \(t\) involves dividing both sides by \(-0.1812\), resulting in \(t = \frac{\ln 0.1 }{-0.1812}\).
• This produces a solution for \(t\) as approximately 16.195 days.

Thus, logarithmic methods are vital for simplifying and solving exponential decay problems.