Question

It takes 5.2 min for a 1.000 -g sample of \({ }^{210} \mathrm{Fr}\) to decay to \(0.250 \mathrm{~g}\). What is the half-life of \({ }^{210} \mathrm{Fr}\) ?

Short answer

The half-life of \({ }^{210} \mathrm{Fr}\) is approximately 3.201 minutes.

Step-by-step solution

Identifying the Equations

In order to find the half-life of the radioactive substance, we will use the radioactive decay equation: \[N = N_0e^{-\lambda t}\] Where: - \(N\) represents the final mass of the substance - \(N_0\) represents the initial mass of the substance - \(e\) is the base of the natural logarithm (approximately 2.71828) - \(\lambda\) is the decay constant - \(t\) is time We also know that the half-life can be calculated using the following equation: \[T_{1/2} = \frac{\ln2}{\lambda}\] Where: - \(T_{1/2}\) = half-life - \(\lambda\) is the decay constant

Finding the Decay Constant

We will first find the decay constant (\(\lambda\)). To do so, we need to rearrange the radioactive decay equation: \(\lambda = -\frac{\ln(\frac{N}{N_0})}{t}\) Using the given information, we have: \(N\) = 0.250 g \(N_0\) = 1.000 g \(t\) = 5.2 minutes Now, we can plug in these values into the rearranged equation to find \(\lambda\): \(\lambda = -\frac{\ln(\frac{0.250}{1.000})}{5.2}\) \[\lambda \approx 0.2164\,\text{min}^{-1}\]

Calculating the Half-life

Now that we have found the decay constant, we can use the equation for half-life to find \(T_{1/2}\): \[T_{1/2} = \frac{\ln2}{\lambda}\] Replacing \(\lambda\) with the calculated value: \[T_{1/2} = \frac{\ln2}{0.2164}\] \[T_{1/2} \approx 3.201\,\text{min}\] So the half-life of \({ }^{210} \mathrm{Fr}\) is approximately 3.201 minutes.

Key concepts

Half-life calculation

The concept of half-life is fundamental when it comes to understanding radioactive decay. Half-life (\[T_{1/2}\]) is the time required for half of the radioactive nuclei in a sample to decay. This allows us to predict the time it takes for a given quantity of radioactive substance to decrease to half of its initial amount.
For any radioactive substance, the half-life remains constant regardless of the initial amount. This means if you start with 1 gram, after one half-life, 0.5 grams will remain, after another half-life only 0.25 grams will be left, and so on.
Calculating the half-life requires knowing the decay constant (\[\lambda\]). The formula \[T_{1/2} = \frac{\ln2}{\lambda}\] helps us find this period. Understanding this enables us to predict how quickly a sample of radioactive material will undergo decay over time.

Decay constant

The decay constant (\[\lambda\]) is a crucial element in the study of radioactive decay. It represents the probability per unit time that a given radioactive atom will decay. A higher decay constant indicates a substance decays more quickly.
To calculate the decay constant, the rearranged form of the radioactive decay equation is used:\[\lambda = -\frac{\ln(\frac{N}{N_0})}{t}\]. Here,

• \[N_0\] is the initial amount of the substance,
• \[N\] is the amount remaining after time \[t\], and
• \[t\] is the time elapsed.

In the provided exercise, using the values \[N_0 = 1.000\, g\], \[N = 0.250\, g\], and \[t = 5.2\, minutes\], you can find \[\lambda\approx 0.2164\, \text{min}^{-1}\]. This indicates that in each minute, around 21.64% of the remaining \[^{210} \mathrm{Fr}\] nuclei in the sample will decay.

Radioactive decay equation

The radioactive decay equation \[N = N_0e^{-\lambda t}\] is central to understanding how radioactive materials decrease over time. This equation allows us to predict how much of a substance remains after a certain period. Here's what each component of the equation represents:

• \[N_0\] is the initial quantity of the substance.
• \[N\] is the remaining quantity after time \[t\].
• \[e\] represents the base of natural logarithms, approximately 2.71828.
• \[\lambda\] is the decay constant, indicating the rate of decay.

In practice, if you know any three of these values, you can find the fourth. For example, in the exercise, starting with a 1-gram sample that decayed to 0.25 grams in 5.2 minutes allowed us to solve for the decay constant. This equation is pivotal in nuclear physics and chemistry, helping us quantify the decay process efficiently.